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instiki_ncatlab.org.atom.xml - sfeed_tests - sfeed tests and RSS and Atom files |
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instiki_ncatlab.org.atom.xml (502820B) |
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1 <feed xmlns="http://www.w3.org/2005/Atom" xml:lang="en"> |
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2 <title>nLab</title> |
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3 <link rel="alternate" type="application/xhtml+xml" href="https://ncatl… |
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4 <link rel="self" href="https://ncatlab.org/nlab/atom_with_content"/> |
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5 <updated>2021-07-02T09:22:44Z</updated> |
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6 <id>tag:ncatlab.org,2008-11-28:nLab</id> |
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7 <subtitle>An Instiki Wiki</subtitle> |
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8 <generator uri="http://golem.ph.utexas.edu/instiki/show/HomePage" vers… |
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9 <entry> |
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10 <title type="html">Urs Frauenfelder</title> |
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11 <link rel="alternate" type="application/xhtml+xml" href="https://nca… |
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12 <updated>2021-07-02T09:22:44Z</updated> |
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13 <published>2021-07-02T09:22:46Z</published> |
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14 <id>tag:ncatlab.org,2021-07-02:nLab,Urs+Frauenfelder</id> |
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15 <author> |
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16 <name>Urs Schreiber</name> |
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17 </author> |
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18 <content type="xhtml" xml:base="https://ncatlab.org/nlab/show/Urs+Fr… |
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19 <div xmlns="http://www.w3.org/1999/xhtml"> |
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20 <ul> |
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21 <li> |
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22 <p><a href='https://www.uni-augsburg.de/en/fakultaet/mntf/math/prof/geom… |
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23 </li> |
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24 |
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25 <li> |
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26 <p><a href='https://www.genealogy.math.ndsu.nodak.edu/id.php?id=120240'>… |
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27 </li> |
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28 </ul> |
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29 |
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30 <h2 id='selected_writings'>Selected writings</h2> |
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31 |
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32 <p>On <a class='existingWikiWord' href='/nlab/show/cyclic+loop+space'>cy… |
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33 |
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34 <ul> |
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35 <li><a class='existingWikiWord' href='/nlab/show/Urs+Frauenfelder'>Urs F… |
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36 </ul> |
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37 |
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38 <p><div class='property'> category: <a class='category_link' href='/nlab… |
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39 </content> |
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40 </entry> |
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41 <entry> |
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42 <title type="html">cyclic loop space</title> |
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43 <link rel="alternate" type="application/xhtml+xml" href="https://nca… |
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44 <updated>2021-07-02T09:20:18Z</updated> |
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45 <published>2017-02-14T09:56:17Z</published> |
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46 <id>tag:ncatlab.org,2017-02-14:nLab,cyclic+loop+space</id> |
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47 <author> |
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48 <name>Urs Schreiber</name> |
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49 </author> |
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50 <content type="xhtml" xml:base="https://ncatlab.org/nlab/show/cyclic… |
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51 <div xmlns="http://www.w3.org/1999/xhtml"> |
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52 <div class='rightHandSide'> |
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53 <div class='toc clickDown' tabindex='0'> |
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54 <h3 id='context'>Context</h3> |
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55 |
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56 <h4 id='topology'>Topology</h4> |
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57 |
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58 <div class='hide'> |
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59 <p><strong><a class='existingWikiWord' href='/nlab/show/topology'>topolo… |
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60 |
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61 <p>see also <em><a class='existingWikiWord' href='/nlab/show/differentia… |
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62 |
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63 <p><a class='existingWikiWord' href='/nlab/show/Introduction+to+Topology… |
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64 |
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65 <p><strong>Basic concepts</strong></p> |
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66 |
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67 <ul> |
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68 <li> |
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69 <p><a class='existingWikiWord' href='/nlab/show/open+subspace'>open subs… |
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70 </li> |
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71 |
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72 <li> |
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73 <p><a class='existingWikiWord' href='/nlab/show/topological+space'>topol… |
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74 </li> |
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75 |
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76 <li> |
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77 <p><a class='existingWikiWord' href='/nlab/show/topological+base'>base f… |
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78 </li> |
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79 |
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80 <li> |
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81 <p><a class='existingWikiWord' href='/nlab/show/finer+topology'>finer/co… |
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82 </li> |
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83 |
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84 <li> |
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85 <p><a class='existingWikiWord' href='/nlab/show/closed+subspace'>closure… |
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86 </li> |
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87 |
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88 <li> |
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89 <p><a class='existingWikiWord' href='/nlab/show/separation+axioms'>separ… |
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90 </li> |
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91 |
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92 <li> |
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93 <p><a class='existingWikiWord' href='/nlab/show/continuous+map'>continuo… |
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94 </li> |
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95 |
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96 <li> |
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97 <p><a class='existingWikiWord' href='/nlab/show/uniformly+continuous+map… |
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98 </li> |
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99 |
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100 <li> |
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101 <p><a class='existingWikiWord' href='/nlab/show/embedding+of+topological… |
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102 </li> |
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103 |
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104 <li> |
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105 <p><a class='existingWikiWord' href='/nlab/show/open+map'>open map</a>, … |
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106 </li> |
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107 |
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108 <li> |
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109 <p><a class='existingWikiWord' href='/nlab/show/sequence'>sequence</a>, … |
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110 </li> |
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111 |
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112 <li> |
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113 <p><a class='existingWikiWord' href='/nlab/show/convergence'>convergence… |
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114 </li> |
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115 |
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116 <li> |
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117 <p><a class='existingWikiWord' href='/nlab/show/category'>category</a> <… |
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118 |
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119 <ul> |
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120 <li><a class='existingWikiWord' href='/nlab/show/convenient+category+of+… |
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121 </ul> |
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122 </li> |
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123 </ul> |
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124 |
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125 <p><strong><a href='Top#UniversalConstructions'>Universal constructions<… |
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126 |
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127 <ul> |
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128 <li> |
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129 <p><a class='existingWikiWord' href='/nlab/show/weak+topology'>initial t… |
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130 </li> |
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131 |
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132 <li> |
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133 <p><a class='existingWikiWord' href='/nlab/show/subspace'>subspace</a>, … |
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134 </li> |
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135 |
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136 <li> |
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137 <p>fiber space, <a class='existingWikiWord' href='/nlab/show/space+attac… |
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138 </li> |
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139 |
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140 <li> |
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141 <p><a class='existingWikiWord' href='/nlab/show/product+topological+spac… |
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142 </li> |
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143 |
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144 <li> |
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145 <p><a class='existingWikiWord' href='/nlab/show/mapping+cylinder'>mappin… |
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146 </li> |
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147 |
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148 <li> |
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149 <p><a class='existingWikiWord' href='/nlab/show/mapping+cone'>mapping co… |
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150 </li> |
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151 |
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152 <li> |
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153 <p><a class='existingWikiWord' href='/nlab/show/mapping+telescope'>mappi… |
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154 </li> |
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155 |
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156 <li> |
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157 <p><a class='existingWikiWord' href='/nlab/show/colimits+of+normal+space… |
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158 </li> |
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159 </ul> |
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160 |
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161 <p><strong><a class='existingWikiWord' href='/nlab/show/stuff%2C+structu… |
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162 |
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163 <ul> |
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164 <li> |
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165 <p><a class='existingWikiWord' href='/nlab/show/nice+topological+space'>… |
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166 </li> |
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167 |
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168 <li> |
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169 <p><a class='existingWikiWord' href='/nlab/show/metric+space'>metric spa… |
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170 </li> |
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171 |
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172 <li> |
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173 <p><a class='existingWikiWord' href='/nlab/show/Kolmogorov+topological+s… |
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174 </li> |
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175 |
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176 <li> |
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177 <p><a class='existingWikiWord' href='/nlab/show/sober+topological+space'… |
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178 </li> |
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179 |
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180 <li> |
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181 <p><a class='existingWikiWord' href='/nlab/show/compact+space'>compact s… |
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182 |
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183 <p><a class='existingWikiWord' href='/nlab/show/sequentially+compact+top… |
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184 </li> |
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185 |
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186 <li> |
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187 <p><a class='existingWikiWord' href='/nlab/show/compactly+generated+topo… |
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188 </li> |
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189 |
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190 <li> |
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191 <p><a class='existingWikiWord' href='/nlab/show/second-countable+space'>… |
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192 </li> |
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193 |
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194 <li> |
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195 <p><a class='existingWikiWord' href='/nlab/show/contractible+space'>cont… |
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196 </li> |
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197 |
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198 <li> |
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199 <p><a class='existingWikiWord' href='/nlab/show/connected+space'>connect… |
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200 </li> |
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201 |
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202 <li> |
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203 <p><a class='existingWikiWord' href='/nlab/show/simply+connected+space'>… |
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204 </li> |
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205 |
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206 <li> |
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207 <p><a class='existingWikiWord' href='/nlab/show/cell+complex'>cell compl… |
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208 </li> |
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209 |
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210 <li> |
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211 <p><a class='existingWikiWord' href='/nlab/show/pointed+topological+spac… |
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212 </li> |
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213 |
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214 <li> |
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215 <p><a class='existingWikiWord' href='/nlab/show/topological+vector+space… |
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216 </li> |
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217 |
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218 <li> |
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219 <p><a class='existingWikiWord' href='/nlab/show/topological+group'>topol… |
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220 </li> |
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221 |
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222 <li> |
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223 <p><a class='existingWikiWord' href='/nlab/show/topological+vector+bundl… |
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224 </li> |
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225 |
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226 <li> |
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227 <p><a class='existingWikiWord' href='/nlab/show/topological+manifold'>to… |
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228 </li> |
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229 </ul> |
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230 |
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231 <p><strong>Examples</strong></p> |
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232 |
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233 <ul> |
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234 <li> |
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235 <p><a class='existingWikiWord' href='/nlab/show/empty+space'>empty space… |
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236 </li> |
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237 |
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238 <li> |
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239 <p><a class='existingWikiWord' href='/nlab/show/discrete+object'>discret… |
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240 </li> |
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241 |
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242 <li> |
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243 <p><a class='existingWikiWord' href='/nlab/show/Sierpinski+space'>Sierpi… |
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244 </li> |
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245 |
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246 <li> |
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247 <p><a class='existingWikiWord' href='/nlab/show/order+topology'>order to… |
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248 </li> |
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249 |
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250 <li> |
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251 <p><a class='existingWikiWord' href='/nlab/show/Euclidean+space'>Euclide… |
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252 |
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253 <ul> |
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254 <li><a class='existingWikiWord' href='/nlab/show/real+number'>real line<… |
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255 </ul> |
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256 </li> |
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257 |
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258 <li> |
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259 <p><a class='existingWikiWord' href='/nlab/show/cylinder+object'>cylinde… |
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260 </li> |
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261 |
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262 <li> |
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263 <p><a class='existingWikiWord' href='/nlab/show/sphere'>sphere</a>, <a c… |
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264 </li> |
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265 |
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266 <li> |
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267 <p><a class='existingWikiWord' href='/nlab/show/circle'>circle</a>, <a c… |
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268 </li> |
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269 |
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270 <li> |
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271 <p><a class='existingWikiWord' href='/nlab/show/polytope'>polytope</a>, … |
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272 </li> |
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273 |
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274 <li> |
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275 <p><a class='existingWikiWord' href='/nlab/show/projective+space'>projec… |
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276 </li> |
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277 |
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278 <li> |
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279 <p><a class='existingWikiWord' href='/nlab/show/classifying+space'>class… |
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280 </li> |
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281 |
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282 <li> |
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283 <p><a class='existingWikiWord' href='/nlab/show/configuration+space+of+p… |
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284 </li> |
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285 |
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286 <li> |
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287 <p><a class='existingWikiWord' href='/nlab/show/path'>path</a>, <a class… |
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288 </li> |
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289 |
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290 <li> |
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291 <p><a class='existingWikiWord' href='/nlab/show/compact-open+topology'>m… |
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292 |
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293 <ul> |
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294 <li><a class='existingWikiWord' href='/nlab/show/loop+space'>loop space<… |
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295 </ul> |
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296 </li> |
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297 |
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298 <li> |
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299 <p><a class='existingWikiWord' href='/nlab/show/Zariski+topology'>Zarisk… |
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300 </li> |
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301 |
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302 <li> |
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303 <p><a class='existingWikiWord' href='/nlab/show/Cantor+space'>Cantor spa… |
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304 </li> |
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305 |
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306 <li> |
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307 <p><a class='existingWikiWord' href='/nlab/show/Peano+curve'>Peano curve… |
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308 </li> |
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309 |
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310 <li> |
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311 <p><a class='existingWikiWord' href='/nlab/show/line+with+two+origins'>l… |
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312 </li> |
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313 |
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314 <li> |
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315 <p><a class='existingWikiWord' href='/nlab/show/K-topology'>K-topology</… |
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316 </li> |
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317 |
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318 <li> |
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319 <p><a class='existingWikiWord' href='/nlab/show/Warsaw+circle'>Warsaw ci… |
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320 </li> |
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321 </ul> |
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322 |
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323 <p><strong>Basic statements</strong></p> |
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324 |
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325 <ul> |
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326 <li> |
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327 <p><a class='existingWikiWord' href='/nlab/show/Hausdorff+implies+sober'… |
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328 </li> |
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329 |
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330 <li> |
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331 <p><a class='existingWikiWord' href='/nlab/show/schemes+are+sober'>schem… |
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332 </li> |
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333 |
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334 <li> |
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335 <p><a class='existingWikiWord' href='/nlab/show/continuous+images+of+com… |
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336 </li> |
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337 |
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338 <li> |
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339 <p><a class='existingWikiWord' href='/nlab/show/closed+subspaces+of+comp… |
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340 </li> |
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341 |
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342 <li> |
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343 <p><a class='existingWikiWord' href='/nlab/show/open+subspaces+of+compac… |
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344 </li> |
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345 |
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346 <li> |
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347 <p><a class='existingWikiWord' href='/nlab/show/quotient+projections+out… |
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348 </li> |
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349 |
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350 <li> |
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351 <p><a class='existingWikiWord' href='/nlab/show/compact+spaces+equivalen… |
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352 |
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353 <ul> |
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354 <li> |
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355 <p><a class='existingWikiWord' href='/nlab/show/Lebesgue+number+lemma'>L… |
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356 </li> |
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357 |
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358 <li> |
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359 <p><a class='existingWikiWord' href='/nlab/show/sequentially+compact+met… |
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360 </li> |
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361 |
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362 <li> |
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363 <p><a class='existingWikiWord' href='/nlab/show/compact+spaces+equivalen… |
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364 </li> |
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365 |
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366 <li> |
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367 <p><a class='existingWikiWord' href='/nlab/show/sequentially+compact+met… |
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368 </li> |
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369 </ul> |
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370 </li> |
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371 |
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372 <li> |
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373 <p><a class='existingWikiWord' href='/nlab/show/continuous+metric+space+… |
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374 </li> |
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375 |
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376 <li> |
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377 <p><a class='existingWikiWord' href='/nlab/show/paracompact+Hausdorff+sp… |
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378 </li> |
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379 |
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380 <li> |
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381 <p><a class='existingWikiWord' href='/nlab/show/paracompact+Hausdorff+sp… |
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382 </li> |
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383 |
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384 <li> |
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385 <p><a class='existingWikiWord' href='/nlab/show/closed+injections+are+em… |
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386 </li> |
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387 |
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388 <li> |
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389 <p><a class='existingWikiWord' href='/nlab/show/proper+maps+to+locally+c… |
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390 </li> |
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391 |
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392 <li> |
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393 <p><a class='existingWikiWord' href='/nlab/show/injective+proper+maps+to… |
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394 </li> |
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395 |
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396 <li> |
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397 <p><a class='existingWikiWord' href='/nlab/show/locally+compact+and+sigm… |
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398 </li> |
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399 |
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400 <li> |
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401 <p><a class='existingWikiWord' href='/nlab/show/locally+compact+and+seco… |
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402 </li> |
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403 |
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404 <li> |
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405 <p><a class='existingWikiWord' href='/nlab/show/second-countable+regular… |
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406 </li> |
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407 |
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408 <li> |
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409 <p><a class='existingWikiWord' href='/nlab/show/CW-complexes+are+paracom… |
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410 </li> |
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411 </ul> |
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412 |
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413 <p><strong>Theorems</strong></p> |
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414 |
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415 <ul> |
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416 <li> |
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417 <p><a class='existingWikiWord' href='/nlab/show/Urysohn%27s+lemma'>Uryso… |
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418 </li> |
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419 |
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420 <li> |
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421 <p><a class='existingWikiWord' href='/nlab/show/Tietze+extension+theorem… |
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422 </li> |
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423 |
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424 <li> |
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425 <p><a class='existingWikiWord' href='/nlab/show/Tychonoff+theorem'>Tycho… |
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426 </li> |
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427 |
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428 <li> |
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429 <p><a class='existingWikiWord' href='/nlab/show/tube+lemma'>tube lemma</… |
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430 </li> |
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431 |
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432 <li> |
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433 <p><a class='existingWikiWord' href='/nlab/show/Michael%27s+theorem'>Mic… |
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434 </li> |
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435 |
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436 <li> |
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437 <p><a class='existingWikiWord' href='/nlab/show/Brouwer%27s+fixed+point+… |
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438 </li> |
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439 |
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440 <li> |
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441 <p><a class='existingWikiWord' href='/nlab/show/topological+invariance+o… |
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442 </li> |
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443 |
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444 <li> |
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445 <p><a class='existingWikiWord' href='/nlab/show/Jordan+curve+theorem'>Jo… |
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446 </li> |
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447 </ul> |
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448 |
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449 <p><strong>Analysis Theorems</strong></p> |
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450 |
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451 <ul> |
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452 <li> |
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453 <p><a class='existingWikiWord' href='/nlab/show/Heine-Borel+theorem'>Hei… |
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454 </li> |
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455 |
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456 <li> |
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457 <p><a class='existingWikiWord' href='/nlab/show/intermediate+value+theor… |
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458 </li> |
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459 |
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460 <li> |
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461 <p><a class='existingWikiWord' href='/nlab/show/extreme+value+theorem'>e… |
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462 </li> |
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463 </ul> |
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464 |
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465 <p><strong><a class='existingWikiWord' href='/nlab/show/topological+homo… |
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466 |
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467 <ul> |
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468 <li> |
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469 <p><a class='existingWikiWord' href='/nlab/show/homotopy'>left homotopy<… |
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470 </li> |
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471 |
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472 <li> |
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473 <p><a class='existingWikiWord' href='/nlab/show/homotopy+equivalence'>ho… |
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474 </li> |
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475 |
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476 <li> |
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477 <p><a class='existingWikiWord' href='/nlab/show/fundamental+group'>funda… |
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478 </li> |
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479 |
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480 <li> |
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481 <p><a class='existingWikiWord' href='/nlab/show/fundamental+theorem+of+c… |
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482 </li> |
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483 |
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484 <li> |
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485 <p><a class='existingWikiWord' href='/nlab/show/homotopy+group'>homotopy… |
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486 </li> |
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487 |
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488 <li> |
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489 <p><a class='existingWikiWord' href='/nlab/show/weak+homotopy+equivalenc… |
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490 </li> |
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491 |
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492 <li> |
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493 <p><a class='existingWikiWord' href='/nlab/show/Whitehead+theorem'>White… |
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494 </li> |
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495 |
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496 <li> |
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497 <p><a class='existingWikiWord' href='/nlab/show/Freudenthal+suspension+t… |
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498 </li> |
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499 |
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500 <li> |
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501 <p><a class='existingWikiWord' href='/nlab/show/nerve+theorem'>nerve the… |
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502 </li> |
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503 |
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504 <li> |
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505 <p><a class='existingWikiWord' href='/nlab/show/homotopy+extension+prope… |
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506 </li> |
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507 |
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508 <li> |
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509 <p><a class='existingWikiWord' href='/nlab/show/topological+cofiber+sequ… |
|
510 </li> |
|
511 |
|
512 <li> |
|
513 <p><a class='existingWikiWord' href='/nlab/show/Str%C3%B8m+model+structu… |
|
514 </li> |
|
515 |
|
516 <li> |
|
517 <p><a class='existingWikiWord' href='/nlab/show/classical+model+structur… |
|
518 </li> |
|
519 </ul> |
|
520 </div> |
|
521 |
|
522 <h4 id='homotopy_theory'>Homotopy theory</h4> |
|
523 |
|
524 <div class='hide'> |
|
525 <p><strong><a class='existingWikiWord' href='/nlab/show/homotopy+theory'… |
|
526 |
|
527 <p>flavors: <a class='existingWikiWord' href='/nlab/show/stable+homotopy… |
|
528 |
|
529 <p>models: <a class='existingWikiWord' href='/nlab/show/topological+homo… |
|
530 |
|
531 <p>see also <strong><a class='existingWikiWord' href='/nlab/show/algebra… |
|
532 |
|
533 <p><strong>Introductions</strong></p> |
|
534 |
|
535 <ul> |
|
536 <li> |
|
537 <p><a class='existingWikiWord' href='/nlab/show/Introduction+to+Topology… |
|
538 </li> |
|
539 |
|
540 <li> |
|
541 <p><a class='existingWikiWord' href='/nlab/show/Introduction+to+Homotopy… |
|
542 </li> |
|
543 |
|
544 <li> |
|
545 <p><a class='existingWikiWord' href='/nlab/show/geometry+of+physics+--+h… |
|
546 </li> |
|
547 </ul> |
|
548 |
|
549 <p><strong>Definitions</strong></p> |
|
550 |
|
551 <ul> |
|
552 <li> |
|
553 <p><a class='existingWikiWord' href='/nlab/show/homotopy'>homotopy</a>, … |
|
554 </li> |
|
555 |
|
556 <li> |
|
557 <p><a class='existingWikiWord' href='/nlab/show/homotopy+type'>homotopy … |
|
558 </li> |
|
559 |
|
560 <li> |
|
561 <p><a class='existingWikiWord' href='/nlab/show/Pi-algebra'>Pi-algebra</… |
|
562 </li> |
|
563 |
|
564 <li> |
|
565 <p><a class='existingWikiWord' href='/nlab/show/homotopy+coherent+catego… |
|
566 |
|
567 <ul> |
|
568 <li> |
|
569 <p><a class='existingWikiWord' href='/nlab/show/homotopical+category'>ho… |
|
570 |
|
571 <ul> |
|
572 <li> |
|
573 <p><a class='existingWikiWord' href='/nlab/show/model+category'>model ca… |
|
574 </li> |
|
575 |
|
576 <li> |
|
577 <p><a class='existingWikiWord' href='/nlab/show/category+of+fibrant+obje… |
|
578 </li> |
|
579 |
|
580 <li> |
|
581 <p><a class='existingWikiWord' href='/nlab/show/Waldhausen+category'>Wal… |
|
582 </li> |
|
583 </ul> |
|
584 </li> |
|
585 |
|
586 <li> |
|
587 <p><a class='existingWikiWord' href='/nlab/show/homotopy+category'>homot… |
|
588 |
|
589 <ul> |
|
590 <li><a class='existingWikiWord' href='/nlab/show/Ho%28Top%29'>Ho(Top)</a… |
|
591 </ul> |
|
592 </li> |
|
593 </ul> |
|
594 </li> |
|
595 |
|
596 <li> |
|
597 <p><a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-categ… |
|
598 |
|
599 <ul> |
|
600 <li><a class='existingWikiWord' href='/nlab/show/homotopy+category+of+an… |
|
601 </ul> |
|
602 </li> |
|
603 </ul> |
|
604 |
|
605 <p><strong>Paths and cylinders</strong></p> |
|
606 |
|
607 <ul> |
|
608 <li> |
|
609 <p><a class='existingWikiWord' href='/nlab/show/homotopy'>left homotopy<… |
|
610 |
|
611 <ul> |
|
612 <li> |
|
613 <p><a class='existingWikiWord' href='/nlab/show/cylinder+object'>cylinde… |
|
614 </li> |
|
615 |
|
616 <li> |
|
617 <p><a class='existingWikiWord' href='/nlab/show/mapping+cone'>mapping co… |
|
618 </li> |
|
619 </ul> |
|
620 </li> |
|
621 |
|
622 <li> |
|
623 <p><a class='existingWikiWord' href='/nlab/show/homotopy'>right homotopy… |
|
624 |
|
625 <ul> |
|
626 <li> |
|
627 <p><a class='existingWikiWord' href='/nlab/show/path+space+object'>path … |
|
628 </li> |
|
629 |
|
630 <li> |
|
631 <p><a class='existingWikiWord' href='/nlab/show/mapping+cocone'>mapping … |
|
632 </li> |
|
633 |
|
634 <li> |
|
635 <p><a class='existingWikiWord' href='/nlab/show/generalized+universal+bu… |
|
636 </li> |
|
637 </ul> |
|
638 </li> |
|
639 |
|
640 <li> |
|
641 <p><a class='existingWikiWord' href='/nlab/show/interval+object'>interva… |
|
642 |
|
643 <ul> |
|
644 <li> |
|
645 <p><a class='existingWikiWord' href='/nlab/show/localization+at+geometri… |
|
646 </li> |
|
647 |
|
648 <li> |
|
649 <p><a class='existingWikiWord' href='/nlab/show/infinitesimal+interval+o… |
|
650 </li> |
|
651 </ul> |
|
652 </li> |
|
653 </ul> |
|
654 |
|
655 <p><strong>Homotopy groups</strong></p> |
|
656 |
|
657 <ul> |
|
658 <li> |
|
659 <p><a class='existingWikiWord' href='/nlab/show/homotopy+group'>homotopy… |
|
660 |
|
661 <ul> |
|
662 <li> |
|
663 <p><a class='existingWikiWord' href='/nlab/show/fundamental+group'>funda… |
|
664 |
|
665 <ul> |
|
666 <li><a class='existingWikiWord' href='/nlab/show/fundamental+group+of+a+… |
|
667 </ul> |
|
668 </li> |
|
669 |
|
670 <li> |
|
671 <p><a class='existingWikiWord' href='/nlab/show/Brown-Grossman+homotopy+… |
|
672 </li> |
|
673 |
|
674 <li> |
|
675 <p><a class='existingWikiWord' href='/nlab/show/categorical+homotopy+gro… |
|
676 </li> |
|
677 |
|
678 <li> |
|
679 <p><a class='existingWikiWord' href='/nlab/show/geometric+homotopy+group… |
|
680 </li> |
|
681 </ul> |
|
682 </li> |
|
683 |
|
684 <li> |
|
685 <p><a class='existingWikiWord' href='/nlab/show/fundamental+infinity-gro… |
|
686 |
|
687 <ul> |
|
688 <li> |
|
689 <p><a class='existingWikiWord' href='/nlab/show/fundamental+groupoid'>fu… |
|
690 |
|
691 <ul> |
|
692 <li><a class='existingWikiWord' href='/nlab/show/path+groupoid'>path gro… |
|
693 </ul> |
|
694 </li> |
|
695 |
|
696 <li> |
|
697 <p><a class='existingWikiWord' href='/nlab/show/fundamental+infinity-gro… |
|
698 </li> |
|
699 |
|
700 <li> |
|
701 <p><a class='existingWikiWord' href='/nlab/show/fundamental+infinity-gro… |
|
702 </li> |
|
703 </ul> |
|
704 </li> |
|
705 |
|
706 <li> |
|
707 <p><a class='existingWikiWord' href='/nlab/show/fundamental+%28infinity%… |
|
708 |
|
709 <ul> |
|
710 <li><a class='existingWikiWord' href='/nlab/show/fundamental+category'>f… |
|
711 </ul> |
|
712 </li> |
|
713 </ul> |
|
714 |
|
715 <p><strong>Basic facts</strong></p> |
|
716 |
|
717 <ul> |
|
718 <li><a class='existingWikiWord' href='/nlab/show/fundamental+group+of+th… |
|
719 </ul> |
|
720 |
|
721 <p><strong>Theorems</strong></p> |
|
722 |
|
723 <ul> |
|
724 <li> |
|
725 <p><a class='existingWikiWord' href='/nlab/show/fundamental+theorem+of+c… |
|
726 </li> |
|
727 |
|
728 <li> |
|
729 <p><a class='existingWikiWord' href='/nlab/show/Freudenthal+suspension+t… |
|
730 </li> |
|
731 |
|
732 <li> |
|
733 <p><a class='existingWikiWord' href='/nlab/show/Blakers-Massey+theorem'>… |
|
734 </li> |
|
735 |
|
736 <li> |
|
737 <p><a class='existingWikiWord' href='/nlab/show/higher+homotopy+van+Kamp… |
|
738 </li> |
|
739 |
|
740 <li> |
|
741 <p><a class='existingWikiWord' href='/nlab/show/nerve+theorem'>nerve the… |
|
742 </li> |
|
743 |
|
744 <li> |
|
745 <p><a class='existingWikiWord' href='/nlab/show/Whitehead+theorem'>White… |
|
746 </li> |
|
747 |
|
748 <li> |
|
749 <p><a class='existingWikiWord' href='/nlab/show/Hurewicz+theorem'>Hurewi… |
|
750 </li> |
|
751 |
|
752 <li> |
|
753 <p><a class='existingWikiWord' href='/nlab/show/Galois+theory'>Galois th… |
|
754 </li> |
|
755 |
|
756 <li> |
|
757 <p><a class='existingWikiWord' href='/nlab/show/homotopy+hypothesis'>hom… |
|
758 </li> |
|
759 </ul> |
|
760 </div> |
|
761 </div> |
|
762 </div> |
|
763 |
|
764 <h1 id='contents'>Contents</h1> |
|
765 <div class='maruku_toc'><ul><li><a href='#idea'>Idea</a></li><li><a href… |
|
766 <h2 id='idea'>Idea</h2> |
|
767 |
|
768 <p>Any <a class='existingWikiWord' href='/nlab/show/free+loop+space'>fre… |
|
769 |
|
770 <p>If <math class='maruku-mathml' display='inline' id='mathml_1b9519ae4b… |
|
771 |
|
772 <p>If <math class='maruku-mathml' display='inline' id='mathml_1b9519ae4b… |
|
773 |
|
774 <h2 id='properties'>Properties</h2> |
|
775 |
|
776 <h3 id='AsRightBaseChange'>As right base change along <math class='maruk… |
|
777 |
|
778 <p>The cyclic loop space <math class='maruku-mathml' display='inline' id… |
|
779 |
|
780 <h3 id='ordinary_cohomology_of__on_cyclic_cohomology_of_'>Ordinary cohom… |
|
781 |
|
782 <p>Let <math class='maruku-mathml' display='inline' id='mathml_1b9519ae4… |
|
783 |
|
784 <p>The <a class='existingWikiWord' href='/nlab/show/ordinary+cohomology'… |
|
785 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
786 H^\bullet(\mathcal{L}X) |
|
787 \simeq |
|
788 HH_\bullet( C^\bullet(X) ) |
|
789 \,. |
|
790 |
|
791 </annotation></semantics></math></div> |
|
792 <p>Moreover the <math class='maruku-mathml' display='inline' id='mathml_… |
|
793 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
794 H^\bullet(\mathcal{L}X \sslash S^1) |
|
795 \simeq |
|
796 HC_\bullet( C^\bullet(X) ) |
|
797 |
|
798 </annotation></semantics></math></div> |
|
799 <p>(<a href='#Jones87'>Jones 87, Thm. A</a>, review in <a href='#Loday92… |
|
800 |
|
801 <p>If the <a class='existingWikiWord' href='/nlab/show/coefficient'>coef… |
|
802 |
|
803 <p>In the special case that the <a class='existingWikiWord' href='/nlab/… |
|
804 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
805 H^\bullet(\mathcal{L}X) |
|
806 \simeq |
|
807 HH_\bullet( \Omega^\bullet(X) ) |
|
808 \,. |
|
809 |
|
810 </annotation></semantics></math></div><div class='maruku-equation'><math… |
|
811 H^\bullet(\mathcal{L}X \sslash S^1) |
|
812 \simeq |
|
813 HC_\bullet( \Omega^\bullet(X) ) |
|
814 \,. |
|
815 |
|
816 </annotation></semantics></math></div> |
|
817 <p>This is known as <em><a class='existingWikiWord' href='/nlab/show/Jon… |
|
818 |
|
819 <p>An <a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-ca… |
|
820 |
|
821 <h3 id='rational_sullivan_model'>Rational Sullivan model</h3> |
|
822 |
|
823 <p>See at <em><a class='existingWikiWord' href='/nlab/show/Sullivan+mode… |
|
824 |
|
825 <h2 id='related_concepts'>Related concepts</h2> |
|
826 |
|
827 <ul> |
|
828 <li> |
|
829 <p><a class='existingWikiWord' href='/nlab/show/double+dimensional+reduc… |
|
830 </li> |
|
831 |
|
832 <li> |
|
833 <p><a class='existingWikiWord' href='/nlab/show/cyclic+loop+stack'>cycli… |
|
834 </li> |
|
835 |
|
836 <li> |
|
837 <p><a class='existingWikiWord' href='/nlab/show/free+loop+space'>free lo… |
|
838 </li> |
|
839 </ul> |
|
840 |
|
841 <h2 id='references'>References</h2> |
|
842 |
|
843 <p>The notion of the cyclic loop space of a topological space appears as… |
|
844 |
|
845 <ul> |
|
846 <li id='Jones87'> |
|
847 <p><a class='existingWikiWord' href='/nlab/show/John+David+Stuart+Jones'… |
|
848 </li> |
|
849 |
|
850 <li> |
|
851 <p><a class='existingWikiWord' href='/nlab/show/Gunnar+Carlsson'>Gunnar … |
|
852 </li> |
|
853 |
|
854 <li id='Witten88'> |
|
855 <p><a class='existingWikiWord' href='/nlab/show/Edward+Witten'>Edward Wi… |
|
856 </li> |
|
857 |
|
858 <li id='Loday92'> |
|
859 <p><a class='existingWikiWord' href='/nlab/show/Jean-Louis+Loday'>Jean-L… |
|
860 |
|
861 <p>Chapter 7 in: <em>Cyclic Homology</em>, Grundlehren <strong>301</stro… |
|
862 </li> |
|
863 |
|
864 <li id='Loday11'> |
|
865 <p><a class='existingWikiWord' href='/nlab/show/Jean-Louis+Loday'>Jean-L… |
|
866 </li> |
|
867 |
|
868 <li id='Stapleton11'> |
|
869 <p><a class='existingWikiWord' href='/nlab/show/Nathaniel+Stapleton'>Nat… |
|
870 </li> |
|
871 </ul> |
|
872 |
|
873 <p>Specifically on cyclic loop spaces of <a class='existingWikiWord' hre… |
|
874 |
|
875 <ul> |
|
876 <li><a class='existingWikiWord' href='/nlab/show/Nancy+Hingston'>Nancy H… |
|
877 </ul> |
|
878 |
|
879 <p>See also:</p> |
|
880 |
|
881 <ul> |
|
882 <li><a class='existingWikiWord' href='/nlab/show/Urs+Frauenfelder'>Urs F… |
|
883 </ul> |
|
884 |
|
885 <p>A version of the cyclic loop space of <a class='existingWikiWord' hre… |
|
886 |
|
887 <ul> |
|
888 <li id='Huan18'><a class='existingWikiWord' href='/nlab/show/Zhen+Huan'>… |
|
889 </ul> |
|
890 |
|
891 <p>following</p> |
|
892 |
|
893 <ul> |
|
894 <li><a class='existingWikiWord' href='/nlab/show/Zhen+Huan'>Zhen Huan</a… |
|
895 </ul> |
|
896 |
|
897 <p>and recalled/expanded on in several followup articles, such as in</p> |
|
898 |
|
899 <ul> |
|
900 <li><a class='existingWikiWord' href='/nlab/show/Zhen+Huan'>Zhen Huan</a… |
|
901 </ul> |
|
902 |
|
903 <p>The above formulation of cyclic loop spaces, in the generality of <a … |
|
904 |
|
905 <ul> |
|
906 <li id='BMSS19'><a class='existingWikiWord' href='/nlab/show/Vincent+Bra… |
|
907 </ul> |
|
908 |
|
909 <p>following the analogous discussion in <a class='existingWikiWord' hre… |
|
910 |
|
911 <ul> |
|
912 <li id='FSS18'><a class='existingWikiWord' href='/nlab/show/Domenico+Fio… |
|
913 </ul> |
|
914 |
|
915 <p>with exposition in</p> |
|
916 |
|
917 <ul> |
|
918 <li><a class='existingWikiWord' href='/nlab/show/Urs+Schreiber'>Urs Schr… |
|
919 </ul> |
|
920 |
|
921 <p> |
|
922 </p> |
|
923 |
|
924 <p> |
|
925 |
|
926 </p> </div> |
|
927 </content> |
|
928 </entry> |
|
929 <entry> |
|
930 <title type="html">Nancy Hingston</title> |
|
931 <link rel="alternate" type="application/xhtml+xml" href="https://nca… |
|
932 <updated>2021-07-02T09:15:40Z</updated> |
|
933 <published>2021-07-02T09:14:11Z</published> |
|
934 <id>tag:ncatlab.org,2021-07-02:nLab,Nancy+Hingston</id> |
|
935 <author> |
|
936 <name>Urs Schreiber</name> |
|
937 </author> |
|
938 <content type="xhtml" xml:base="https://ncatlab.org/nlab/show/Nancy+… |
|
939 <div xmlns="http://www.w3.org/1999/xhtml"> |
|
940 <ul> |
|
941 <li> |
|
942 <p><a href='https://en.wikipedia.org/wiki/Nancy_Hingston'>Wikipedia entr… |
|
943 </li> |
|
944 |
|
945 <li> |
|
946 <p><a href='https://science.tcnj.edu/school-information/women-in-science… |
|
947 </li> |
|
948 |
|
949 <li> |
|
950 <p><a href='https://science.tcnj.edu/school-information/women-in-science… |
|
951 </li> |
|
952 </ul> |
|
953 |
|
954 <h2 id='selected_writings'>Selected writings</h2> |
|
955 |
|
956 <p>On the <a class='existingWikiWord' href='/nlab/show/cyclic+loop+space… |
|
957 |
|
958 <ul> |
|
959 <li><a class='existingWikiWord' href='/nlab/show/Nancy+Hingston'>Nancy H… |
|
960 </ul> |
|
961 |
|
962 <p><div class='property'> category: <a class='category_link' href='/nlab… |
|
963 </content> |
|
964 </entry> |
|
965 <entry> |
|
966 <title type="html">Ralph Cohen</title> |
|
967 <link rel="alternate" type="application/xhtml+xml" href="https://nca… |
|
968 <updated>2021-07-02T08:58:22Z</updated> |
|
969 <published>2010-11-26T21:46:21Z</published> |
|
970 <id>tag:ncatlab.org,2010-11-26:nLab,Ralph+Cohen</id> |
|
971 <author> |
|
972 <name>Urs Schreiber</name> |
|
973 </author> |
|
974 <content type="xhtml" xml:base="https://ncatlab.org/nlab/show/Ralph+… |
|
975 <div xmlns="http://www.w3.org/1999/xhtml"> |
|
976 <ul> |
|
977 <li><a href='http://math.stanford.edu/~ralph/'>website</a></li> |
|
978 </ul> |
|
979 |
|
980 <h2 id='selected_writings'>Selected writings</h2> |
|
981 |
|
982 <p>On <a class='existingWikiWord' href='/nlab/show/cyclic+loop+space'>cy… |
|
983 |
|
984 <ul> |
|
985 <li><a class='existingWikiWord' href='/nlab/show/Gunnar+Carlsson'>Gunnar… |
|
986 </ul> |
|
987 |
|
988 <p>On <a class='existingWikiWord' href='/nlab/show/string+topology'>stri… |
|
989 |
|
990 <ul> |
|
991 <li> |
|
992 <p><a class='existingWikiWord' href='/nlab/show/Ralph+Cohen'>Ralph Cohen… |
|
993 </li> |
|
994 |
|
995 <li> |
|
996 <p><a class='existingWikiWord' href='/nlab/show/Ralph+Cohen'>Ralph Cohen… |
|
997 </li> |
|
998 |
|
999 <li id='CohenJones'> |
|
1000 <p><a class='existingWikiWord' href='/nlab/show/Ralph+Cohen'>Ralph Cohen… |
|
1001 </li> |
|
1002 |
|
1003 <li id='CohenGodin03'> |
|
1004 <p><a class='existingWikiWord' href='/nlab/show/Ralph+Cohen'>Ralph Cohen… |
|
1005 </li> |
|
1006 |
|
1007 <li> |
|
1008 <p><a class='existingWikiWord' href='/nlab/show/Ralph+Cohen'>Ralph Cohen… |
|
1009 </li> |
|
1010 |
|
1011 <li> |
|
1012 <p><a class='existingWikiWord' href='/nlab/show/Ralph+Cohen'>Ralph Cohen… |
|
1013 </li> |
|
1014 </ul> |
|
1015 |
|
1016 <p>On <a class='existingWikiWord' href='/nlab/show/moduli+space+of+monop… |
|
1017 |
|
1018 <ul> |
|
1019 <li> |
|
1020 <p><a class='existingWikiWord' href='/nlab/show/Fred+Cohen'>Fred Cohen</… |
|
1021 </li> |
|
1022 |
|
1023 <li> |
|
1024 <p><a class='existingWikiWord' href='/nlab/show/Ralph+Cohen'>Ralph Cohen… |
|
1025 </li> |
|
1026 </ul> |
|
1027 |
|
1028 <p>and more generally on <a class='existingWikiWord' href='/nlab/show/mo… |
|
1029 |
|
1030 <ul> |
|
1031 <li><a class='existingWikiWord' href='/nlab/show/Ralph+Cohen'>Ralph Cohe… |
|
1032 </ul> |
|
1033 |
|
1034 <p><div class='property'> category: <a class='category_link' href='/nlab… |
|
1035 </content> |
|
1036 </entry> |
|
1037 <entry> |
|
1038 <title type="html">Gunnar Carlsson</title> |
|
1039 <link rel="alternate" type="application/xhtml+xml" href="https://nca… |
|
1040 <updated>2021-07-02T08:57:51Z</updated> |
|
1041 <published>2014-04-13T09:19:17Z</published> |
|
1042 <id>tag:ncatlab.org,2014-04-13:nLab,Gunnar+Carlsson</id> |
|
1043 <author> |
|
1044 <name>Urs Schreiber</name> |
|
1045 </author> |
|
1046 <content type="xhtml" xml:base="https://ncatlab.org/nlab/show/Gunnar… |
|
1047 <div xmlns="http://www.w3.org/1999/xhtml"> |
|
1048 <ul> |
|
1049 <li><a href='http://math.stanford.edu/~gunnar/'>webpage</a></li> |
|
1050 </ul> |
|
1051 |
|
1052 <h2 id='selected_writings'>Selected writings</h2> |
|
1053 |
|
1054 <p>On <a class='existingWikiWord' href='/nlab/show/cyclic+loop+space'>cy… |
|
1055 |
|
1056 <ul> |
|
1057 <li><a class='existingWikiWord' href='/nlab/show/Gunnar+Carlsson'>Gunnar… |
|
1058 </ul> |
|
1059 |
|
1060 <p>On <a class='existingWikiWord' href='/nlab/show/topological+data+anal… |
|
1061 |
|
1062 <ul> |
|
1063 <li><a class='existingWikiWord' href='/nlab/show/Gunnar+Carlsson'>Gunnar… |
|
1064 </ul> |
|
1065 |
|
1066 <p>On <a class='existingWikiWord' href='/nlab/show/persistent+homology'>… |
|
1067 |
|
1068 <ul> |
|
1069 <li> |
|
1070 <p>A. Zomorodian, <a class='existingWikiWord' href='/nlab/show/Gunnar+Ca… |
|
1071 </li> |
|
1072 |
|
1073 <li> |
|
1074 <p><a class='existingWikiWord' href='/nlab/show/Gunnar+Carlsson'>Gunnar … |
|
1075 </li> |
|
1076 |
|
1077 <li> |
|
1078 <p><a class='existingWikiWord' href='/nlab/show/Gunnar+Carlsson'>Gunnar … |
|
1079 </li> |
|
1080 </ul> |
|
1081 |
|
1082 <h2 id='related_lab_entries'>Related <math class='maruku-mathml' display… |
|
1083 |
|
1084 <ul> |
|
1085 <li> |
|
1086 <p><a class='existingWikiWord' href='/nlab/show/equivariant+stable+homot… |
|
1087 </li> |
|
1088 |
|
1089 <li> |
|
1090 <p><a class='existingWikiWord' href='/nlab/show/Segal-Carlsson+completio… |
|
1091 |
|
1092 <p><a class='existingWikiWord' href='/nlab/show/Burnside+ring'>Burnside … |
|
1093 </li> |
|
1094 |
|
1095 <li> |
|
1096 <p><a class='existingWikiWord' href='/nlab/show/persistent+homology'>per… |
|
1097 </li> |
|
1098 </ul> |
|
1099 |
|
1100 <p><div class='property'> category: <a class='category_link' href='/nlab… |
|
1101 |
|
1102 <p> |
|
1103 </p> </div> |
|
1104 </content> |
|
1105 </entry> |
|
1106 <entry> |
|
1107 <title type="html">Doug Ravenel</title> |
|
1108 <link rel="alternate" type="application/xhtml+xml" href="https://nca… |
|
1109 <updated>2021-07-02T08:31:03Z</updated> |
|
1110 <published>2012-08-14T17:36:28Z</published> |
|
1111 <id>tag:ncatlab.org,2012-08-14:nLab,Doug+Ravenel</id> |
|
1112 <author> |
|
1113 <name>Urs Schreiber</name> |
|
1114 </author> |
|
1115 <content type="xhtml" xml:base="https://ncatlab.org/nlab/show/Doug+R… |
|
1116 <div xmlns="http://www.w3.org/1999/xhtml"> |
|
1117 <ul> |
|
1118 <li> |
|
1119 <p><a href='http://www.math.rochester.edu/people/faculty/doug/'>webpage<… |
|
1120 </li> |
|
1121 |
|
1122 <li> |
|
1123 <p><a class='existingWikiWord' href='/nlab/show/Michael+Hopkins'>Michael… |
|
1124 </li> |
|
1125 </ul> |
|
1126 |
|
1127 <h2 id='selected_writings'>Selected writings</h2> |
|
1128 |
|
1129 <p>On the <a class='existingWikiWord' href='/nlab/show/Hopf+algebra'>Hop… |
|
1130 |
|
1131 <ul> |
|
1132 <li><a class='existingWikiWord' href='/nlab/show/Doug+Ravenel'>Douglas R… |
|
1133 </ul> |
|
1134 |
|
1135 <p>On the <a class='existingWikiWord' href='/nlab/show/Adams%E2%80%93Nov… |
|
1136 |
|
1137 <ul> |
|
1138 <li id='Ravenel78'><a class='existingWikiWord' href='/nlab/show/Doug+Rav… |
|
1139 </ul> |
|
1140 |
|
1141 <p>On <a class='existingWikiWord' href='/nlab/show/chromatic+homotopy+th… |
|
1142 |
|
1143 <ul> |
|
1144 <li id='Ravenel84'><a class='existingWikiWord' href='/nlab/show/Doug+Rav… |
|
1145 </ul> |
|
1146 |
|
1147 <p>On <a class='existingWikiWord' href='/nlab/show/homotopy+groups+of+sp… |
|
1148 |
|
1149 <ul> |
|
1150 <li id='MahowaldRavenel87'><a class='existingWikiWord' href='/nlab/show/… |
|
1151 </ul> |
|
1152 |
|
1153 <p>On <a class='existingWikiWord' href='/nlab/show/elliptic+genus'>ellip… |
|
1154 |
|
1155 <ul> |
|
1156 <li id='LandweberRavenelStong93'><a class='existingWikiWord' href='/nlab… |
|
1157 </ul> |
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1158 |
|
1159 <p>On <a class='existingWikiWord' href='/nlab/show/chromatic+homotopy+th… |
|
1160 |
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1161 <ul> |
|
1162 <li><a class='existingWikiWord' href='/nlab/show/Doug+Ravenel'>Doug Rave… |
|
1163 </ul> |
|
1164 |
|
1165 <p>On <a class='existingWikiWord' href='/nlab/show/iterated+loop+space'>… |
|
1166 |
|
1167 <ul> |
|
1168 <li><a class='existingWikiWord' href='/nlab/show/Doug+Ravenel'>Douglas R… |
|
1169 </ul> |
|
1170 |
|
1171 <p><div class='property'> category: <a class='category_link' href='/nlab… |
|
1172 |
|
1173 <p> |
|
1174 </p> </div> |
|
1175 </content> |
|
1176 </entry> |
|
1177 <entry> |
|
1178 <title type="html">stable splitting of mapping spaces</title> |
|
1179 <link rel="alternate" type="application/xhtml+xml" href="https://nca… |
|
1180 <updated>2021-07-02T08:30:06Z</updated> |
|
1181 <published>2018-10-28T12:06:36Z</published> |
|
1182 <id>tag:ncatlab.org,2018-10-28:nLab,stable+splitting+of+mapping+spac… |
|
1183 <author> |
|
1184 <name>Urs Schreiber</name> |
|
1185 </author> |
|
1186 <content type="xhtml" xml:base="https://ncatlab.org/nlab/show/stable… |
|
1187 <div xmlns="http://www.w3.org/1999/xhtml"> |
|
1188 <div class='rightHandSide'> |
|
1189 <div class='toc clickDown' tabindex='0'> |
|
1190 <h3 id='context'>Context</h3> |
|
1191 |
|
1192 <h4 id='stable_homotopy_theory'>Stable Homotopy theory</h4> |
|
1193 |
|
1194 <div class='hide'> |
|
1195 <p><strong><a class='existingWikiWord' href='/nlab/show/stable+homotopy+… |
|
1196 |
|
1197 <ul> |
|
1198 <li><a class='existingWikiWord' href='/nlab/show/homological+algebra'>ho… |
|
1199 </ul> |
|
1200 |
|
1201 <p><em><a class='existingWikiWord' href='/nlab/show/Introduction+to+Stab… |
|
1202 |
|
1203 <h1 id='ingredients'>Ingredients</h1> |
|
1204 |
|
1205 <ul> |
|
1206 <li><a class='existingWikiWord' href='/nlab/show/homotopy+theory'>homoto… |
|
1207 </ul> |
|
1208 |
|
1209 <h1 id='contents'>Contents</h1> |
|
1210 |
|
1211 <ul> |
|
1212 <li> |
|
1213 <p><a class='existingWikiWord' href='/nlab/show/loop+space+object'>loop … |
|
1214 </li> |
|
1215 |
|
1216 <li> |
|
1217 <p><a class='existingWikiWord' href='/nlab/show/suspension+object'>suspe… |
|
1218 </li> |
|
1219 |
|
1220 <li> |
|
1221 <p><a class='existingWikiWord' href='/nlab/show/looping'>looping and del… |
|
1222 </li> |
|
1223 |
|
1224 <li> |
|
1225 <p><a class='existingWikiWord' href='/nlab/show/stable+%28infinity%2C1%2… |
|
1226 |
|
1227 <ul> |
|
1228 <li> |
|
1229 <p><a class='existingWikiWord' href='/nlab/show/stabilization'>stabiliza… |
|
1230 |
|
1231 <ul> |
|
1232 <li><a class='existingWikiWord' href='/nlab/show/spectrum+object'>spectr… |
|
1233 </ul> |
|
1234 </li> |
|
1235 |
|
1236 <li> |
|
1237 <p><a class='existingWikiWord' href='/nlab/show/stable+derivator'>stable… |
|
1238 </li> |
|
1239 |
|
1240 <li> |
|
1241 <p><a class='existingWikiWord' href='/nlab/show/triangulated+category'>t… |
|
1242 </li> |
|
1243 </ul> |
|
1244 </li> |
|
1245 |
|
1246 <li> |
|
1247 <p><a class='existingWikiWord' href='/nlab/show/stable+%28infinity%2C1%2… |
|
1248 |
|
1249 <ul> |
|
1250 <li> |
|
1251 <p><a class='existingWikiWord' href='/nlab/show/spectrum'>spectrum</a></… |
|
1252 </li> |
|
1253 |
|
1254 <li> |
|
1255 <p><a class='existingWikiWord' href='/nlab/show/stable+homotopy+category… |
|
1256 </li> |
|
1257 </ul> |
|
1258 </li> |
|
1259 |
|
1260 <li> |
|
1261 <p><a class='existingWikiWord' href='/nlab/show/smash+product+of+spectra… |
|
1262 |
|
1263 <ul> |
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1264 <li> |
|
1265 <p><a class='existingWikiWord' href='/nlab/show/symmetric+smash+product+… |
|
1266 </li> |
|
1267 |
|
1268 <li> |
|
1269 <p><a class='existingWikiWord' href='/nlab/show/Spanier-Whitehead+dualit… |
|
1270 </li> |
|
1271 |
|
1272 <li> |
|
1273 <p><a class='existingWikiWord' href='/nlab/show/A-infinity-ring'>A-∞ r… |
|
1274 </li> |
|
1275 |
|
1276 <li> |
|
1277 <p><a class='existingWikiWord' href='/nlab/show/E-infinity-ring'>E-∞ r… |
|
1278 </li> |
|
1279 </ul> |
|
1280 </li> |
|
1281 </ul> |
|
1282 <div> |
|
1283 <p> |
|
1284 <a href='/nlab/edit/stable+homotopy+theory+-+contents'>Edit this sideb… |
|
1285 </p> |
|
1286 </div></div> |
|
1287 |
|
1288 <h4 id='goodwillie_calculus'>Goodwillie calculus</h4> |
|
1289 |
|
1290 <div class='hide'> |
|
1291 <p><strong><a class='existingWikiWord' href='/nlab/show/Goodwillie+calcu… |
|
1292 |
|
1293 <ul> |
|
1294 <li> |
|
1295 <p><a class='existingWikiWord' href='/nlab/show/Goodwillie-differentiabl… |
|
1296 </li> |
|
1297 |
|
1298 <li> |
|
1299 <p><a class='existingWikiWord' href='/nlab/show/excisive+%28%E2%88%9E%2C… |
|
1300 |
|
1301 <ul> |
|
1302 <li> |
|
1303 <p><a class='existingWikiWord' href='/nlab/show/spectrum+object'>spectru… |
|
1304 </li> |
|
1305 |
|
1306 <li> |
|
1307 <p><a class='existingWikiWord' href='/nlab/show/tangent+%28infinity%2C1%… |
|
1308 </li> |
|
1309 </ul> |
|
1310 </li> |
|
1311 |
|
1312 <li> |
|
1313 <p><a class='existingWikiWord' href='/nlab/show/n-excisive+%28%E2%88%9E%… |
|
1314 |
|
1315 <ul> |
|
1316 <li> |
|
1317 <p><a class='existingWikiWord' href='/nlab/show/jet+%28infinity%2C1%29-c… |
|
1318 </li> |
|
1319 |
|
1320 <li> |
|
1321 <p><a class='existingWikiWord' href='/nlab/show/polynomial+%28%E2%88%9E%… |
|
1322 </li> |
|
1323 </ul> |
|
1324 </li> |
|
1325 |
|
1326 <li> |
|
1327 <p><a class='existingWikiWord' href='/nlab/show/Goodwillie-Taylor+tower'… |
|
1328 |
|
1329 <ul> |
|
1330 <li> |
|
1331 <p><a class='existingWikiWord' href='/nlab/show/analytic+%28%E2%88%9E%2C… |
|
1332 </li> |
|
1333 |
|
1334 <li> |
|
1335 <p><a class='existingWikiWord' href='/nlab/show/Goodwillie+spectral+sequ… |
|
1336 </li> |
|
1337 </ul> |
|
1338 </li> |
|
1339 </ul> |
|
1340 </div> |
|
1341 |
|
1342 <h4 id='mapping_space'>Mapping space</h4> |
|
1343 |
|
1344 <div class='hide'> |
|
1345 <p><strong><a class='existingWikiWord' href='/nlab/show/compact-open+top… |
|
1346 |
|
1347 <h3 id='general_abstract'>General abstract</h3> |
|
1348 |
|
1349 <ul> |
|
1350 <li> |
|
1351 <p><a class='existingWikiWord' href='/nlab/show/hom-set'>hom-set</a>, <a… |
|
1352 </li> |
|
1353 |
|
1354 <li> |
|
1355 <p><a class='existingWikiWord' href='/nlab/show/loop+space+object'>loop … |
|
1356 </li> |
|
1357 </ul> |
|
1358 |
|
1359 <h3 id='topology'>Topology</h3> |
|
1360 |
|
1361 <ul> |
|
1362 <li> |
|
1363 <p><a class='existingWikiWord' href='/nlab/show/topology+of+mapping+spac… |
|
1364 |
|
1365 <ul> |
|
1366 <li><a class='existingWikiWord' href='/nlab/show/compact-open+topology'>… |
|
1367 </ul> |
|
1368 </li> |
|
1369 |
|
1370 <li> |
|
1371 <p><a class='existingWikiWord' href='/nlab/show/evaluation+fibration+of+… |
|
1372 </li> |
|
1373 |
|
1374 <li> |
|
1375 <p><a class='existingWikiWord' href='/nlab/show/loop+space'>loop space</… |
|
1376 </li> |
|
1377 </ul> |
|
1378 |
|
1379 <h3 id='differential_topology'>Differential topology</h3> |
|
1380 |
|
1381 <ul> |
|
1382 <li> |
|
1383 <p><a class='existingWikiWord' href='/nlab/show/differential+topology+of… |
|
1384 |
|
1385 <ul> |
|
1386 <li><a class='existingWikiWord' href='/nlab/show/C-infinity+topology'>C-… |
|
1387 </ul> |
|
1388 </li> |
|
1389 |
|
1390 <li> |
|
1391 <p><a class='existingWikiWord' href='/nlab/show/manifold+structure+of+ma… |
|
1392 |
|
1393 <ul> |
|
1394 <li><a class='existingWikiWord' href='/nlab/show/tangent+spaces+of+mappi… |
|
1395 </ul> |
|
1396 </li> |
|
1397 |
|
1398 <li> |
|
1399 <p><a class='existingWikiWord' href='/nlab/show/smooth+loop+space'>smoot… |
|
1400 </li> |
|
1401 </ul> |
|
1402 |
|
1403 <h3 id='stable_homotopy_theory_2'>Stable homotopy theory</h3> |
|
1404 |
|
1405 <ul> |
|
1406 <li><a class='existingWikiWord' href='/nlab/show/function+spectrum'>mapp… |
|
1407 </ul> |
|
1408 <div> |
|
1409 <p> |
|
1410 <a href='/nlab/edit/mapping+space+-+contents'>Edit this sidebar</a> |
|
1411 </p> |
|
1412 </div></div> |
|
1413 </div> |
|
1414 </div> |
|
1415 |
|
1416 <h1 id='contents_2'>Contents</h1> |
|
1417 <div class='maruku_toc'><ul><li><a href='#idea'>Idea</a></li><li><a href… |
|
1418 <h2 id='idea'>Idea</h2> |
|
1419 |
|
1420 <p>The <a class='existingWikiWord' href='/nlab/show/stabilization'>stabi… |
|
1421 |
|
1422 <h2 id='Definition'>Definition</h2> |
|
1423 |
|
1424 <p>The stable splitting of mapping spaces discussed <a href='#StableSpli… |
|
1425 |
|
1426 <p>The following Def. <a class='maruku-ref' href='#ConfigurationSpacesOf… |
|
1427 |
|
1428 <div class='num_defn' id='ConfigurationSpacesOfnPoints'> |
|
1429 <h6 id='definition_2'>Definition</h6> |
|
1430 |
|
1431 <p><strong>(<a class='existingWikiWord' href='/nlab/show/configuration+s… |
|
1432 |
|
1433 <p>Let <math class='maruku-mathml' display='inline' id='mathml_088ef31fd… |
|
1434 |
|
1435 <p>For <math class='maruku-mathml' display='inline' id='mathml_088ef31fd… |
|
1436 <div class='maruku-equation' id='eq:DistinguishableConfigurationSpaceJus… |
|
1437 |
|
1438 \mathrm{Conf}^{ord}_{n}(X) |
|
1439 \;\coloneqq\; |
|
1440 \big( |
|
1441 X^n \setminus \mathbf{\Delta}_X^n |
|
1442 \big) |
|
1443 / \partial(X^n) |
|
1444 |
|
1445 </annotation></semantics></math></div> |
|
1446 <p>which is the <a class='existingWikiWord' href='/nlab/show/complement'… |
|
1447 |
|
1448 <p>Then the <em><strong>configuration space of <math class='maruku-mathm… |
|
1449 <div class='maruku-equation' id='eq:ConfigurationSpaceJustForX'><span cl… |
|
1450 |
|
1451 \mathrm{Conf}_{n}(X) |
|
1452 \;\coloneqq\; |
|
1453 Conf_n^{ord}(X)/\Sigma_n |
|
1454 \;=\; |
|
1455 \Big( |
|
1456 \big( |
|
1457 X^n \setminus \mathbf{\Delta}_X^n |
|
1458 \big) |
|
1459 / \partial(X^n) |
|
1460 \Big) |
|
1461 /\Sigma(n) |
|
1462 \,, |
|
1463 |
|
1464 </annotation></semantics></math></div> |
|
1465 <p>where <math class='maruku-mathml' display='inline' id='mathml_088ef31… |
|
1466 |
|
1467 <p>More generally, let <math class='maruku-mathml' display='inline' id='… |
|
1468 <div class='maruku-equation' id='eq:ConfigurationSpaceWithXAndY'><span c… |
|
1469 |
|
1470 \mathrm{Conf}_{n}(X,Y) |
|
1471 \;\coloneqq\; |
|
1472 \Big( |
|
1473 \big( |
|
1474 ( |
|
1475 X^n \setminus \mathbf{\Delta}_X^n |
|
1476 ) |
|
1477 \times |
|
1478 Y^n |
|
1479 \big) |
|
1480 / \partial(X^n \times Y^n) |
|
1481 \Big) |
|
1482 /\Sigma(n) |
|
1483 |
|
1484 </annotation></semantics></math></div> |
|
1485 <p>where now <math class='maruku-mathml' display='inline' id='mathml_088… |
|
1486 |
|
1487 <p>This more general definition reduces to the previous case for <math c… |
|
1488 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
1489 \mathrm{Conf}_n(X) |
|
1490 \;=\; |
|
1491 \mathrm{Conf}_n(X,\ast) |
|
1492 \,. |
|
1493 |
|
1494 </annotation></semantics></math></div> |
|
1495 <p>Finally the <em><strong>configuration space of an arbitrary number of… |
|
1496 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
1497 Conf\left( X, Y\right) |
|
1498 \;\coloneqq\; |
|
1499 \left( |
|
1500 \underset{n \in \mathbb{n}}{\sqcup} |
|
1501 \big( |
|
1502 ( |
|
1503 X^n \setminus \mathbf{\Delta}_X^n |
|
1504 ) |
|
1505 \times |
|
1506 Y^k |
|
1507 \big) |
|
1508 /\Sigma(n) |
|
1509 \right)/\sim |
|
1510 |
|
1511 </annotation></semantics></math></div> |
|
1512 <p>by the <a class='existingWikiWord' href='/nlab/show/equivalence+relat… |
|
1513 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
1514 \big( |
|
1515 (x_1, y_1), \cdots, (x_{n-1}, y_{n-1}), (x_n, y_n) |
|
1516 \big) |
|
1517 \;\sim\; |
|
1518 \big( |
|
1519 (x_1, y_1), \cdots, (x_{n-1}, y_{n-1}) |
|
1520 \big) |
|
1521 \;\;\;\; \Leftrightarrow |
|
1522 \;\;\;\; (x_n, y_n) \in \partial (X \times Y) |
|
1523 \,. |
|
1524 |
|
1525 </annotation></semantics></math></div> |
|
1526 <p>This is naturally a <a class='existingWikiWord' href='/nlab/show/filt… |
|
1527 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
1528 Conf_{\leq n}\left( X, Y\right) |
|
1529 \;\coloneqq\; |
|
1530 \left( |
|
1531 \underset{k \in \{1, \cdots, n\}}{\sqcup} |
|
1532 \big( |
|
1533 ( |
|
1534 X^k \setminus \mathbf{\Delta}_X^k |
|
1535 ) |
|
1536 \times |
|
1537 Y^k |
|
1538 \big) |
|
1539 /\Sigma(k) |
|
1540 \right)/\sim |
|
1541 \,. |
|
1542 |
|
1543 </annotation></semantics></math></div> |
|
1544 <p>The corresponding <a class='existingWikiWord' href='/nlab/show/quotie… |
|
1545 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
1546 Conf_n(X,Y) |
|
1547 \;\simeq\; |
|
1548 Conf_{\leq n}(X,Y) / Conf_{\leq (n-1)}(X,Y) |
|
1549 \,. |
|
1550 |
|
1551 </annotation></semantics></math></div></div> |
|
1552 |
|
1553 <div class='num_remark' id='ComparisonToNotationInLiterature'> |
|
1554 <h6 id='remark'>Remark</h6> |
|
1555 |
|
1556 <p><strong>(comparison to notation in the literature)</strong></p> |
|
1557 |
|
1558 <p>The above Def. <a class='maruku-ref' href='#ConfigurationSpacesOfnPoi… |
|
1559 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
1560 \array{ |
|
1561 \text{ here: } |
|
1562 && |
|
1563 \array{ \text{ Segal 73,} \\ \text{ Snaith 74}: } |
|
1564 && |
|
1565 \text{ Bödigheimer 87: } |
|
1566 \\ |
|
1567 \\ |
|
1568 Conf(\mathbb{R}^d,Y) |
|
1569 &=& |
|
1570 C_d( Y/\partial Y ) |
|
1571 &=& |
|
1572 C( \mathbb{R}^d, \emptyset; Y ) |
|
1573 \\ |
|
1574 \mathrm{Conf}_n\left( \mathbb{R}^d \right) |
|
1575 & = & |
|
1576 F_n C_d( S^0 ) / F_{n-1} C_d( S^0 ) |
|
1577 & = & |
|
1578 D_n\left( \mathbb{R}^d, \emptyset; S^0 \right) |
|
1579 \\ |
|
1580 \mathrm{Conf}_n\left( \mathbb{R}^d, Y \right) |
|
1581 & = & |
|
1582 F_n C_d( Y/\partial Y ) / F_{n-1} C_d( Y/\partial Y ) |
|
1583 & = & |
|
1584 D_n\left( \mathbb{R}^d, \emptyset; Y/\partial Y \right) |
|
1585 \\ |
|
1586 \mathrm{Conf}_n( X ) && &=& D_n\left( X, \partial … |
|
1587 \\ |
|
1588 \mathrm{Conf}_n( X, Y ) && &=& D_n\left( X, \part… |
|
1589 } |
|
1590 |
|
1591 </annotation></semantics></math></div> |
|
1592 <p>Notice here that when <math class='maruku-mathml' display='inline' id… |
|
1593 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
1594 Y / \partial Y \coloneqq Y \underset{\partial Y}{\sqcup} \ast |
|
1595 |
|
1596 </annotation></semantics></math></div> |
|
1597 <p>is <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc… |
|
1598 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
1599 \ast/\partial \ast = S^0 |
|
1600 |
|
1601 </annotation></semantics></math></div> |
|
1602 <p>is the <a class='existingWikiWord' href='/nlab/show/0-sphere'>0-spher… |
|
1603 </div> |
|
1604 |
|
1605 <h2 id='Statements'>Statements</h2> |
|
1606 |
|
1607 <h3 id='prelude_equivalence_to_the_infinite_configuration_space'>Prelude… |
|
1608 |
|
1609 <p>First recall the following equivalence already before <a class='exist… |
|
1610 |
|
1611 <div class='num_prop' id='ScanningMapEquivalenceOverCartesianSpace'> |
|
1612 <h6 id='proposition'>Proposition</h6> |
|
1613 |
|
1614 <p>For</p> |
|
1615 |
|
1616 <ol> |
|
1617 <li> |
|
1618 <p><math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e1… |
|
1619 </li> |
|
1620 |
|
1621 <li> |
|
1622 <p><math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e1… |
|
1623 </li> |
|
1624 </ol> |
|
1625 |
|
1626 <p>the <a class='existingWikiWord' href='/nlab/show/cohomotopy+charge+ma… |
|
1627 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
1628 Conf\left( |
|
1629 \mathbb{R}^d, Y |
|
1630 \right) |
|
1631 \overset{scan}{\longrightarrow} |
|
1632 \Omega^d \Sigma^d (Y/\partial Y) |
|
1633 |
|
1634 </annotation></semantics></math></div> |
|
1635 <p>between</p> |
|
1636 |
|
1637 <ol> |
|
1638 <li> |
|
1639 <p>the configuration space of arbitrary points in <math class='maruku-ma… |
|
1640 </li> |
|
1641 |
|
1642 <li> |
|
1643 <p>the <math class='maruku-mathml' display='inline' id='mathml_088ef31fd… |
|
1644 </li> |
|
1645 </ol> |
|
1646 |
|
1647 <p>In particular when <math class='maruku-mathml' display='inline' id='m… |
|
1648 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
1649 Conf\left( |
|
1650 \mathbb{R}^d, \mathbb{D}^k |
|
1651 \right) |
|
1652 \overset{scan}{\longrightarrow} |
|
1653 \Omega^d S^{ d + k } |
|
1654 |
|
1655 </annotation></semantics></math></div> |
|
1656 <p>with the <math class='maruku-mathml' display='inline' id='mathml_088e… |
|
1657 </div> |
|
1658 |
|
1659 <p>(<a href='#May72'>May 72, Theorem 2.7</a>, <a href='#Segal73'>Segal 7… |
|
1660 |
|
1661 <h3 id='StableSplittings'>Stable splitting of mapping spaces</h3> |
|
1662 |
|
1663 <div class='num_prop' id='StableSplittingOfMappingSpacesOutOfEuclideanSp… |
|
1664 <h6 id='proposition_2'>Proposition</h6> |
|
1665 |
|
1666 <p><strong>(<a class='existingWikiWord' href='/nlab/show/stable+splittin… |
|
1667 |
|
1668 <p>For</p> |
|
1669 |
|
1670 <ol> |
|
1671 <li> |
|
1672 <p><math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e1… |
|
1673 </li> |
|
1674 |
|
1675 <li> |
|
1676 <p><math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e1… |
|
1677 </li> |
|
1678 </ol> |
|
1679 |
|
1680 <p>there is a <a class='existingWikiWord' href='/nlab/show/stable+weak+h… |
|
1681 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
1682 \Sigma^\infty Conf(\mathbb{R}^d, Y) |
|
1683 \overset{\simeq}{\longrightarrow} |
|
1684 \underset{n \in \mathbb{N}}{\oplus} \Sigma^\infty Conf_n(\mathbb{R}^d,… |
|
1685 |
|
1686 </annotation></semantics></math></div> |
|
1687 <p>between</p> |
|
1688 |
|
1689 <ol> |
|
1690 <li> |
|
1691 <p>the <a class='existingWikiWord' href='/nlab/show/suspension+spectrum'… |
|
1692 </li> |
|
1693 |
|
1694 <li> |
|
1695 <p>the <a class='existingWikiWord' href='/nlab/show/direct+sum'>direct s… |
|
1696 </li> |
|
1697 </ol> |
|
1698 |
|
1699 <p>Combined with the <a class='existingWikiWord' href='/nlab/show/stabil… |
|
1700 <div class='maruku-equation' id='eq:StableSplittingOfMappingSpacesOutOfS… |
|
1701 |
|
1702 Maps_{cp}(\mathbb{R}^d, \Sigma^d (Y / \partial Y)) |
|
1703 = |
|
1704 Maps^{\ast/}( S^d, \Sigma^d (Y / \partial Y)) |
|
1705 = |
|
1706 \Omega^d \Sigma^d (Y/\partial Y) |
|
1707 \underoverset{\Sigma^\infty scan}{\simeq}{\longrightarrow} |
|
1708 \Sigma^\infty Conf(\mathbb{R}^d, Y) |
|
1709 \overset{\simeq}{\longrightarrow} |
|
1710 \underset{n \in \mathbb{N}}{\oplus} \Sigma^\infty Conf_n(\mathbb{R}^d,… |
|
1711 |
|
1712 </annotation></semantics></math></div> |
|
1713 <p>between the latter direct sum and the <a class='existingWikiWord' hre… |
|
1714 </div> |
|
1715 |
|
1716 <p>(<a href='#Snaith74'>Snaith 74, theorem 1.1</a>, <a href='#Boedigheim… |
|
1717 |
|
1718 <p>In fact by <a href='#Boedigheimer87'>Bödigheimer 87, Example 5</a> t… |
|
1719 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
1720 Maps_{cp}(\mathbb{R}^d, \Sigma^d (Y / \partial Y)) |
|
1721 = |
|
1722 Maps^{\ast/}( S^d, \Sigma^d (Y / \partial Y)) |
|
1723 \overset{\simeq}{\longrightarrow} |
|
1724 \underset{n \in \mathbb{N}}{\oplus} \Sigma^\infty Conf_n(\mathbb{R}^d … |
|
1725 |
|
1726 </annotation></semantics></math></div> |
|
1727 <h3 id='InTermsOfGoodwillieTowers'>In terms of Goodwillie-Taylor towers<… |
|
1728 |
|
1729 <p>We discuss the interpretation of the above stable splitting of mappin… |
|
1730 |
|
1731 <p>Observe that the <a class='existingWikiWord' href='/nlab/show/configu… |
|
1732 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
1733 Conf_n(X,Y) |
|
1734 \;\coloneqq\; |
|
1735 \Big( |
|
1736 \big( |
|
1737 ( |
|
1738 X^n \setminus \mathbf{\Delta}_X^n |
|
1739 ) |
|
1740 \times |
|
1741 Y^n |
|
1742 \big) |
|
1743 / \partial(X^n \times Y^n) |
|
1744 \Big) |
|
1745 /\Sigma(n) |
|
1746 |
|
1747 </annotation></semantics></math></div> |
|
1748 <p>is the <a class='existingWikiWord' href='/nlab/show/quotient+object'>… |
|
1749 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
1750 Y^{\times_n}/\partial (Y^{\times_n}) |
|
1751 \;\simeq\; |
|
1752 ( Y/\partial Y )^{\wedge_n} |
|
1753 \,. |
|
1754 |
|
1755 </annotation></semantics></math></div> |
|
1756 <p>Hence in summary:</p> |
|
1757 <div class='maruku-equation' id='eq:ConfSplitsAsSmashProduct'><span clas… |
|
1758 |
|
1759 Conf_n(X, Y) |
|
1760 \;\simeq\; |
|
1761 Conf^{ord}_n(X) \wedge_{\Sigma(n)} \left( Y/\partial Y \right)^{\wedge… |
|
1762 \,, |
|
1763 |
|
1764 </annotation></semantics></math></div> |
|
1765 <p>where</p> |
|
1766 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
1767 Conf_n^{ord}(X) |
|
1768 \;\coloneqq\; |
|
1769 \left( |
|
1770 X^{\times_n} \setminus \mathbf{\Delta}_X^n |
|
1771 \right)/ \partial(X^n) |
|
1772 |
|
1773 </annotation></semantics></math></div> |
|
1774 <p>is the ordered configuration space <a class='maruku-eqref' href='#eq:… |
|
1775 |
|
1776 <p>This construction, regarded as a <a class='existingWikiWord' href='/n… |
|
1777 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
1778 \array{ |
|
1779 Top^{\ast/} |
|
1780 &\longrightarrow& |
|
1781 Spectra |
|
1782 \\ |
|
1783 Z |
|
1784 &\mapsto& |
|
1785 \Sigma^\infty Conf^{ord}_n(X) \wedge_{\Sigma(n)} Z^{\wedge_n} |
|
1786 } |
|
1787 |
|
1788 </annotation></semantics></math></div> |
|
1789 <p>is an <a class='existingWikiWord' href='/nlab/show/n-homogeneous+%28%… |
|
1790 <div class='maruku-equation' id='eq:IdentifyingTheGoodwillieTaylorStage'… |
|
1791 |
|
1792 Z |
|
1793 \;\mapsto\; |
|
1794 \underset{k \in \{1, \cdot, n\}}{\bigoplus} |
|
1795 \Sigma^\infty Conf^{ord}_k(X) \wedge_{\Sigma(k)} Z^{\wedge_k} |
|
1796 |
|
1797 </annotation></semantics></math></div> |
|
1798 <p>are <a class='existingWikiWord' href='/nlab/show/n-excisive+%28%E2%88… |
|
1799 <div class='maruku-equation' id='eq:ProjectionMaps'><span class='maruku-… |
|
1800 |
|
1801 \array{ |
|
1802 Maps_{cp}(\mathbb{R}^d, \Sigma^d Z) |
|
1803 &=& |
|
1804 Maps^{\ast/}( S^d, \Sigma^d Z) |
|
1805 &\simeq& |
|
1806 \underset{k \in \mathbb{N}}{\oplus} |
|
1807 \Sigma^\infty Conf^{ord}_k(\mathbb{R}^d) \wedge_{\Sigma(k)} Z^{\we… |
|
1808 \\ |
|
1809 && |
|
1810 && |
|
1811 \Big\downarrow {}^{\mathrlap{ p_n }} |
|
1812 \\ |
|
1813 && |
|
1814 && |
|
1815 \underset{k \in \{1, \cdot, n\}}{\bigoplus} |
|
1816 \Sigma^\infty Conf^{ord}_k( \mathbb{R}^d ) \wedge_{\Sigma(k)} Z^{\w… |
|
1817 } |
|
1818 |
|
1819 </annotation></semantics></math></div> |
|
1820 <p>and this is <a class='existingWikiWord' href='/nlab/show/n-connected+… |
|
1821 |
|
1822 <p>By <a class='existingWikiWord' href='/nlab/show/Goodwillie+calculus'>… |
|
1823 <div class='maruku-equation' id='eq:TheGoodwillieStagesOfTheMappingSpace… |
|
1824 |
|
1825 P_n Maps^{\ast/}( S^d, \Sigma^d (-)) |
|
1826 \;\colon\; |
|
1827 Z \mapsto |
|
1828 \underset{k \in \{1, \cdot, n\}}{\bigoplus} |
|
1829 \Sigma^\infty Conf^{ord}_k(S^d, Z) |
|
1830 |
|
1831 </annotation></semantics></math></div> |
|
1832 <p>at <math class='maruku-mathml' display='inline' id='mathml_088ef31fdc… |
|
1833 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
1834 Maps_{cp}(\mathbb{R}^d, \Sigma^d (-)) |
|
1835 = |
|
1836 Maps^{\ast/}( S^d, \Sigma^d (-)) |
|
1837 \;\colon\; |
|
1838 Top^{\ast/} \longrightarrow Top^{\ast/} |
|
1839 \,. |
|
1840 |
|
1841 </annotation></semantics></math></div> |
|
1842 <p>Therefore the stable splitting theorem <a class='maruku-ref' href='#S… |
|
1843 |
|
1844 <p>(<a href='#Arone99'>Arone 99, p. 1-2</a>, <a href='#Goodwillie03'>Goo… |
|
1845 |
|
1846 <p><math class='maruku-mathml' display='inline' id='mathml_088ef31fdc8e1… |
|
1847 |
|
1848 <h3 id='lax_closed_structure_on_'>Lax closed structure on <math class='m… |
|
1849 |
|
1850 <p>Notice that the first stage in the <a class='existingWikiWord' href='… |
|
1851 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
1852 \begin{aligned} |
|
1853 P_1 Maps^{\ast/}( S^d, \Sigma^d (Y / \partial Y) ) |
|
1854 & = |
|
1855 \Sigma^\infty Conf^{ord}_1( \mathbb{R}^d , Y ) |
|
1856 \\ |
|
1857 & \simeq |
|
1858 \Sigma^\infty |
|
1859 \underset{\simeq S^0}{\underbrace{Conf^{ord}_1( \mathbb{R}^d )}} |
|
1860 \wedge (Y/\partial Y) |
|
1861 \\ |
|
1862 & \simeq |
|
1863 \Sigma^\infty (Y/\partial Y) |
|
1864 \\ |
|
1865 & \simeq |
|
1866 \Omega^d \Sigma^d \Sigma^\infty (Y/\partial Y) |
|
1867 \\ |
|
1868 & \simeq |
|
1869 Maps\left( \Sigma^\infty S^d, \Sigma^d (Y/\partial Y) \right) |
|
1870 \end{aligned} |
|
1871 |
|
1872 </annotation></semantics></math></div> |
|
1873 <p>Here in the first step we used <a class='maruku-eqref' href='#eq:TheG… |
|
1874 |
|
1875 <p>Therefore the projection <math class='maruku-mathml' display='inline'… |
|
1876 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
1877 p_1 |
|
1878 \;\colon\; |
|
1879 \Sigma^\infty Maps\left( S^d , \Sigma^d (Y /\partial Y) \right) |
|
1880 \longrightarrow |
|
1881 Maps |
|
1882 \left( |
|
1883 \Sigma^\infty S^d, \Sigma^\infty \Sigma^d (Y / \partial Y) |
|
1884 \right) |
|
1885 \,. |
|
1886 |
|
1887 </annotation></semantics></math></div> |
|
1888 <p>Since <math class='maruku-mathml' display='inline' id='mathml_088ef31… |
|
1889 |
|
1890 <blockquote> |
|
1891 <p>Does it?</p> |
|
1892 </blockquote> |
|
1893 |
|
1894 <h2 id='related_concepts'>Related concepts</h2> |
|
1895 |
|
1896 <ul> |
|
1897 <li><a class='existingWikiWord' href='/nlab/show/function+spectrum'>mapp… |
|
1898 </ul> |
|
1899 |
|
1900 <h2 id='references'>References</h2> |
|
1901 |
|
1902 <p>The theorem is originally due to</p> |
|
1903 |
|
1904 <ul> |
|
1905 <li id='Snaith74'><a class='existingWikiWord' href='/nlab/show/Victor+Sn… |
|
1906 </ul> |
|
1907 |
|
1908 <p>using the homotopy equivalence before stabilization due to</p> |
|
1909 |
|
1910 <ul> |
|
1911 <li id='May72'> |
|
1912 <p><a class='existingWikiWord' href='/nlab/show/Peter+May'>Peter May</a>… |
|
1913 </li> |
|
1914 |
|
1915 <li id='Segal73'> |
|
1916 <p><a class='existingWikiWord' href='/nlab/show/Graeme+Segal'>Graeme Seg… |
|
1917 </li> |
|
1918 </ul> |
|
1919 |
|
1920 <p>An alternative proof is due to</p> |
|
1921 |
|
1922 <ul> |
|
1923 <li id='Cohen80'><a class='existingWikiWord' href='/nlab/show/Ralph+Cohe… |
|
1924 </ul> |
|
1925 |
|
1926 <p>Review and generalization is due to</p> |
|
1927 |
|
1928 <ul> |
|
1929 <li id='Boedigheimer87'><a class='existingWikiWord' href='/nlab/show/Car… |
|
1930 </ul> |
|
1931 |
|
1932 <p>Interpretation in terms of the <a class='existingWikiWord' href='/nla… |
|
1933 |
|
1934 <ul> |
|
1935 <li id='Arone99'> |
|
1936 <p><a class='existingWikiWord' href='/nlab/show/Gregory+Arone'>Greg Aron… |
|
1937 </li> |
|
1938 |
|
1939 <li id='Ching05'> |
|
1940 <p><a class='existingWikiWord' href='/nlab/show/Michael+Ching'>Michael C… |
|
1941 </li> |
|
1942 |
|
1943 <li id='Goodwillie03'> |
|
1944 <p><a class='existingWikiWord' href='/nlab/show/Thomas+Goodwillie'>Thoma… |
|
1945 </li> |
|
1946 </ul> |
|
1947 |
|
1948 <p>A proof via <a class='existingWikiWord' href='/nlab/show/nonabelian+P… |
|
1949 |
|
1950 <ul> |
|
1951 <li>Lauren Bandklayder, <em>Stable splitting of mapping spaces via nonab… |
|
1952 </ul> |
|
1953 |
|
1954 <p>See also:</p> |
|
1955 |
|
1956 <ul> |
|
1957 <li><a class='existingWikiWord' href='/nlab/show/Doug+Ravenel'>Douglas R… |
|
1958 </ul> |
|
1959 |
|
1960 <p> |
|
1961 </p> </div> |
|
1962 </content> |
|
1963 </entry> |
|
1964 <entry> |
|
1965 <title type="html">Tarmo Uustalu</title> |
|
1966 <link rel="alternate" type="application/xhtml+xml" href="https://nca… |
|
1967 <updated>2021-07-02T00:18:56Z</updated> |
|
1968 <published>2021-07-01T06:30:08Z</published> |
|
1969 <id>tag:ncatlab.org,2021-07-01:nLab,Tarmo+Uustalu</id> |
|
1970 <author> |
|
1971 <name>Dmitri Pavlov</name> |
|
1972 </author> |
|
1973 <content type="xhtml" xml:base="https://ncatlab.org/nlab/show/Tarmo+… |
|
1974 <div xmlns="http://www.w3.org/1999/xhtml"> |
|
1975 <p>Tarmo Uustalu is a professor at the Dept. of Computer Science of Reyk… |
|
1976 |
|
1977 <ul> |
|
1978 <li><a href='https://www.ioc.ee/~tarmo/'>Home page</a></li> |
|
1979 </ul> |
|
1980 |
|
1981 <p><div class='property'> category: <a class='category_link' href='/nlab… |
|
1982 |
|
1983 <p> |
|
1984 </p> </div> |
|
1985 </content> |
|
1986 </entry> |
|
1987 <entry> |
|
1988 <title type="html">Monster group</title> |
|
1989 <link rel="alternate" type="application/xhtml+xml" href="https://nca… |
|
1990 <updated>2021-07-01T21:40:24Z</updated> |
|
1991 <published>2010-05-19T06:28:51Z</published> |
|
1992 <id>tag:ncatlab.org,2010-05-19:nLab,Monster+group</id> |
|
1993 <author> |
|
1994 <name>Urs Schreiber</name> |
|
1995 </author> |
|
1996 <content type="xhtml" xml:base="https://ncatlab.org/nlab/show/Monste… |
|
1997 <div xmlns="http://www.w3.org/1999/xhtml"> |
|
1998 <div class='rightHandSide'> |
|
1999 <div class='toc clickDown' tabindex='0'> |
|
2000 <h3 id='context'>Context</h3> |
|
2001 |
|
2002 <h4 id='exceptional_structures'>Exceptional structures</h4> |
|
2003 |
|
2004 <div class='hide'> |
|
2005 <p><strong><a class='existingWikiWord' href='/nlab/show/exceptional+stru… |
|
2006 |
|
2007 <h2 id='examples'>Examples</h2> |
|
2008 |
|
2009 <ul> |
|
2010 <li> |
|
2011 <p><a class='existingWikiWord' href='/nlab/show/sporadic+finite+simple+g… |
|
2012 |
|
2013 <ul> |
|
2014 <li> |
|
2015 <p><a class='existingWikiWord' href='/nlab/show/Monster+group'>monster g… |
|
2016 </li> |
|
2017 |
|
2018 <li> |
|
2019 <p><a class='existingWikiWord' href='/nlab/show/Mathieu+group'>Mathieu g… |
|
2020 </li> |
|
2021 |
|
2022 <li> |
|
2023 <p><a class='existingWikiWord' href='/nlab/show/Conway+group'>Conway gro… |
|
2024 </li> |
|
2025 </ul> |
|
2026 </li> |
|
2027 |
|
2028 <li> |
|
2029 <p>exceptional <a class='existingWikiWord' href='/nlab/show/finite+rotat… |
|
2030 |
|
2031 <ul> |
|
2032 <li> |
|
2033 <p><a class='existingWikiWord' href='/nlab/show/tetrahedral+group'>tetra… |
|
2034 </li> |
|
2035 |
|
2036 <li> |
|
2037 <p><a class='existingWikiWord' href='/nlab/show/octahedral+group'>octahe… |
|
2038 </li> |
|
2039 |
|
2040 <li> |
|
2041 <p><a class='existingWikiWord' href='/nlab/show/icosahedral+group'>icosa… |
|
2042 </li> |
|
2043 </ul> |
|
2044 </li> |
|
2045 |
|
2046 <li> |
|
2047 <p><a class='existingWikiWord' href='/nlab/show/exceptional+Lie+group'>e… |
|
2048 |
|
2049 <ul> |
|
2050 <li> |
|
2051 <p><a class='existingWikiWord' href='/nlab/show/G2'>G2</a></p> |
|
2052 </li> |
|
2053 |
|
2054 <li> |
|
2055 <p><a class='existingWikiWord' href='/nlab/show/F4'>F4</a></p> |
|
2056 </li> |
|
2057 |
|
2058 <li> |
|
2059 <p><a class='existingWikiWord' href='/nlab/show/E6'>E6</a>, <a class='ex… |
|
2060 </li> |
|
2061 </ul> |
|
2062 |
|
2063 <p>and <a class='existingWikiWord' href='/nlab/show/Kac-Moody+group'>Kac… |
|
2064 |
|
2065 <ul> |
|
2066 <li><a class='existingWikiWord' href='/nlab/show/E9'>E9</a>, <a class='e… |
|
2067 </ul> |
|
2068 </li> |
|
2069 |
|
2070 <li> |
|
2071 <p><a class='existingWikiWord' href='/nlab/show/Dwyer-Wilkerson+H-space'… |
|
2072 </li> |
|
2073 |
|
2074 <li> |
|
2075 <p><a class='existingWikiWord' href='/nlab/show/exceptional+Lie+algebra'… |
|
2076 </li> |
|
2077 |
|
2078 <li> |
|
2079 <p><a class='existingWikiWord' href='/nlab/show/Albert+algebra'>exceptio… |
|
2080 |
|
2081 <ul> |
|
2082 <li><a class='existingWikiWord' href='/nlab/show/Albert+algebra'>Albert … |
|
2083 </ul> |
|
2084 </li> |
|
2085 |
|
2086 <li> |
|
2087 <p>exceptional <a class='existingWikiWord' href='/nlab/show/Jordan+super… |
|
2088 </li> |
|
2089 |
|
2090 <li> |
|
2091 <p><a class='existingWikiWord' href='/nlab/show/Leech+lattice'>Leech lat… |
|
2092 </li> |
|
2093 |
|
2094 <li> |
|
2095 <p><a class='existingWikiWord' href='/nlab/show/Cayley+plane'>Cayley pla… |
|
2096 </li> |
|
2097 </ul> |
|
2098 |
|
2099 <h2 id='interrelations'>Interrelations</h2> |
|
2100 |
|
2101 <ul> |
|
2102 <li> |
|
2103 <p><a class='existingWikiWord' href='/nlab/show/division+algebra+and+sup… |
|
2104 </li> |
|
2105 |
|
2106 <li> |
|
2107 <p><a class='existingWikiWord' href='/nlab/show/Freudenthal+magic+square… |
|
2108 </li> |
|
2109 |
|
2110 <li> |
|
2111 <p><a class='existingWikiWord' href='/nlab/show/Moonshine'>moonshine</a>… |
|
2112 |
|
2113 <ul> |
|
2114 <li> |
|
2115 <p><a class='existingWikiWord' href='/nlab/show/Mathieu+moonshine'>Mathi… |
|
2116 </li> |
|
2117 |
|
2118 <li> |
|
2119 <p><a class='existingWikiWord' href='/nlab/show/umbral+moonshine'>umbral… |
|
2120 </li> |
|
2121 |
|
2122 <li> |
|
2123 <p><a class='existingWikiWord' href='/nlab/show/O%27Nan+moonshine'>O'… |
|
2124 </li> |
|
2125 </ul> |
|
2126 </li> |
|
2127 </ul> |
|
2128 |
|
2129 <h2 id='applications'>Applications</h2> |
|
2130 |
|
2131 <ul> |
|
2132 <li> |
|
2133 <p><a class='existingWikiWord' href='/nlab/show/exceptional+geometry'>ex… |
|
2134 </li> |
|
2135 |
|
2136 <li> |
|
2137 <p><a class='existingWikiWord' href='/nlab/show/exceptional+field+theory… |
|
2138 </li> |
|
2139 </ul> |
|
2140 |
|
2141 <h2 id='philosophy'>Philosophy</h2> |
|
2142 |
|
2143 <ul> |
|
2144 <li><a class='existingWikiWord' href='/nlab/show/universal+exceptionalis… |
|
2145 </ul> |
|
2146 </div> |
|
2147 |
|
2148 <h4 id='group_theory'>Group Theory</h4> |
|
2149 |
|
2150 <div class='hide'> |
|
2151 <p><strong><a class='existingWikiWord' href='/nlab/show/group+theory'>gr… |
|
2152 |
|
2153 <ul> |
|
2154 <li><a class='existingWikiWord' href='/nlab/show/group'>group</a>, <a cl… |
|
2155 |
|
2156 <li><a class='existingWikiWord' href='/nlab/show/group+object'>group obj… |
|
2157 |
|
2158 <li><a class='existingWikiWord' href='/nlab/show/abelian+group'>abelian … |
|
2159 |
|
2160 <li><a class='existingWikiWord' href='/nlab/show/action'>group action</a… |
|
2161 |
|
2162 <li><a class='existingWikiWord' href='/nlab/show/representation'>represe… |
|
2163 |
|
2164 <li><a class='existingWikiWord' href='/nlab/show/progroup'>progroup</a><… |
|
2165 |
|
2166 <li><a class='existingWikiWord' href='/nlab/show/homogeneous+space'>homo… |
|
2167 </ul> |
|
2168 |
|
2169 <h3 id='classical_groups'>Classical groups</h3> |
|
2170 |
|
2171 <ul> |
|
2172 <li> |
|
2173 <p><a class='existingWikiWord' href='/nlab/show/general+linear+group'>ge… |
|
2174 </li> |
|
2175 |
|
2176 <li> |
|
2177 <p><a class='existingWikiWord' href='/nlab/show/unitary+group'>unitary g… |
|
2178 |
|
2179 <ul> |
|
2180 <li><a class='existingWikiWord' href='/nlab/show/special+unitary+group'>… |
|
2181 </ul> |
|
2182 </li> |
|
2183 |
|
2184 <li> |
|
2185 <p><a class='existingWikiWord' href='/nlab/show/orthogonal+group'>orthog… |
|
2186 |
|
2187 <ul> |
|
2188 <li><a class='existingWikiWord' href='/nlab/show/special+orthogonal+grou… |
|
2189 </ul> |
|
2190 </li> |
|
2191 |
|
2192 <li> |
|
2193 <p><a class='existingWikiWord' href='/nlab/show/symplectic+group'>symple… |
|
2194 </li> |
|
2195 </ul> |
|
2196 |
|
2197 <h3 id='finite_groups'>Finite groups</h3> |
|
2198 |
|
2199 <ul> |
|
2200 <li> |
|
2201 <p><a class='existingWikiWord' href='/nlab/show/finite+group'>finite gro… |
|
2202 </li> |
|
2203 |
|
2204 <li> |
|
2205 <p><a class='existingWikiWord' href='/nlab/show/symmetric+group'>symmetr… |
|
2206 </li> |
|
2207 |
|
2208 <li> |
|
2209 <p><a class='existingWikiWord' href='/nlab/show/classification+of+finite… |
|
2210 </li> |
|
2211 |
|
2212 <li> |
|
2213 <p><a class='existingWikiWord' href='/nlab/show/sporadic+finite+simple+g… |
|
2214 |
|
2215 <ul> |
|
2216 <li><a class='existingWikiWord' href='/nlab/show/Monster+group'>Monster … |
|
2217 </ul> |
|
2218 </li> |
|
2219 </ul> |
|
2220 |
|
2221 <h3 id='group_schemes'>Group schemes</h3> |
|
2222 |
|
2223 <ul> |
|
2224 <li><a class='existingWikiWord' href='/nlab/show/algebraic+group'>algebr… |
|
2225 |
|
2226 <li><a class='existingWikiWord' href='/nlab/show/abelian+variety'>abelia… |
|
2227 </ul> |
|
2228 |
|
2229 <h3 id='topological_groups'>Topological groups</h3> |
|
2230 |
|
2231 <ul> |
|
2232 <li> |
|
2233 <p><a class='existingWikiWord' href='/nlab/show/topological+group'>topol… |
|
2234 </li> |
|
2235 |
|
2236 <li> |
|
2237 <p><a class='existingWikiWord' href='/nlab/show/compact+topological+grou… |
|
2238 </li> |
|
2239 |
|
2240 <li> |
|
2241 <p><a class='existingWikiWord' href='/nlab/show/maximal+compact+subgroup… |
|
2242 </li> |
|
2243 |
|
2244 <li> |
|
2245 <p><a class='existingWikiWord' href='/nlab/show/string+group'>string gro… |
|
2246 </li> |
|
2247 </ul> |
|
2248 |
|
2249 <h3 id='lie_groups'>Lie groups</h3> |
|
2250 |
|
2251 <ul> |
|
2252 <li> |
|
2253 <p><a class='existingWikiWord' href='/nlab/show/Lie+group'>Lie group</a>… |
|
2254 </li> |
|
2255 |
|
2256 <li> |
|
2257 <p><a class='existingWikiWord' href='/nlab/show/compact+Lie+group'>compa… |
|
2258 </li> |
|
2259 |
|
2260 <li> |
|
2261 <p><a class='existingWikiWord' href='/nlab/show/Kac-Moody+group'>Kac-Moo… |
|
2262 </li> |
|
2263 </ul> |
|
2264 |
|
2265 <h3 id='superlie_groups'>Super-Lie groups</h3> |
|
2266 |
|
2267 <ul> |
|
2268 <li> |
|
2269 <p><a class='existingWikiWord' href='/nlab/show/supergroup'>super Lie gr… |
|
2270 </li> |
|
2271 |
|
2272 <li> |
|
2273 <p><a class='existingWikiWord' href='/nlab/show/super+Euclidean+group'>s… |
|
2274 </li> |
|
2275 </ul> |
|
2276 |
|
2277 <h3 id='higher_groups'>Higher groups</h3> |
|
2278 |
|
2279 <ul> |
|
2280 <li> |
|
2281 <p><a class='existingWikiWord' href='/nlab/show/2-group'>2-group</a></p> |
|
2282 |
|
2283 <ul> |
|
2284 <li><a class='existingWikiWord' href='/nlab/show/crossed+module'>crossed… |
|
2285 </ul> |
|
2286 </li> |
|
2287 |
|
2288 <li> |
|
2289 <p><a class='existingWikiWord' href='/nlab/show/n-group'>n-group</a></p> |
|
2290 </li> |
|
2291 |
|
2292 <li> |
|
2293 <p><a class='existingWikiWord' href='/nlab/show/infinity-group'>∞-grou… |
|
2294 |
|
2295 <ul> |
|
2296 <li> |
|
2297 <p><a class='existingWikiWord' href='/nlab/show/simplicial+group'>simpli… |
|
2298 </li> |
|
2299 |
|
2300 <li> |
|
2301 <p><a class='existingWikiWord' href='/nlab/show/crossed+complex'>crossed… |
|
2302 </li> |
|
2303 |
|
2304 <li> |
|
2305 <p><a class='existingWikiWord' href='/nlab/show/k-tuply+groupal+n-groupo… |
|
2306 </li> |
|
2307 |
|
2308 <li> |
|
2309 <p><a class='existingWikiWord' href='/nlab/show/spectrum'>spectrum</a></… |
|
2310 </li> |
|
2311 </ul> |
|
2312 </li> |
|
2313 |
|
2314 <li> |
|
2315 <p><a class='existingWikiWord' href='/nlab/show/circle+n-group'>circle n… |
|
2316 </li> |
|
2317 </ul> |
|
2318 |
|
2319 <h3 id='cohomology_and_extensions'>Cohomology and Extensions</h3> |
|
2320 |
|
2321 <ul> |
|
2322 <li> |
|
2323 <p><a class='existingWikiWord' href='/nlab/show/group+cohomology'>group … |
|
2324 </li> |
|
2325 |
|
2326 <li> |
|
2327 <p><a class='existingWikiWord' href='/nlab/show/group+extension'>group e… |
|
2328 </li> |
|
2329 |
|
2330 <li> |
|
2331 <p><a class='existingWikiWord' href='/nlab/show/infinity-group+extension… |
|
2332 </li> |
|
2333 </ul> |
|
2334 |
|
2335 <h3 id='_related_concepts'>Related concepts</h3> |
|
2336 |
|
2337 <ul> |
|
2338 <li><a class='existingWikiWord' href='/nlab/show/quantum+group'>quantum … |
|
2339 </ul> |
|
2340 <div> |
|
2341 <p> |
|
2342 <a href='/nlab/edit/group+theory+-+contents'>Edit this sidebar</a> |
|
2343 </p> |
|
2344 </div></div> |
|
2345 </div> |
|
2346 </div> |
|
2347 |
|
2348 <h1 id='contents'>Contents</h1> |
|
2349 <div class='maruku_toc'><ul><li><a href='#idea'>Idea</a></li><li><a href… |
|
2350 <h2 id='idea'>Idea</h2> |
|
2351 |
|
2352 <p>The <strong>Monster group</strong> <math class='maruku-mathml' displa… |
|
2353 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
2354 \begin{aligned} |
|
2355 & 2^{46}\cdot 3^{20}\cdot 5^9\cdot 7^6\cdot 11^2\cdot 13^3\cdot 17\c… |
|
2356 \\ |
|
2357 & = 808017424794512875886459904961710757005754368000000000 |
|
2358 \end{aligned} |
|
2359 |
|
2360 </annotation></semantics></math></div> |
|
2361 <p>and contains all but six (the ‘<a class='existingWikiWord' href='/n… |
|
2362 |
|
2363 <p>See also <a class='existingWikiWord' href='/nlab/show/Moonshine'>Moon… |
|
2364 |
|
2365 <h2 id='history'>History</h2> |
|
2366 |
|
2367 <p>The Monster group was predicted to exist by <a class='existingWikiWor… |
|
2368 |
|
2369 <ul> |
|
2370 <li><a class='existingWikiWord' href='/nlab/show/Robert+Griess'>Robert G… |
|
2371 </ul> |
|
2372 |
|
2373 <p>Griess proved the existence of the largest simple sporadic group. The… |
|
2374 |
|
2375 <p>After Griess found this algebra <a class='existingWikiWord' href='/nl… |
|
2376 |
|
2377 <p>There is a school of thought, going back to at least <a class='existi… |
|
2378 |
|
2379 <h2 id='presentation'>Presentation</h2> |
|
2380 |
|
2381 <h3 id='via_coxeter_groups'>Via Coxeter groups</h3> |
|
2382 |
|
2383 <p>The Monster admits a reasonably succinct description in terms of <a c… |
|
2384 |
|
2385 <p>Regard <math class='maruku-mathml' display='inline' id='mathml_b35f69… |
|
2386 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
2387 <p>the resulting quotient <math class='maruku-mathml' display='inline' i… |
|
2388 |
|
2389 <p>It turns out that <math class='maruku-mathml' display='inline' id='ma… |
|
2390 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
2391 <p>This implicitly describes the Monster in terms of 12 generators and 8… |
|
2392 |
|
2393 <p>Such “<math class='maruku-mathml' display='inline' id='mathml_b35f6… |
|
2394 |
|
2395 <h3 id='ViaAutomorphisms'>Via automorphisms of a super vertex operator a… |
|
2396 |
|
2397 <p>There is a <a class='existingWikiWord' href='/nlab/show/super+vertex+… |
|
2398 |
|
2399 <p>(<a href='#FrenkelLepowskiMeurman89'>Frenkel-Lepowski-Meurman 89</a>,… |
|
2400 |
|
2401 <h2 id='related_concepts_2'>Related concepts</h2> |
|
2402 |
|
2403 <ul> |
|
2404 <li> |
|
2405 <p><a class='existingWikiWord' href='/nlab/show/Moonshine'>Moonshine</a>… |
|
2406 </li> |
|
2407 |
|
2408 <li> |
|
2409 <p><a class='existingWikiWord' href='/nlab/show/Monster+vertex+operator+… |
|
2410 </li> |
|
2411 |
|
2412 <li> |
|
2413 <p><a class='existingWikiWord' href='/nlab/show/Mathieu+group'>Mathieu g… |
|
2414 </li> |
|
2415 </ul> |
|
2416 |
|
2417 <h2 id='references'>References</h2> |
|
2418 |
|
2419 <ul> |
|
2420 <li> |
|
2421 <p><a href='http://mathoverflow.net/users/39521/adam-p-goucher'>Adam P. … |
|
2422 </li> |
|
2423 |
|
2424 <li id='Iv'> |
|
2425 <p>Alexander Ivanov, <em>Y-groups via transitive extension</em>, Journal… |
|
2426 </li> |
|
2427 |
|
2428 <li id='Iv2'> |
|
2429 <p>A. A. Ivanov, <em>Constructing the Monster via its Y-presentation</em… |
|
2430 </li> |
|
2431 |
|
2432 <li id='FrenkelLepowskiMeurman89'> |
|
2433 <p><a class='existingWikiWord' href='/nlab/show/Igor+Frenkel'>Igor Frenk… |
|
2434 </li> |
|
2435 |
|
2436 <li id='GriessLam11'> |
|
2437 <p><a class='existingWikiWord' href='/nlab/show/Robert+Griess'>Robert Gr… |
|
2438 </li> |
|
2439 |
|
2440 <li> |
|
2441 <p><a class='existingWikiWord' href='/nlab/show/Andr%C3%A9+Henriques'>An… |
|
2442 </li> |
|
2443 </ul> |
|
2444 |
|
2445 <p>Possible relation to <a class='existingWikiWord' href='/nlab/show/bos… |
|
2446 |
|
2447 <ul> |
|
2448 <li><a class='existingWikiWord' href='/nlab/show/Alessio+Marrani'>Alessi… |
|
2449 </ul> |
|
2450 <div style='float: right; margin: 0 20px 10px 20px;'><img alt='The Monst… |
|
2451 <p> |
|
2452 </p> |
|
2453 |
|
2454 <p> |
|
2455 </p> |
|
2456 |
|
2457 <p> |
|
2458 </p> </div> |
|
2459 </content> |
|
2460 </entry> |
|
2461 <entry> |
|
2462 <title type="html">Moonshine</title> |
|
2463 <link rel="alternate" type="application/xhtml+xml" href="https://nca… |
|
2464 <updated>2021-07-01T21:06:02Z</updated> |
|
2465 <published>2010-05-18T20:54:08Z</published> |
|
2466 <id>tag:ncatlab.org,2010-05-18:nLab,Moonshine</id> |
|
2467 <author> |
|
2468 <name>Urs Schreiber</name> |
|
2469 </author> |
|
2470 <content type="xhtml" xml:base="https://ncatlab.org/nlab/show/Moonsh… |
|
2471 <div xmlns="http://www.w3.org/1999/xhtml"> |
|
2472 <div class='rightHandSide'> |
|
2473 <div class='toc clickDown' tabindex='0'> |
|
2474 <h3 id='context'>Context</h3> |
|
2475 |
|
2476 <h4 id='exceptional_structures'>Exceptional structures</h4> |
|
2477 |
|
2478 <div class='hide'> |
|
2479 <p><strong><a class='existingWikiWord' href='/nlab/show/exceptional+stru… |
|
2480 |
|
2481 <h2 id='examples'>Examples</h2> |
|
2482 |
|
2483 <ul> |
|
2484 <li> |
|
2485 <p><a class='existingWikiWord' href='/nlab/show/sporadic+finite+simple+g… |
|
2486 |
|
2487 <ul> |
|
2488 <li> |
|
2489 <p><a class='existingWikiWord' href='/nlab/show/Monster+group'>monster g… |
|
2490 </li> |
|
2491 |
|
2492 <li> |
|
2493 <p><a class='existingWikiWord' href='/nlab/show/Mathieu+group'>Mathieu g… |
|
2494 </li> |
|
2495 |
|
2496 <li> |
|
2497 <p><a class='existingWikiWord' href='/nlab/show/Conway+group'>Conway gro… |
|
2498 </li> |
|
2499 </ul> |
|
2500 </li> |
|
2501 |
|
2502 <li> |
|
2503 <p>exceptional <a class='existingWikiWord' href='/nlab/show/finite+rotat… |
|
2504 |
|
2505 <ul> |
|
2506 <li> |
|
2507 <p><a class='existingWikiWord' href='/nlab/show/tetrahedral+group'>tetra… |
|
2508 </li> |
|
2509 |
|
2510 <li> |
|
2511 <p><a class='existingWikiWord' href='/nlab/show/octahedral+group'>octahe… |
|
2512 </li> |
|
2513 |
|
2514 <li> |
|
2515 <p><a class='existingWikiWord' href='/nlab/show/icosahedral+group'>icosa… |
|
2516 </li> |
|
2517 </ul> |
|
2518 </li> |
|
2519 |
|
2520 <li> |
|
2521 <p><a class='existingWikiWord' href='/nlab/show/exceptional+Lie+group'>e… |
|
2522 |
|
2523 <ul> |
|
2524 <li> |
|
2525 <p><a class='existingWikiWord' href='/nlab/show/G2'>G2</a></p> |
|
2526 </li> |
|
2527 |
|
2528 <li> |
|
2529 <p><a class='existingWikiWord' href='/nlab/show/F4'>F4</a></p> |
|
2530 </li> |
|
2531 |
|
2532 <li> |
|
2533 <p><a class='existingWikiWord' href='/nlab/show/E6'>E6</a>, <a class='ex… |
|
2534 </li> |
|
2535 </ul> |
|
2536 |
|
2537 <p>and <a class='existingWikiWord' href='/nlab/show/Kac-Moody+group'>Kac… |
|
2538 |
|
2539 <ul> |
|
2540 <li><a class='existingWikiWord' href='/nlab/show/E9'>E9</a>, <a class='e… |
|
2541 </ul> |
|
2542 </li> |
|
2543 |
|
2544 <li> |
|
2545 <p><a class='existingWikiWord' href='/nlab/show/Dwyer-Wilkerson+H-space'… |
|
2546 </li> |
|
2547 |
|
2548 <li> |
|
2549 <p><a class='existingWikiWord' href='/nlab/show/exceptional+Lie+algebra'… |
|
2550 </li> |
|
2551 |
|
2552 <li> |
|
2553 <p><a class='existingWikiWord' href='/nlab/show/Albert+algebra'>exceptio… |
|
2554 |
|
2555 <ul> |
|
2556 <li><a class='existingWikiWord' href='/nlab/show/Albert+algebra'>Albert … |
|
2557 </ul> |
|
2558 </li> |
|
2559 |
|
2560 <li> |
|
2561 <p>exceptional <a class='existingWikiWord' href='/nlab/show/Jordan+super… |
|
2562 </li> |
|
2563 |
|
2564 <li> |
|
2565 <p><a class='existingWikiWord' href='/nlab/show/Leech+lattice'>Leech lat… |
|
2566 </li> |
|
2567 |
|
2568 <li> |
|
2569 <p><a class='existingWikiWord' href='/nlab/show/Cayley+plane'>Cayley pla… |
|
2570 </li> |
|
2571 </ul> |
|
2572 |
|
2573 <h2 id='interrelations'>Interrelations</h2> |
|
2574 |
|
2575 <ul> |
|
2576 <li> |
|
2577 <p><a class='existingWikiWord' href='/nlab/show/division+algebra+and+sup… |
|
2578 </li> |
|
2579 |
|
2580 <li> |
|
2581 <p><a class='existingWikiWord' href='/nlab/show/Freudenthal+magic+square… |
|
2582 </li> |
|
2583 |
|
2584 <li> |
|
2585 <p><a class='existingWikiWord' href='/nlab/show/Moonshine'>moonshine</a>… |
|
2586 |
|
2587 <ul> |
|
2588 <li> |
|
2589 <p><a class='existingWikiWord' href='/nlab/show/Mathieu+moonshine'>Mathi… |
|
2590 </li> |
|
2591 |
|
2592 <li> |
|
2593 <p><a class='existingWikiWord' href='/nlab/show/umbral+moonshine'>umbral… |
|
2594 </li> |
|
2595 |
|
2596 <li> |
|
2597 <p><a class='existingWikiWord' href='/nlab/show/O%27Nan+moonshine'>O'… |
|
2598 </li> |
|
2599 </ul> |
|
2600 </li> |
|
2601 </ul> |
|
2602 |
|
2603 <h2 id='applications'>Applications</h2> |
|
2604 |
|
2605 <ul> |
|
2606 <li> |
|
2607 <p><a class='existingWikiWord' href='/nlab/show/exceptional+geometry'>ex… |
|
2608 </li> |
|
2609 |
|
2610 <li> |
|
2611 <p><a class='existingWikiWord' href='/nlab/show/exceptional+field+theory… |
|
2612 </li> |
|
2613 </ul> |
|
2614 |
|
2615 <h2 id='philosophy'>Philosophy</h2> |
|
2616 |
|
2617 <ul> |
|
2618 <li><a class='existingWikiWord' href='/nlab/show/universal+exceptionalis… |
|
2619 </ul> |
|
2620 </div> |
|
2621 </div> |
|
2622 </div> |
|
2623 |
|
2624 <h1 id='contents'>Contents</h1> |
|
2625 <div class='maruku_toc'><ul><li><a href='#idea'>Idea</a></li><li><a href… |
|
2626 <h2 id='idea'>Idea</h2> |
|
2627 |
|
2628 <p>Moonshine usually refers to the mysterious connections between the <a… |
|
2629 |
|
2630 <p>The whole idea of moonshine began with <a class='existingWikiWord' hr… |
|
2631 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
2632 j(\tau) = q^{-1} + 744 + 196884q + 21493760q^{2} + \dots |
|
2633 |
|
2634 </annotation></semantics></math></div> |
|
2635 <p>where <math class='maruku-mathml' display='inline' id='mathml_5fa8578… |
|
2636 |
|
2637 <p>But the monster was merely <em>conjectured</em> to exist until Griess… |
|
2638 |
|
2639 <p><a href='#FrenkelLepowskiMeurman89'>Frenkel-Lepowski-Meurman 89</a> c… |
|
2640 |
|
2641 <p>To cut the story short, we end up getting from the Monster group to a… |
|
2642 |
|
2643 <h2 id='AutomorphismGroupsOfVertexOperatorAlgebras'>Automorphism groups … |
|
2644 |
|
2645 <p>Realizations of <a class='existingWikiWord' href='/nlab/show/sporadic… |
|
2646 |
|
2647 <ul> |
|
2648 <li> |
|
2649 <p>The <a class='existingWikiWord' href='/nlab/show/Conway+group'>Conway… |
|
2650 |
|
2651 <p>(<a href='#Duncan05'>Duncan 05</a>, see also <a href='#PaquettePersso… |
|
2652 </li> |
|
2653 |
|
2654 <li> |
|
2655 <p>similarly, there is a super VOA, the <em><a class='existingWikiWord' … |
|
2656 |
|
2657 <p>(<a href='#FrenkelLepowskiMeurman89'>Frenkel-Lepowski-Meurman 89</a>,… |
|
2658 </li> |
|
2659 </ul> |
|
2660 |
|
2661 <h2 id='related_concepts'>Related concepts</h2> |
|
2662 |
|
2663 <ul> |
|
2664 <li> |
|
2665 <p><a class='existingWikiWord' href='/nlab/show/Monster+group'>Monster</… |
|
2666 </li> |
|
2667 |
|
2668 <li> |
|
2669 <p>moonshine</p> |
|
2670 |
|
2671 <ul> |
|
2672 <li> |
|
2673 <p><a class='existingWikiWord' href='/nlab/show/Mathieu+moonshine'>Mathi… |
|
2674 </li> |
|
2675 |
|
2676 <li> |
|
2677 <p><a class='existingWikiWord' href='/nlab/show/umbral+moonshine'>umbral… |
|
2678 </li> |
|
2679 |
|
2680 <li> |
|
2681 <p><a class='existingWikiWord' href='/nlab/show/O%27Nan+moonshine'>O'… |
|
2682 </li> |
|
2683 </ul> |
|
2684 </li> |
|
2685 |
|
2686 <li> |
|
2687 <p><a class='existingWikiWord' href='/nlab/show/automorphism+of+a+vertex… |
|
2688 </li> |
|
2689 </ul> |
|
2690 |
|
2691 <h2 id='references'>References</h2> |
|
2692 |
|
2693 <h3 id='general'>General</h3> |
|
2694 |
|
2695 <ul> |
|
2696 <li> |
|
2697 <p><a class='existingWikiWord' href='/nlab/show/Richard+Borcherds'>Richa… |
|
2698 </li> |
|
2699 |
|
2700 <li> |
|
2701 <p>John F. R. Duncan, Michael J. Griffin, Ken Ono, <em>Moonshine</em> (<… |
|
2702 </li> |
|
2703 |
|
2704 <li id='GriessLam11'> |
|
2705 <p><a class='existingWikiWord' href='/nlab/show/Robert+Griess'>Robert Gr… |
|
2706 </li> |
|
2707 |
|
2708 <li id='FrenkelLepowskiMeurman89'> |
|
2709 <p><a class='existingWikiWord' href='/nlab/show/Igor+Frenkel'>Igor Frenk… |
|
2710 </li> |
|
2711 |
|
2712 <li> |
|
2713 <p><a class='existingWikiWord' href='/nlab/show/Terry+Gannon'>Terry Gann… |
|
2714 </li> |
|
2715 |
|
2716 <li> |
|
2717 <p><a class='existingWikiWord' href='/nlab/show/Terry+Gannon'>Terry Gann… |
|
2718 </li> |
|
2719 |
|
2720 <li> |
|
2721 <p>Koichiro Harada, <em>“Moonshine” of finite groups</em>. EMS Serie… |
|
2722 </li> |
|
2723 |
|
2724 <li> |
|
2725 <p>Griess, Robert L., Jr.; Lam, Ching Hung <em>A moonshine path from E8 … |
|
2726 </li> |
|
2727 |
|
2728 <li> |
|
2729 <p>Jae-Hyun Yang “Kac-Moody algebras, the Monstrous Moonshine, Jacobi … |
|
2730 </li> |
|
2731 |
|
2732 <li> |
|
2733 <p>Vassilis Anagiannis, <a class='existingWikiWord' href='/nlab/show/Mir… |
|
2734 </li> |
|
2735 </ul> |
|
2736 |
|
2737 <h3 id='historical_references'>Historical References</h3> |
|
2738 |
|
2739 <ul> |
|
2740 <li> |
|
2741 <p><a class='existingWikiWord' href='/nlab/show/John+Horton+Conway'>John… |
|
2742 </li> |
|
2743 |
|
2744 <li id='FrenkelLepowskiMeurman89'> |
|
2745 <p><a class='existingWikiWord' href='/nlab/show/Igor+Frenkel'>Igor Frenk… |
|
2746 </li> |
|
2747 |
|
2748 <li> |
|
2749 <p><a class='existingWikiWord' href='/nlab/show/Robert+Griess'>Robert Gr… |
|
2750 </li> |
|
2751 |
|
2752 <li> |
|
2753 <p>John G. Thompson, “Some numerology between the Fischer-Griess Monst… |
|
2754 </li> |
|
2755 </ul> |
|
2756 |
|
2757 <h3 id='FurtherDevelomentsReferences'>Further developments</h3> |
|
2758 |
|
2759 <ul> |
|
2760 <li> |
|
2761 <p><a class='existingWikiWord' href='/nlab/show/Miranda+Cheng'>Miranda C… |
|
2762 </li> |
|
2763 |
|
2764 <li> |
|
2765 <p>John F. R. Duncan, Michael J. Griffin and Ken Ono, <em>Proof of the U… |
|
2766 </li> |
|
2767 |
|
2768 <li> |
|
2769 <p><a class='existingWikiWord' href='/nlab/show/Scott+Carnahan'>Scott Ca… |
|
2770 </li> |
|
2771 </ul> |
|
2772 |
|
2773 <h3 id='realization_in_superstring_theory'>Realization in superstring th… |
|
2774 |
|
2775 <p>Discussion of possible realizations in <a class='existingWikiWord' hr… |
|
2776 |
|
2777 <ul> |
|
2778 <li> |
|
2779 <p>S. Chaudhuri, D.A. Lowe, <em>Monstrous String-String Duality</em>, Nu… |
|
2780 </li> |
|
2781 |
|
2782 <li id='Duncan05'> |
|
2783 <p>John F. Duncan, <em>Super-moonshine for Conway’s largest sporadic g… |
|
2784 </li> |
|
2785 |
|
2786 <li id='PaquettePerssonVolpato16'> |
|
2787 <p><a class='existingWikiWord' href='/nlab/show/Natalie+Paquette'>Natali… |
|
2788 </li> |
|
2789 |
|
2790 <li id='KachruPaquetteVolpato16'> |
|
2791 <p><a class='existingWikiWord' href='/nlab/show/Shamit+Kachru'>Shamit Ka… |
|
2792 </li> |
|
2793 |
|
2794 <li id='PaquettePerssonVolpato17'> |
|
2795 <p><a class='existingWikiWord' href='/nlab/show/Natalie+Paquette'>Natali… |
|
2796 </li> |
|
2797 |
|
2798 <li> |
|
2799 <p><a class='existingWikiWord' href='/nlab/show/Shamit+Kachru'>Shamit Ka… |
|
2800 </li> |
|
2801 </ul> |
|
2802 |
|
2803 <p>Specifically in relation to <a class='existingWikiWord' href='/nlab/s… |
|
2804 |
|
2805 <ul> |
|
2806 <li id='ChengHarrisonVolpatoZimet16'><a class='existingWikiWord' href='/… |
|
2807 </ul> |
|
2808 |
|
2809 <p>Possible relation to <a class='existingWikiWord' href='/nlab/show/bos… |
|
2810 |
|
2811 <ul> |
|
2812 <li><a class='existingWikiWord' href='/nlab/show/Alessio+Marrani'>Alessi… |
|
2813 </ul> |
|
2814 |
|
2815 <p> |
|
2816 |
|
2817 </p> |
|
2818 |
|
2819 <p> |
|
2820 |
|
2821 </p> |
|
2822 |
|
2823 <p> |
|
2824 |
|
2825 </p> </div> |
|
2826 </content> |
|
2827 </entry> |
|
2828 <entry> |
|
2829 <title type="html">geometric realization of categories</title> |
|
2830 <link rel="alternate" type="application/xhtml+xml" href="https://nca… |
|
2831 <updated>2021-07-01T17:07:36Z</updated> |
|
2832 <published>2011-05-30T18:19:35Z</published> |
|
2833 <id>tag:ncatlab.org,2011-05-30:nLab,geometric+realization+of+categor… |
|
2834 <author> |
|
2835 <name>Dmitri Pavlov</name> |
|
2836 </author> |
|
2837 <content type="xhtml" xml:base="https://ncatlab.org/nlab/show/geomet… |
|
2838 <div xmlns="http://www.w3.org/1999/xhtml"> |
|
2839 <div class='rightHandSide'> |
|
2840 <div class='toc clickDown' tabindex='0'> |
|
2841 <h3 id='context'>Context</h3> |
|
2842 |
|
2843 <h4 id='homotopy_theory'>Homotopy theory</h4> |
|
2844 |
|
2845 <div class='hide'> |
|
2846 <p><strong><a class='existingWikiWord' href='/nlab/show/homotopy+theory'… |
|
2847 |
|
2848 <p>flavors: <a class='existingWikiWord' href='/nlab/show/stable+homotopy… |
|
2849 |
|
2850 <p>models: <a class='existingWikiWord' href='/nlab/show/topological+homo… |
|
2851 |
|
2852 <p>see also <strong><a class='existingWikiWord' href='/nlab/show/algebra… |
|
2853 |
|
2854 <p><strong>Introductions</strong></p> |
|
2855 |
|
2856 <ul> |
|
2857 <li> |
|
2858 <p><a class='existingWikiWord' href='/nlab/show/Introduction+to+Topology… |
|
2859 </li> |
|
2860 |
|
2861 <li> |
|
2862 <p><a class='existingWikiWord' href='/nlab/show/Introduction+to+Homotopy… |
|
2863 </li> |
|
2864 |
|
2865 <li> |
|
2866 <p><a class='existingWikiWord' href='/nlab/show/geometry+of+physics+--+h… |
|
2867 </li> |
|
2868 </ul> |
|
2869 |
|
2870 <p><strong>Definitions</strong></p> |
|
2871 |
|
2872 <ul> |
|
2873 <li> |
|
2874 <p><a class='existingWikiWord' href='/nlab/show/homotopy'>homotopy</a>, … |
|
2875 </li> |
|
2876 |
|
2877 <li> |
|
2878 <p><a class='existingWikiWord' href='/nlab/show/homotopy+type'>homotopy … |
|
2879 </li> |
|
2880 |
|
2881 <li> |
|
2882 <p><a class='existingWikiWord' href='/nlab/show/Pi-algebra'>Pi-algebra</… |
|
2883 </li> |
|
2884 |
|
2885 <li> |
|
2886 <p><a class='existingWikiWord' href='/nlab/show/homotopy+coherent+catego… |
|
2887 |
|
2888 <ul> |
|
2889 <li> |
|
2890 <p><a class='existingWikiWord' href='/nlab/show/homotopical+category'>ho… |
|
2891 |
|
2892 <ul> |
|
2893 <li> |
|
2894 <p><a class='existingWikiWord' href='/nlab/show/model+category'>model ca… |
|
2895 </li> |
|
2896 |
|
2897 <li> |
|
2898 <p><a class='existingWikiWord' href='/nlab/show/category+of+fibrant+obje… |
|
2899 </li> |
|
2900 |
|
2901 <li> |
|
2902 <p><a class='existingWikiWord' href='/nlab/show/Waldhausen+category'>Wal… |
|
2903 </li> |
|
2904 </ul> |
|
2905 </li> |
|
2906 |
|
2907 <li> |
|
2908 <p><a class='existingWikiWord' href='/nlab/show/homotopy+category'>homot… |
|
2909 |
|
2910 <ul> |
|
2911 <li><a class='existingWikiWord' href='/nlab/show/Ho%28Top%29'>Ho(Top)</a… |
|
2912 </ul> |
|
2913 </li> |
|
2914 </ul> |
|
2915 </li> |
|
2916 |
|
2917 <li> |
|
2918 <p><a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-categ… |
|
2919 |
|
2920 <ul> |
|
2921 <li><a class='existingWikiWord' href='/nlab/show/homotopy+category+of+an… |
|
2922 </ul> |
|
2923 </li> |
|
2924 </ul> |
|
2925 |
|
2926 <p><strong>Paths and cylinders</strong></p> |
|
2927 |
|
2928 <ul> |
|
2929 <li> |
|
2930 <p><a class='existingWikiWord' href='/nlab/show/homotopy'>left homotopy<… |
|
2931 |
|
2932 <ul> |
|
2933 <li> |
|
2934 <p><a class='existingWikiWord' href='/nlab/show/cylinder+object'>cylinde… |
|
2935 </li> |
|
2936 |
|
2937 <li> |
|
2938 <p><a class='existingWikiWord' href='/nlab/show/mapping+cone'>mapping co… |
|
2939 </li> |
|
2940 </ul> |
|
2941 </li> |
|
2942 |
|
2943 <li> |
|
2944 <p><a class='existingWikiWord' href='/nlab/show/homotopy'>right homotopy… |
|
2945 |
|
2946 <ul> |
|
2947 <li> |
|
2948 <p><a class='existingWikiWord' href='/nlab/show/path+space+object'>path … |
|
2949 </li> |
|
2950 |
|
2951 <li> |
|
2952 <p><a class='existingWikiWord' href='/nlab/show/mapping+cocone'>mapping … |
|
2953 </li> |
|
2954 |
|
2955 <li> |
|
2956 <p><a class='existingWikiWord' href='/nlab/show/generalized+universal+bu… |
|
2957 </li> |
|
2958 </ul> |
|
2959 </li> |
|
2960 |
|
2961 <li> |
|
2962 <p><a class='existingWikiWord' href='/nlab/show/interval+object'>interva… |
|
2963 |
|
2964 <ul> |
|
2965 <li> |
|
2966 <p><a class='existingWikiWord' href='/nlab/show/localization+at+geometri… |
|
2967 </li> |
|
2968 |
|
2969 <li> |
|
2970 <p><a class='existingWikiWord' href='/nlab/show/infinitesimal+interval+o… |
|
2971 </li> |
|
2972 </ul> |
|
2973 </li> |
|
2974 </ul> |
|
2975 |
|
2976 <p><strong>Homotopy groups</strong></p> |
|
2977 |
|
2978 <ul> |
|
2979 <li> |
|
2980 <p><a class='existingWikiWord' href='/nlab/show/homotopy+group'>homotopy… |
|
2981 |
|
2982 <ul> |
|
2983 <li> |
|
2984 <p><a class='existingWikiWord' href='/nlab/show/fundamental+group'>funda… |
|
2985 |
|
2986 <ul> |
|
2987 <li><a class='existingWikiWord' href='/nlab/show/fundamental+group+of+a+… |
|
2988 </ul> |
|
2989 </li> |
|
2990 |
|
2991 <li> |
|
2992 <p><a class='existingWikiWord' href='/nlab/show/Brown-Grossman+homotopy+… |
|
2993 </li> |
|
2994 |
|
2995 <li> |
|
2996 <p><a class='existingWikiWord' href='/nlab/show/categorical+homotopy+gro… |
|
2997 </li> |
|
2998 |
|
2999 <li> |
|
3000 <p><a class='existingWikiWord' href='/nlab/show/geometric+homotopy+group… |
|
3001 </li> |
|
3002 </ul> |
|
3003 </li> |
|
3004 |
|
3005 <li> |
|
3006 <p><a class='existingWikiWord' href='/nlab/show/fundamental+infinity-gro… |
|
3007 |
|
3008 <ul> |
|
3009 <li> |
|
3010 <p><a class='existingWikiWord' href='/nlab/show/fundamental+groupoid'>fu… |
|
3011 |
|
3012 <ul> |
|
3013 <li><a class='existingWikiWord' href='/nlab/show/path+groupoid'>path gro… |
|
3014 </ul> |
|
3015 </li> |
|
3016 |
|
3017 <li> |
|
3018 <p><a class='existingWikiWord' href='/nlab/show/fundamental+infinity-gro… |
|
3019 </li> |
|
3020 |
|
3021 <li> |
|
3022 <p><a class='existingWikiWord' href='/nlab/show/fundamental+infinity-gro… |
|
3023 </li> |
|
3024 </ul> |
|
3025 </li> |
|
3026 |
|
3027 <li> |
|
3028 <p><a class='existingWikiWord' href='/nlab/show/fundamental+%28infinity%… |
|
3029 |
|
3030 <ul> |
|
3031 <li><a class='existingWikiWord' href='/nlab/show/fundamental+category'>f… |
|
3032 </ul> |
|
3033 </li> |
|
3034 </ul> |
|
3035 |
|
3036 <p><strong>Basic facts</strong></p> |
|
3037 |
|
3038 <ul> |
|
3039 <li><a class='existingWikiWord' href='/nlab/show/fundamental+group+of+th… |
|
3040 </ul> |
|
3041 |
|
3042 <p><strong>Theorems</strong></p> |
|
3043 |
|
3044 <ul> |
|
3045 <li> |
|
3046 <p><a class='existingWikiWord' href='/nlab/show/fundamental+theorem+of+c… |
|
3047 </li> |
|
3048 |
|
3049 <li> |
|
3050 <p><a class='existingWikiWord' href='/nlab/show/Freudenthal+suspension+t… |
|
3051 </li> |
|
3052 |
|
3053 <li> |
|
3054 <p><a class='existingWikiWord' href='/nlab/show/Blakers-Massey+theorem'>… |
|
3055 </li> |
|
3056 |
|
3057 <li> |
|
3058 <p><a class='existingWikiWord' href='/nlab/show/higher+homotopy+van+Kamp… |
|
3059 </li> |
|
3060 |
|
3061 <li> |
|
3062 <p><a class='existingWikiWord' href='/nlab/show/nerve+theorem'>nerve the… |
|
3063 </li> |
|
3064 |
|
3065 <li> |
|
3066 <p><a class='existingWikiWord' href='/nlab/show/Whitehead+theorem'>White… |
|
3067 </li> |
|
3068 |
|
3069 <li> |
|
3070 <p><a class='existingWikiWord' href='/nlab/show/Hurewicz+theorem'>Hurewi… |
|
3071 </li> |
|
3072 |
|
3073 <li> |
|
3074 <p><a class='existingWikiWord' href='/nlab/show/Galois+theory'>Galois th… |
|
3075 </li> |
|
3076 |
|
3077 <li> |
|
3078 <p><a class='existingWikiWord' href='/nlab/show/homotopy+hypothesis'>hom… |
|
3079 </li> |
|
3080 </ul> |
|
3081 </div> |
|
3082 |
|
3083 <h4 id='category_theory'>Category theory</h4> |
|
3084 |
|
3085 <div class='hide'> |
|
3086 <p><strong><a class='existingWikiWord' href='/nlab/show/category+theory'… |
|
3087 |
|
3088 <h2 id='concepts'>Concepts</h2> |
|
3089 |
|
3090 <ul> |
|
3091 <li> |
|
3092 <p><a class='existingWikiWord' href='/nlab/show/category'>category</a></… |
|
3093 </li> |
|
3094 |
|
3095 <li> |
|
3096 <p><a class='existingWikiWord' href='/nlab/show/functor'>functor</a></p> |
|
3097 </li> |
|
3098 |
|
3099 <li> |
|
3100 <p><a class='existingWikiWord' href='/nlab/show/natural+transformation'>… |
|
3101 </li> |
|
3102 |
|
3103 <li> |
|
3104 <p><a class='existingWikiWord' href='/nlab/show/Cat'>Cat</a></p> |
|
3105 </li> |
|
3106 </ul> |
|
3107 |
|
3108 <h2 id='universal_constructions'>Universal constructions</h2> |
|
3109 |
|
3110 <ul> |
|
3111 <li> |
|
3112 <p><a class='existingWikiWord' href='/nlab/show/universal+construction'>… |
|
3113 |
|
3114 <ul> |
|
3115 <li> |
|
3116 <p><a class='existingWikiWord' href='/nlab/show/representable+functor'>r… |
|
3117 </li> |
|
3118 |
|
3119 <li> |
|
3120 <p><a class='existingWikiWord' href='/nlab/show/adjoint+functor'>adjoint… |
|
3121 </li> |
|
3122 |
|
3123 <li> |
|
3124 <p><a class='existingWikiWord' href='/nlab/show/limit'>limit</a>/<a clas… |
|
3125 </li> |
|
3126 |
|
3127 <li> |
|
3128 <p><a class='existingWikiWord' href='/nlab/show/weighted+limit'>weighted… |
|
3129 </li> |
|
3130 |
|
3131 <li> |
|
3132 <p><a class='existingWikiWord' href='/nlab/show/end'>end</a>/<a class='e… |
|
3133 </li> |
|
3134 |
|
3135 <li> |
|
3136 <p><a class='existingWikiWord' href='/nlab/show/Kan+extension'>Kan exten… |
|
3137 </li> |
|
3138 </ul> |
|
3139 </li> |
|
3140 </ul> |
|
3141 |
|
3142 <h2 id='theorems'>Theorems</h2> |
|
3143 |
|
3144 <ul> |
|
3145 <li> |
|
3146 <p><a class='existingWikiWord' href='/nlab/show/Yoneda+lemma'>Yoneda lem… |
|
3147 </li> |
|
3148 |
|
3149 <li> |
|
3150 <p><a class='existingWikiWord' href='/nlab/show/Isbell+duality'>Isbell d… |
|
3151 </li> |
|
3152 |
|
3153 <li> |
|
3154 <p><a class='existingWikiWord' href='/nlab/show/Grothendieck+constructio… |
|
3155 </li> |
|
3156 |
|
3157 <li> |
|
3158 <p><a class='existingWikiWord' href='/nlab/show/adjoint+functor+theorem'… |
|
3159 </li> |
|
3160 |
|
3161 <li> |
|
3162 <p><a class='existingWikiWord' href='/nlab/show/monadicity+theorem'>mona… |
|
3163 </li> |
|
3164 |
|
3165 <li> |
|
3166 <p><a class='existingWikiWord' href='/nlab/show/adjoint+lifting+theorem'… |
|
3167 </li> |
|
3168 |
|
3169 <li> |
|
3170 <p><a class='existingWikiWord' href='/nlab/show/Tannaka+duality'>Tannaka… |
|
3171 </li> |
|
3172 |
|
3173 <li> |
|
3174 <p><a class='existingWikiWord' href='/nlab/show/Gabriel-Ulmer+duality'>G… |
|
3175 </li> |
|
3176 |
|
3177 <li> |
|
3178 <p><a class='existingWikiWord' href='/nlab/show/small+object+argument'>s… |
|
3179 </li> |
|
3180 |
|
3181 <li> |
|
3182 <p><a class='existingWikiWord' href='/nlab/show/Freyd-Mitchell+embedding… |
|
3183 </li> |
|
3184 |
|
3185 <li> |
|
3186 <p><a class='existingWikiWord' href='/nlab/show/relation+between+type+th… |
|
3187 </li> |
|
3188 </ul> |
|
3189 |
|
3190 <h2 id='extensions'>Extensions</h2> |
|
3191 |
|
3192 <ul> |
|
3193 <li> |
|
3194 <p><a class='existingWikiWord' href='/nlab/show/sheaf+and+topos+theory'>… |
|
3195 </li> |
|
3196 |
|
3197 <li> |
|
3198 <p><a class='existingWikiWord' href='/nlab/show/enriched+category+theory… |
|
3199 </li> |
|
3200 |
|
3201 <li> |
|
3202 <p><a class='existingWikiWord' href='/nlab/show/higher+category+theory'>… |
|
3203 </li> |
|
3204 </ul> |
|
3205 |
|
3206 <h2 id='applications'>Applications</h2> |
|
3207 |
|
3208 <ul> |
|
3209 <li><a class='existingWikiWord' href='/nlab/show/applications+of+%28high… |
|
3210 </ul> |
|
3211 <div> |
|
3212 <p> |
|
3213 <a href='/nlab/edit/category+theory+-+contents'>Edit this sidebar</a> |
|
3214 </p> |
|
3215 </div></div> |
|
3216 </div> |
|
3217 </div> |
|
3218 |
|
3219 <h1 id='contents'>Contents</h1> |
|
3220 <div class='maruku_toc'><ul><li><a href='#idea'>Idea</a></li><li><a href… |
|
3221 <h2 id='idea'>Idea</h2> |
|
3222 |
|
3223 <p>What is called <em>geometric realization of categories</em> is a <a c… |
|
3224 |
|
3225 <p>By the <a class='existingWikiWord' href='/nlab/show/homotopy+hypothes… |
|
3226 |
|
3227 <p>Therefore an equivalent perspective on geometric realization of categ… |
|
3228 |
|
3229 <p>Geometric realization of categories has various good properties:</p> |
|
3230 |
|
3231 <p>It sends <a class='existingWikiWord' href='/nlab/show/equivalence+of+… |
|
3232 |
|
3233 <p>The existence of the <a class='existingWikiWord' href='/nlab/show/Tho… |
|
3234 |
|
3235 <h2 id='definition'>Definition</h2> |
|
3236 |
|
3237 <p>Write</p> |
|
3238 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
3239 N \colon Cat \to sSet |
|
3240 |
|
3241 </annotation></semantics></math></div> |
|
3242 <p>for the <a href='nerve#NerveOfACategory'>nerve functor</a> from <a cl… |
|
3243 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
3244 {\vert - \vert} : sSet \to Top |
|
3245 |
|
3246 </annotation></semantics></math></div> |
|
3247 <p>for the <a class='existingWikiWord' href='/nlab/show/geometric+realiz… |
|
3248 |
|
3249 <p>The <em>geometric realization of categories</em> is the <a class='exi… |
|
3250 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
3251 {\vert - \vert} \coloneqq {\vert N(-)\vert} \;\colon\; Cat \to Top |
|
3252 |
|
3253 </annotation></semantics></math></div> |
|
3254 <h2 id='properties'>Properties</h2> |
|
3255 |
|
3256 <h3 id='ThomasonModelStructure'>Thomason model structure</h3> |
|
3257 |
|
3258 <p>There is a <a class='existingWikiWord' href='/nlab/show/model+categor… |
|
3259 |
|
3260 <p>The existence of the Thomas model structure implies that every <a cla… |
|
3261 |
|
3262 <div class='num_defn' id='PosetOfSimplicesInNerveOfCategory'> |
|
3263 <h6 id='definition_2'>Definition</h6> |
|
3264 |
|
3265 <p>For <math class='maruku-mathml' display='inline' id='mathml_ea68c383f… |
|
3266 </div> |
|
3267 |
|
3268 <div class='num_prop'> |
|
3269 <h6 id='proposition'>Proposition</h6> |
|
3270 |
|
3271 <p>For every category <math class='maruku-mathml' display='inline' id='m… |
|
3272 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
3273 {\vert N(\nabla \mathcal{C}) \vert} \simeq_{wh} {\vert \mathcal{C} \ve… |
|
3274 \,. |
|
3275 |
|
3276 </annotation></semantics></math></div></div> |
|
3277 |
|
3278 <h3 id='recognizing_weak_equivalences_quillens_theorem_a_and_b'>Recogniz… |
|
3279 |
|
3280 <p>Let <math class='maruku-mathml' display='inline' id='mathml_ea68c383f… |
|
3281 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
3282 F \;\colon\; \mathcal{C} \longrightarrow \mathcal{D} |
|
3283 |
|
3284 </annotation></semantics></math></div> |
|
3285 <p>be a <a class='existingWikiWord' href='/nlab/show/functor'>functor</a… |
|
3286 |
|
3287 <div class='num_theorem' id='QuillenTheoremA'> |
|
3288 <h6 id='theorem'>Theorem</h6> |
|
3289 |
|
3290 <p><strong>(<a href='#Quillen72'>Quillen 72</a>, theorem A)</strong></p> |
|
3291 |
|
3292 <p>If for all <a class='existingWikiWord' href='/nlab/show/object'>objec… |
|
3293 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
3294 {\vert N(F) \vert} |
|
3295 \;\colon\; |
|
3296 {\vert N(\mathcal{C}) \vert} |
|
3297 \longrightarrow |
|
3298 {\vert N(\mathcal{D}) \vert} |
|
3299 |
|
3300 </annotation></semantics></math></div> |
|
3301 <p>is a <a class='existingWikiWord' href='/nlab/show/weak+homotopy+equiv… |
|
3302 </div> |
|
3303 |
|
3304 <div class='num_theorem' id='QuillenTheoremB'> |
|
3305 <h6 id='theorem_2'>Theorem</h6> |
|
3306 |
|
3307 <p><strong>(<a href='#Quillen72'>Quillen 72</a> theorem B)</strong></p> |
|
3308 |
|
3309 <p>If for all <a class='existingWikiWord' href='/nlab/show/object'>objec… |
|
3310 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
3311 X |
|
3312 \longrightarrow |
|
3313 {\vert N(\mathcal{C}) \vert} |
|
3314 \overset{\vert N(F) \vert }{\longrightarrow} |
|
3315 {\vert N(\mathcal{D}) \vert} |
|
3316 \,. |
|
3317 |
|
3318 </annotation></semantics></math></div></div> |
|
3319 |
|
3320 <p>As a consequence:</p> |
|
3321 |
|
3322 <div class='num_prop'> |
|
3323 <h6 id='proposition_2'>Proposition</h6> |
|
3324 |
|
3325 <p><strong>(<a href='#McCord66'>McCord 66, theorem 6</a>, <a href='#Quil… |
|
3326 |
|
3327 <p>Let <math class='maruku-mathml' display='inline' id='mathml_ea68c383f… |
|
3328 |
|
3329 <p>If for each element/object <math class='maruku-mathml' display='inlin… |
|
3330 </div> |
|
3331 |
|
3332 <p>An alternative proof is given in (<a href='#Barmak10'>Barmak 10</a>).… |
|
3333 |
|
3334 <h3 id='natural_transformations_and_homotopies'>Natural transformations … |
|
3335 |
|
3336 <div class='num_prop' id='NaturalTrafoMapsToHomotopy'> |
|
3337 <h6 id='proposition_3'>Proposition</h6> |
|
3338 |
|
3339 <p>A <a class='existingWikiWord' href='/nlab/show/natural+transformation… |
|
3340 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
3341 {|N(\eta)|} \colon {\vert N(F)\vert} \longrightarrow {\vert N(G) \vert} |
|
3342 \,. |
|
3343 |
|
3344 </annotation></semantics></math></div></div> |
|
3345 |
|
3346 <div class='proof'> |
|
3347 <h6 id='proof'>Proof</h6> |
|
3348 |
|
3349 <p>The natural transformation is equivalently a functor of the form</p> |
|
3350 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
3351 \eta \;\colon\; \mathcal{C} \times \{0 \to 1\} \to \mathcal{D} |
|
3352 |
|
3353 </annotation></semantics></math></div> |
|
3354 <p>out of the <a class='existingWikiWord' href='/nlab/show/product+categ… |
|
3355 |
|
3356 <p>Since <a class='existingWikiWord' href='/nlab/show/geometric+realizat… |
|
3357 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
3358 {\vert N( \mathcal{C} \times \{0,1\} ) \vert} |
|
3359 \;\simeq_{iso}\; |
|
3360 {\vert N(\mathcal{C}) \vert} \times {\vert N(\{0 \to 1\}) \vert} |
|
3361 |
|
3362 </annotation></semantics></math></div> |
|
3363 <p>But this is a <a class='existingWikiWord' href='/nlab/show/cylinder+o… |
|
3364 </div> |
|
3365 |
|
3366 <div class='num_cor' id='RealizationOfEquivalenceIsHomotopyEquivalence'> |
|
3367 <h6 id='corollary'>Corollary</h6> |
|
3368 |
|
3369 <p>An <a class='existingWikiWord' href='/nlab/show/equivalence+of+catego… |
|
3370 </div> |
|
3371 |
|
3372 <div class='num_remark'> |
|
3373 <h6 id='remark'>Remark</h6> |
|
3374 |
|
3375 <p>The statement still remains true for a pair of <a class='existingWiki… |
|
3376 </div> |
|
3377 |
|
3378 <div class='num_remark'> |
|
3379 <h6 id='remark_2'>Remark</h6> |
|
3380 |
|
3381 <p>Notice that the converse is far from true: Very different categories … |
|
3382 </div> |
|
3383 |
|
3384 <div class='num_cor' id='RealizationWithTerminalObjectIsContractible'> |
|
3385 <h6 id='corollary_2'>Corollary</h6> |
|
3386 |
|
3387 <p>If a <a class='existingWikiWord' href='/nlab/show/category'>category<… |
|
3388 </div> |
|
3389 |
|
3390 <div class='proof'> |
|
3391 <h6 id='proof_2'>Proof</h6> |
|
3392 |
|
3393 <p>Assume the case of a terminal object, the other case works <a class='… |
|
3394 |
|
3395 <p>Then we have an equality of functors</p> |
|
3396 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
3397 Id_* = (* \stackrel{\bottom}{\to} C \to *) |
|
3398 \,, |
|
3399 |
|
3400 </annotation></semantics></math></div> |
|
3401 <p>where the first functor on the right picks the terminal object, and w… |
|
3402 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
3403 Id_C \Rightarrow (C \to * \stackrel{\bottom}{\to} C) |
|
3404 |
|
3405 </annotation></semantics></math></div> |
|
3406 <p>whose components are the unique morphisms into the terminal object.</… |
|
3407 |
|
3408 <p>By prop. <a class='maruku-ref' href='#NaturalTrafoMapsToHomotopy'>3</… |
|
3409 </div> |
|
3410 |
|
3411 <h3 id='behaviour_under_homotopy_colimits'>Behaviour under homotopy coli… |
|
3412 |
|
3413 <div class='num_prop'> |
|
3414 <h6 id='proposition_4'>Proposition</h6> |
|
3415 |
|
3416 <p>For <math class='maruku-mathml' display='inline' id='mathml_ea68c383f… |
|
3417 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
3418 {\vert N(F(-))\vert} |
|
3419 \;\colon\; |
|
3420 \mathcal{D} |
|
3421 \overset{F}{\longrightarrow} |
|
3422 Cat |
|
3423 \stackrel{\vert N(-) \vert}{\to} |
|
3424 Top |
|
3425 |
|
3426 </annotation></semantics></math></div> |
|
3427 <p>be its postcomposition with geometric realization of categories</p> |
|
3428 |
|
3429 <p>Then we have a <a class='existingWikiWord' href='/nlab/show/weak+homo… |
|
3430 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
3431 {\left\vert N\left(\int F \right) \right\vert} |
|
3432 \simeq |
|
3433 hocolim {\vert F(N(-)) \vert} |
|
3434 |
|
3435 </annotation></semantics></math></div> |
|
3436 <p>exhibiting the <a class='existingWikiWord' href='/nlab/show/homotopy+… |
|
3437 </div> |
|
3438 |
|
3439 <p>This is due to (<a href='#Thomason79'>Thomason 79</a>).</p> |
|
3440 |
|
3441 <h2 id='related_concepts'>Related concepts</h2> |
|
3442 |
|
3443 <ul> |
|
3444 <li> |
|
3445 <p><a class='existingWikiWord' href='/nlab/show/geometric+realization'>g… |
|
3446 |
|
3447 <ul> |
|
3448 <li><strong>of categories</strong>, <a class='existingWikiWord' href='/n… |
|
3449 </ul> |
|
3450 </li> |
|
3451 </ul> |
|
3452 |
|
3453 <h2 id='references'>References</h2> |
|
3454 |
|
3455 <h3 id='general'>General</h3> |
|
3456 |
|
3457 <p>For general references see also <em><a class='existingWikiWord' href=… |
|
3458 |
|
3459 <h3 id='quillens_theorems_a_and_b'>Quillen’s theorems A and B</h3> |
|
3460 |
|
3461 <p>The original articles are</p> |
|
3462 |
|
3463 <ul> |
|
3464 <li id='McCord66'> |
|
3465 <p><a class='existingWikiWord' href='/nlab/show/Michael+C.+McCord'>Micha… |
|
3466 </li> |
|
3467 |
|
3468 <li id='Quillen72'> |
|
3469 <p><a class='existingWikiWord' href='/nlab/show/Daniel+Quillen'>Daniel Q… |
|
3470 </li> |
|
3471 |
|
3472 <li id='Quillen78'> |
|
3473 <p><a class='existingWikiWord' href='/nlab/show/Daniel+Quillen'>Daniel Q… |
|
3474 </li> |
|
3475 </ul> |
|
3476 |
|
3477 <p>The geometric realization of <a class='existingWikiWord' href='/nlab/… |
|
3478 |
|
3479 <ul> |
|
3480 <li id='Thomason79'><a class='existingWikiWord' href='/nlab/show/Robert+… |
|
3481 </ul> |
|
3482 |
|
3483 <p>Review is in</p> |
|
3484 |
|
3485 <ul> |
|
3486 <li id='Barmak10'><a class='existingWikiWord' href='/nlab/show/Jonathan+… |
|
3487 </ul> |
|
3488 |
|
3489 <p>Further development includes</p> |
|
3490 |
|
3491 <ul> |
|
3492 <li> |
|
3493 <p><a class='existingWikiWord' href='/nlab/show/Clark+Barwick'>Clark Bar… |
|
3494 </li> |
|
3495 |
|
3496 <li> |
|
3497 <p><a class='existingWikiWord' href='/nlab/show/David+Michael+Roberts'>D… |
|
3498 </li> |
|
3499 </ul> |
|
3500 |
|
3501 <p> |
|
3502 </p> |
|
3503 |
|
3504 <p> |
|
3505 |
|
3506 |
|
3507 |
|
3508 |
|
3509 |
|
3510 |
|
3511 |
|
3512 </p> </div> |
|
3513 </content> |
|
3514 </entry> |
|
3515 <entry> |
|
3516 <title type="html">Borel model structure</title> |
|
3517 <link rel="alternate" type="application/xhtml+xml" href="https://nca… |
|
3518 <updated>2021-07-01T16:26:36Z</updated> |
|
3519 <published>2014-04-15T05:41:15Z</published> |
|
3520 <id>tag:ncatlab.org,2014-04-15:nLab,Borel+model+structure</id> |
|
3521 <author> |
|
3522 <name>Urs Schreiber</name> |
|
3523 </author> |
|
3524 <content type="xhtml" xml:base="https://ncatlab.org/nlab/show/Borel+… |
|
3525 <div xmlns="http://www.w3.org/1999/xhtml"> |
|
3526 <div class='rightHandSide'> |
|
3527 <div class='toc clickDown' tabindex='0'> |
|
3528 <h3 id='context'>Context</h3> |
|
3529 |
|
3530 <h4 id='model_category_theory'>Model category theory</h4> |
|
3531 |
|
3532 <div class='hide'> |
|
3533 <p><strong><a class='existingWikiWord' href='/nlab/show/model+category'>… |
|
3534 |
|
3535 <h2 id='definitions'>Definitions</h2> |
|
3536 |
|
3537 <ul> |
|
3538 <li> |
|
3539 <p><a class='existingWikiWord' href='/nlab/show/category+with+weak+equiv… |
|
3540 </li> |
|
3541 |
|
3542 <li> |
|
3543 <p><a class='existingWikiWord' href='/nlab/show/weak+factorization+syste… |
|
3544 </li> |
|
3545 |
|
3546 <li> |
|
3547 <p><a class='existingWikiWord' href='/nlab/show/homotopy+%28as+an+operat… |
|
3548 |
|
3549 <ul> |
|
3550 <li><a class='existingWikiWord' href='/nlab/show/homotopy+category'>homo… |
|
3551 </ul> |
|
3552 </li> |
|
3553 |
|
3554 <li> |
|
3555 <p><a class='existingWikiWord' href='/nlab/show/small+object+argument'>s… |
|
3556 </li> |
|
3557 |
|
3558 <li> |
|
3559 <p><a class='existingWikiWord' href='/nlab/show/resolution'>resolution</… |
|
3560 </li> |
|
3561 </ul> |
|
3562 |
|
3563 <h2 id='morphisms'>Morphisms</h2> |
|
3564 |
|
3565 <ul> |
|
3566 <li> |
|
3567 <p><a class='existingWikiWord' href='/nlab/show/Quillen+adjunction'>Quil… |
|
3568 |
|
3569 <ul> |
|
3570 <li> |
|
3571 <p><a class='existingWikiWord' href='/nlab/show/Quillen+equivalence'>Qui… |
|
3572 </li> |
|
3573 |
|
3574 <li> |
|
3575 <p><a class='existingWikiWord' href='/nlab/show/Quillen+bifunctor'>Quill… |
|
3576 </li> |
|
3577 |
|
3578 <li> |
|
3579 <p><a class='existingWikiWord' href='/nlab/show/derived+functor'>derived… |
|
3580 </li> |
|
3581 </ul> |
|
3582 </li> |
|
3583 </ul> |
|
3584 |
|
3585 <h2 id='universal_constructions'>Universal constructions</h2> |
|
3586 |
|
3587 <ul> |
|
3588 <li> |
|
3589 <p><a class='existingWikiWord' href='/nlab/show/homotopy+Kan+extension'>… |
|
3590 </li> |
|
3591 |
|
3592 <li> |
|
3593 <p><a class='existingWikiWord' href='/nlab/show/homotopy+limit'>homotopy… |
|
3594 </li> |
|
3595 |
|
3596 <li> |
|
3597 <p><a class='existingWikiWord' href='/nlab/show/Bousfield-Kan+map'>Bousf… |
|
3598 </li> |
|
3599 </ul> |
|
3600 |
|
3601 <h2 id='refinements'>Refinements</h2> |
|
3602 |
|
3603 <ul> |
|
3604 <li> |
|
3605 <p><a class='existingWikiWord' href='/nlab/show/monoidal+model+category'… |
|
3606 |
|
3607 <ul> |
|
3608 <li><a class='existingWikiWord' href='/nlab/show/monoidal+Quillen+adjunc… |
|
3609 </ul> |
|
3610 </li> |
|
3611 |
|
3612 <li> |
|
3613 <p><a class='existingWikiWord' href='/nlab/show/enriched+model+category'… |
|
3614 |
|
3615 <ul> |
|
3616 <li><a class='existingWikiWord' href='/nlab/show/enriched+Quillen+adjunc… |
|
3617 </ul> |
|
3618 </li> |
|
3619 |
|
3620 <li> |
|
3621 <p><a class='existingWikiWord' href='/nlab/show/simplicial+model+categor… |
|
3622 |
|
3623 <ul> |
|
3624 <li><a class='existingWikiWord' href='/nlab/show/simplicial+Quillen+adju… |
|
3625 </ul> |
|
3626 </li> |
|
3627 |
|
3628 <li> |
|
3629 <p><a class='existingWikiWord' href='/nlab/show/cofibrantly+generated+mo… |
|
3630 |
|
3631 <ul> |
|
3632 <li> |
|
3633 <p><a class='existingWikiWord' href='/nlab/show/combinatorial+model+cate… |
|
3634 </li> |
|
3635 |
|
3636 <li> |
|
3637 <p><a class='existingWikiWord' href='/nlab/show/cellular+model+category'… |
|
3638 </li> |
|
3639 </ul> |
|
3640 </li> |
|
3641 |
|
3642 <li> |
|
3643 <p><a class='existingWikiWord' href='/nlab/show/algebraic+model+category… |
|
3644 </li> |
|
3645 |
|
3646 <li> |
|
3647 <p><a class='existingWikiWord' href='/nlab/show/compactly+generated+mode… |
|
3648 </li> |
|
3649 |
|
3650 <li> |
|
3651 <p><a class='existingWikiWord' href='/nlab/show/proper+model+category'>p… |
|
3652 </li> |
|
3653 |
|
3654 <li> |
|
3655 <p><a class='existingWikiWord' href='/nlab/show/cartesian+model+category… |
|
3656 </li> |
|
3657 |
|
3658 <li> |
|
3659 <p><a class='existingWikiWord' href='/nlab/show/stable+model+category'>s… |
|
3660 </li> |
|
3661 </ul> |
|
3662 |
|
3663 <h2 id='producing_new_model_structures'>Producing new model structures</… |
|
3664 |
|
3665 <ul> |
|
3666 <li> |
|
3667 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+funct… |
|
3668 |
|
3669 <ul> |
|
3670 <li><a class='existingWikiWord' href='/nlab/show/Reedy+model+structure'>… |
|
3671 </ul> |
|
3672 </li> |
|
3673 |
|
3674 <li> |
|
3675 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+an+ov… |
|
3676 </li> |
|
3677 |
|
3678 <li> |
|
3679 <p><a class='existingWikiWord' href='/nlab/show/Bousfield+localization+o… |
|
3680 </li> |
|
3681 |
|
3682 <li> |
|
3683 <p><a class='existingWikiWord' href='/nlab/show/transferred+model+struct… |
|
3684 |
|
3685 <ul> |
|
3686 <li><a class='existingWikiWord' href='/nlab/show/model+structure+on+alge… |
|
3687 </ul> |
|
3688 </li> |
|
3689 |
|
3690 <li> |
|
3691 <p><a class='existingWikiWord' href='/nlab/show/Grothendieck+constructio… |
|
3692 </li> |
|
3693 </ul> |
|
3694 |
|
3695 <h2 id='presentation_of_categories'>Presentation of <math class='maruku-… |
|
3696 |
|
3697 <ul> |
|
3698 <li> |
|
3699 <p><a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-categ… |
|
3700 </li> |
|
3701 |
|
3702 <li> |
|
3703 <p><a class='existingWikiWord' href='/nlab/show/simplicial+localization'… |
|
3704 </li> |
|
3705 |
|
3706 <li> |
|
3707 <p><a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-categ… |
|
3708 </li> |
|
3709 |
|
3710 <li> |
|
3711 <p><a class='existingWikiWord' href='/nlab/show/locally+presentable+%28i… |
|
3712 </li> |
|
3713 </ul> |
|
3714 |
|
3715 <h2 id='model_structures'>Model structures</h2> |
|
3716 |
|
3717 <ul> |
|
3718 <li><a class='existingWikiWord' href='/nlab/show/Cisinski+model+structur… |
|
3719 </ul> |
|
3720 |
|
3721 <h3 id='for_groupoids'>for <math class='maruku-mathml' display='inline' … |
|
3722 |
|
3723 <p><a class='existingWikiWord' href='/nlab/show/model+structure+for+infi… |
|
3724 |
|
3725 <ul> |
|
3726 <li> |
|
3727 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+topol… |
|
3728 |
|
3729 <ul> |
|
3730 <li> |
|
3731 <p><a class='existingWikiWord' href='/nlab/show/classical+model+structur… |
|
3732 </li> |
|
3733 |
|
3734 <li> |
|
3735 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+Delta… |
|
3736 </li> |
|
3737 |
|
3738 <li> |
|
3739 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+diffe… |
|
3740 </li> |
|
3741 |
|
3742 <li> |
|
3743 <p><a class='existingWikiWord' href='/nlab/show/Str%C3%B8m+model+structu… |
|
3744 </li> |
|
3745 </ul> |
|
3746 </li> |
|
3747 |
|
3748 <li> |
|
3749 <p><a class='existingWikiWord' href='/nlab/show/Thomason+model+structure… |
|
3750 </li> |
|
3751 |
|
3752 <li> |
|
3753 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+presh… |
|
3754 </li> |
|
3755 |
|
3756 <li> |
|
3757 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+simpl… |
|
3758 |
|
3759 <ul> |
|
3760 <li> |
|
3761 <p><a class='existingWikiWord' href='/nlab/show/classical+model+structur… |
|
3762 </li> |
|
3763 |
|
3764 <li> |
|
3765 <p><a class='existingWikiWord' href='/nlab/show/constructive+model+struc… |
|
3766 </li> |
|
3767 |
|
3768 <li> |
|
3769 <p><a class='existingWikiWord' href='/nlab/show/model+structure+for+left… |
|
3770 </li> |
|
3771 </ul> |
|
3772 </li> |
|
3773 |
|
3774 <li> |
|
3775 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+simpl… |
|
3776 </li> |
|
3777 |
|
3778 <li> |
|
3779 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+cubic… |
|
3780 </li> |
|
3781 |
|
3782 <li> |
|
3783 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+stric… |
|
3784 </li> |
|
3785 |
|
3786 <li> |
|
3787 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+chain… |
|
3788 |
|
3789 <p>related by the <a class='existingWikiWord' href='/nlab/show/Dold-Kan+… |
|
3790 </li> |
|
3791 |
|
3792 <li> |
|
3793 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+cosim… |
|
3794 </li> |
|
3795 </ul> |
|
3796 |
|
3797 <h3 id='for_rational_groupoids'>for rational <math class='maruku-mathml'… |
|
3798 |
|
3799 <ul> |
|
3800 <li> |
|
3801 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+dg-al… |
|
3802 </li> |
|
3803 |
|
3804 <li> |
|
3805 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+equiv… |
|
3806 |
|
3807 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+equiv… |
|
3808 </li> |
|
3809 </ul> |
|
3810 |
|
3811 <h3 id='for_groupoids_2'>for <math class='maruku-mathml' display='inline… |
|
3812 |
|
3813 <ul> |
|
3814 <li> |
|
3815 <p><a class='existingWikiWord' href='/nlab/show/model+structure+for+homo… |
|
3816 </li> |
|
3817 |
|
3818 <li> |
|
3819 <p><a class='existingWikiWord' href='/nlab/show/canonical+model+structur… |
|
3820 </li> |
|
3821 </ul> |
|
3822 |
|
3823 <h3 id='for_groups'>for <math class='maruku-mathml' display='inline' id=… |
|
3824 |
|
3825 <ul> |
|
3826 <li> |
|
3827 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+simpl… |
|
3828 </li> |
|
3829 |
|
3830 <li> |
|
3831 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+reduc… |
|
3832 </li> |
|
3833 </ul> |
|
3834 |
|
3835 <h3 id='for_algebras'>for <math class='maruku-mathml' display='inline' i… |
|
3836 |
|
3837 <h4 id='general'>general</h4> |
|
3838 |
|
3839 <ul> |
|
3840 <li> |
|
3841 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+monoi… |
|
3842 </li> |
|
3843 |
|
3844 <li> |
|
3845 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+simpl… |
|
3846 </li> |
|
3847 |
|
3848 <li> |
|
3849 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+algeb… |
|
3850 </li> |
|
3851 |
|
3852 <li> |
|
3853 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+algeb… |
|
3854 </li> |
|
3855 </ul> |
|
3856 |
|
3857 <h4 id='specific'>specific</h4> |
|
3858 |
|
3859 <ul> |
|
3860 <li> |
|
3861 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+dg-al… |
|
3862 </li> |
|
3863 |
|
3864 <li> |
|
3865 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+diffe… |
|
3866 </li> |
|
3867 |
|
3868 <li> |
|
3869 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+dg-al… |
|
3870 |
|
3871 <ul> |
|
3872 <li> |
|
3873 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+dg-al… |
|
3874 |
|
3875 <p>related by the <a class='existingWikiWord' href='/nlab/show/monoidal+… |
|
3876 </li> |
|
3877 |
|
3878 <li> |
|
3879 <p><a class='existingWikiWord' href='/nlab/show/model+structure+for+L-in… |
|
3880 </li> |
|
3881 </ul> |
|
3882 </li> |
|
3883 |
|
3884 <li> |
|
3885 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+dg-mo… |
|
3886 </li> |
|
3887 </ul> |
|
3888 |
|
3889 <h3 id='for_stablespectrum_objects'>for stable/spectrum objects</h3> |
|
3890 |
|
3891 <ul> |
|
3892 <li> |
|
3893 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+spect… |
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3894 </li> |
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3895 |
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3896 <li> |
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3897 <p><a class='existingWikiWord' href='/nlab/show/model+structure+for+ring… |
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3898 </li> |
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3899 |
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3900 <li> |
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3901 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+presh… |
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3902 </li> |
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3903 </ul> |
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3904 |
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3905 <h3 id='for_categories'>for <math class='maruku-mathml' display='inline'… |
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3906 |
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3907 <ul> |
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3908 <li> |
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3909 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+relat… |
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3910 </li> |
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3911 |
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3912 <li> |
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3913 <p><a class='existingWikiWord' href='/nlab/show/model+structure+for+quas… |
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3914 </li> |
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3915 |
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3916 <li> |
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3917 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+sSet-… |
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3918 </li> |
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3919 |
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3920 <li> |
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3921 <p><a class='existingWikiWord' href='/nlab/show/model+structure+for+comp… |
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3922 </li> |
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3923 |
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3924 <li> |
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3925 <p><a class='existingWikiWord' href='/nlab/show/model+structure+for+Cart… |
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3926 </li> |
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3927 </ul> |
|
3928 |
|
3929 <h3 id='for_stable_categories'>for stable <math class='maruku-mathml' di… |
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3930 |
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3931 <ul> |
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3932 <li><a class='existingWikiWord' href='/nlab/show/model+structure+on+dg-c… |
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3933 </ul> |
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3934 |
|
3935 <h3 id='for_operads'>for <math class='maruku-mathml' display='inline' id… |
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3936 |
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3937 <ul> |
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3938 <li> |
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3939 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+opera… |
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3940 |
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3941 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+algeb… |
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3942 </li> |
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3943 |
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3944 <li> |
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3945 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+dendr… |
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3946 </li> |
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3947 </ul> |
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3948 |
|
3949 <h3 id='for_categories_2'>for <math class='maruku-mathml' display='inlin… |
|
3950 |
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3951 <ul> |
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3952 <li> |
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3953 <p><a class='existingWikiWord' href='/nlab/show/Theta-space'>for (n,r)-c… |
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3954 </li> |
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3955 |
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3956 <li> |
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3957 <p><a class='existingWikiWord' href='/nlab/show/model+structure+for+weak… |
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3958 </li> |
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3959 |
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3960 <li> |
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3961 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+cellu… |
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3962 </li> |
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3963 |
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3964 <li> |
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3965 <p><a class='existingWikiWord' href='/nlab/show/canonical+model+structur… |
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3966 </li> |
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3967 |
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3968 <li> |
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3969 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+stric… |
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3970 </li> |
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3971 </ul> |
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3972 |
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3973 <h3 id='for_sheaves__stacks'>for <math class='maruku-mathml' display='in… |
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3974 |
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3975 <ul> |
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3976 <li> |
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3977 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+homot… |
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3978 |
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3979 <ul> |
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3980 <li> |
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3981 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+simpl… |
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3982 |
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3983 <p><a class='existingWikiWord' href='/nlab/show/global+model+structure+o… |
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3984 |
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3985 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+simpl… |
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3986 |
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3987 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+presh… |
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3988 |
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3989 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+sSet-… |
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3990 </li> |
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3991 </ul> |
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3992 </li> |
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3993 |
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3994 <li> |
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3995 <p><a class='existingWikiWord' href='/nlab/show/model+structure+for+%282… |
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3996 </li> |
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3997 </ul> |
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3998 <div> |
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3999 <p> |
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4000 <a href='/nlab/edit/model+category+theory+-+contents'>Edit this sideba… |
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4001 </p> |
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4002 </div></div> |
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4003 |
|
4004 <h4 id='group_theory'>Group Theory</h4> |
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4005 |
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4006 <div class='hide'> |
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4007 <p><strong><a class='existingWikiWord' href='/nlab/show/group+theory'>gr… |
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4008 |
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4009 <ul> |
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4010 <li><a class='existingWikiWord' href='/nlab/show/group'>group</a>, <a cl… |
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4011 |
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4012 <li><a class='existingWikiWord' href='/nlab/show/group+object'>group obj… |
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4013 |
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4014 <li><a class='existingWikiWord' href='/nlab/show/abelian+group'>abelian … |
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4015 |
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4016 <li><a class='existingWikiWord' href='/nlab/show/action'>group action</a… |
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4017 |
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4018 <li><a class='existingWikiWord' href='/nlab/show/representation'>represe… |
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4019 |
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4020 <li><a class='existingWikiWord' href='/nlab/show/progroup'>progroup</a><… |
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4021 |
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4022 <li><a class='existingWikiWord' href='/nlab/show/homogeneous+space'>homo… |
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4023 </ul> |
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4024 |
|
4025 <h3 id='classical_groups'>Classical groups</h3> |
|
4026 |
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4027 <ul> |
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4028 <li> |
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4029 <p><a class='existingWikiWord' href='/nlab/show/general+linear+group'>ge… |
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4030 </li> |
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4031 |
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4032 <li> |
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4033 <p><a class='existingWikiWord' href='/nlab/show/unitary+group'>unitary g… |
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4034 |
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4035 <ul> |
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4036 <li><a class='existingWikiWord' href='/nlab/show/special+unitary+group'>… |
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4037 </ul> |
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4038 </li> |
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4039 |
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4040 <li> |
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4041 <p><a class='existingWikiWord' href='/nlab/show/orthogonal+group'>orthog… |
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4042 |
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4043 <ul> |
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4044 <li><a class='existingWikiWord' href='/nlab/show/special+orthogonal+grou… |
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4045 </ul> |
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4046 </li> |
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4047 |
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4048 <li> |
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4049 <p><a class='existingWikiWord' href='/nlab/show/symplectic+group'>symple… |
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4050 </li> |
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4051 </ul> |
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4052 |
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4053 <h3 id='finite_groups'>Finite groups</h3> |
|
4054 |
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4055 <ul> |
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4056 <li> |
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4057 <p><a class='existingWikiWord' href='/nlab/show/finite+group'>finite gro… |
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4058 </li> |
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4059 |
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4060 <li> |
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4061 <p><a class='existingWikiWord' href='/nlab/show/symmetric+group'>symmetr… |
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4062 </li> |
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4063 |
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4064 <li> |
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4065 <p><a class='existingWikiWord' href='/nlab/show/classification+of+finite… |
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4066 </li> |
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4067 |
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4068 <li> |
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4069 <p><a class='existingWikiWord' href='/nlab/show/sporadic+finite+simple+g… |
|
4070 |
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4071 <ul> |
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4072 <li><a class='existingWikiWord' href='/nlab/show/Monster+group'>Monster … |
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4073 </ul> |
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4074 </li> |
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4075 </ul> |
|
4076 |
|
4077 <h3 id='group_schemes'>Group schemes</h3> |
|
4078 |
|
4079 <ul> |
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4080 <li><a class='existingWikiWord' href='/nlab/show/algebraic+group'>algebr… |
|
4081 |
|
4082 <li><a class='existingWikiWord' href='/nlab/show/abelian+variety'>abelia… |
|
4083 </ul> |
|
4084 |
|
4085 <h3 id='topological_groups'>Topological groups</h3> |
|
4086 |
|
4087 <ul> |
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4088 <li> |
|
4089 <p><a class='existingWikiWord' href='/nlab/show/topological+group'>topol… |
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4090 </li> |
|
4091 |
|
4092 <li> |
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4093 <p><a class='existingWikiWord' href='/nlab/show/compact+topological+grou… |
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4094 </li> |
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4095 |
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4096 <li> |
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4097 <p><a class='existingWikiWord' href='/nlab/show/maximal+compact+subgroup… |
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4098 </li> |
|
4099 |
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4100 <li> |
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4101 <p><a class='existingWikiWord' href='/nlab/show/string+group'>string gro… |
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4102 </li> |
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4103 </ul> |
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4104 |
|
4105 <h3 id='lie_groups'>Lie groups</h3> |
|
4106 |
|
4107 <ul> |
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4108 <li> |
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4109 <p><a class='existingWikiWord' href='/nlab/show/Lie+group'>Lie group</a>… |
|
4110 </li> |
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4111 |
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4112 <li> |
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4113 <p><a class='existingWikiWord' href='/nlab/show/compact+Lie+group'>compa… |
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4114 </li> |
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4115 |
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4116 <li> |
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4117 <p><a class='existingWikiWord' href='/nlab/show/Kac-Moody+group'>Kac-Moo… |
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4118 </li> |
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4119 </ul> |
|
4120 |
|
4121 <h3 id='superlie_groups'>Super-Lie groups</h3> |
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4122 |
|
4123 <ul> |
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4124 <li> |
|
4125 <p><a class='existingWikiWord' href='/nlab/show/supergroup'>super Lie gr… |
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4126 </li> |
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4127 |
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4128 <li> |
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4129 <p><a class='existingWikiWord' href='/nlab/show/super+Euclidean+group'>s… |
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4130 </li> |
|
4131 </ul> |
|
4132 |
|
4133 <h3 id='higher_groups'>Higher groups</h3> |
|
4134 |
|
4135 <ul> |
|
4136 <li> |
|
4137 <p><a class='existingWikiWord' href='/nlab/show/2-group'>2-group</a></p> |
|
4138 |
|
4139 <ul> |
|
4140 <li><a class='existingWikiWord' href='/nlab/show/crossed+module'>crossed… |
|
4141 </ul> |
|
4142 </li> |
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4143 |
|
4144 <li> |
|
4145 <p><a class='existingWikiWord' href='/nlab/show/n-group'>n-group</a></p> |
|
4146 </li> |
|
4147 |
|
4148 <li> |
|
4149 <p><a class='existingWikiWord' href='/nlab/show/infinity-group'>∞-grou… |
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4150 |
|
4151 <ul> |
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4152 <li> |
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4153 <p><a class='existingWikiWord' href='/nlab/show/simplicial+group'>simpli… |
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4154 </li> |
|
4155 |
|
4156 <li> |
|
4157 <p><a class='existingWikiWord' href='/nlab/show/crossed+complex'>crossed… |
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4158 </li> |
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4159 |
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4160 <li> |
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4161 <p><a class='existingWikiWord' href='/nlab/show/k-tuply+groupal+n-groupo… |
|
4162 </li> |
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4163 |
|
4164 <li> |
|
4165 <p><a class='existingWikiWord' href='/nlab/show/spectrum'>spectrum</a></… |
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4166 </li> |
|
4167 </ul> |
|
4168 </li> |
|
4169 |
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4170 <li> |
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4171 <p><a class='existingWikiWord' href='/nlab/show/circle+n-group'>circle n… |
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4172 </li> |
|
4173 </ul> |
|
4174 |
|
4175 <h3 id='cohomology_and_extensions'>Cohomology and Extensions</h3> |
|
4176 |
|
4177 <ul> |
|
4178 <li> |
|
4179 <p><a class='existingWikiWord' href='/nlab/show/group+cohomology'>group … |
|
4180 </li> |
|
4181 |
|
4182 <li> |
|
4183 <p><a class='existingWikiWord' href='/nlab/show/group+extension'>group e… |
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4184 </li> |
|
4185 |
|
4186 <li> |
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4187 <p><a class='existingWikiWord' href='/nlab/show/infinity-group+extension… |
|
4188 </li> |
|
4189 </ul> |
|
4190 |
|
4191 <h3 id='_related_concepts'>Related concepts</h3> |
|
4192 |
|
4193 <ul> |
|
4194 <li><a class='existingWikiWord' href='/nlab/show/quantum+group'>quantum … |
|
4195 </ul> |
|
4196 <div> |
|
4197 <p> |
|
4198 <a href='/nlab/edit/group+theory+-+contents'>Edit this sidebar</a> |
|
4199 </p> |
|
4200 </div></div> |
|
4201 </div> |
|
4202 </div> |
|
4203 |
|
4204 <h1 id='contents'>Contents</h1> |
|
4205 <div class='maruku_toc'><ul><li><a href='#idea'>Idea</a></li><li><a href… |
|
4206 <h2 id='idea'>Idea</h2> |
|
4207 |
|
4208 <p>Given a <a class='existingWikiWord' href='/nlab/show/simplicial+group… |
|
4209 |
|
4210 <p>In the context of <a class='existingWikiWord' href='/nlab/show/equiva… |
|
4211 |
|
4212 <h2 id='definition'>Definition</h2> |
|
4213 |
|
4214 <p>\begin{defn}\label{BorelModelStructure}</p> |
|
4215 |
|
4216 <p>For <math class='maruku-mathml' display='inline' id='mathml_6bd24f229… |
|
4217 |
|
4218 <ul> |
|
4219 <li> |
|
4220 <p><math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d9… |
|
4221 </li> |
|
4222 |
|
4223 <li> |
|
4224 <p><math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d9… |
|
4225 </li> |
|
4226 |
|
4227 <li> |
|
4228 <p><math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d9… |
|
4229 </li> |
|
4230 </ul> |
|
4231 |
|
4232 <p>This is the <math class='maruku-mathml' display='inline' id='mathml_6… |
|
4233 |
|
4234 <p>\end{defn}</p> |
|
4235 |
|
4236 <h2 id='properties'>Properties</h2> |
|
4237 |
|
4238 <h3 id='CofibrantReplacementAndHomotopyQuotientsFixedPoints'>Cofibrant r… |
|
4239 |
|
4240 <p>\begin{prop}\label{CofibrationsOfSimplicialActions} <strong>(cofibrat… |
|
4241 |
|
4242 <ol> |
|
4243 <li> |
|
4244 <p>the underlying morphism of <a class='existingWikiWord' href='/nlab/sh… |
|
4245 </li> |
|
4246 |
|
4247 <li> |
|
4248 <p>the <math class='maruku-mathml' display='inline' id='mathml_6bd24f229… |
|
4249 </li> |
|
4250 </ol> |
|
4251 |
|
4252 <p>\end{prop}</p> |
|
4253 |
|
4254 <p>This is (<a href='#DDK80'>DDK 80, Prop. 2.2. (ii)</a>, <a href='#Guil… |
|
4255 |
|
4256 <p>\begin{remark} In particular this means that an object is <a class='e… |
|
4257 |
|
4258 <p>Hence <a class='existingWikiWord' href='/nlab/show/fibrant+replacemen… |
|
4259 |
|
4260 <p>\begin{remark} It follows that for <math class='maruku-mathml' displa… |
|
4261 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
4262 R Hom_G(X,A) |
|
4263 |
|
4264 </annotation></semantics></math></div> |
|
4265 <p>models the Borel <math class='maruku-mathml' display='inline' id='mat… |
|
4266 |
|
4267 <p>In particular,if <math class='maruku-mathml' display='inline' id='mat… |
|
4268 |
|
4269 <ol> |
|
4270 <li> |
|
4271 <p>if the <math class='maruku-mathml' display='inline' id='mathml_6bd24f… |
|
4272 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
4273 R Hom_G(X,A) |
|
4274 \simeq |
|
4275 Hom_G(W G \times X , A) |
|
4276 \simeq |
|
4277 Hom(W G \times_G X, A) |
|
4278 |
|
4279 </annotation></semantics></math></div> |
|
4280 <p>is equivalently maps of <a class='existingWikiWord' href='/nlab/show/… |
|
4281 </li> |
|
4282 |
|
4283 <li> |
|
4284 <p>if <math class='maruku-mathml' display='inline' id='mathml_6bd24f2291… |
|
4285 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
4286 R Hom_G(X,A) |
|
4287 \simeq |
|
4288 Hom_G(W G, A) |
|
4289 \simeq |
|
4290 Hom(\overline{W} G , A) |
|
4291 \simeq |
|
4292 A^{h G} |
|
4293 |
|
4294 </annotation></semantics></math></div> |
|
4295 <p>is the <a class='existingWikiWord' href='/nlab/show/homotopy+fixed+po… |
|
4296 </li> |
|
4297 </ol> |
|
4298 |
|
4299 <p>\end{remark}</p> |
|
4300 |
|
4301 <h3 id='RelationToSliceOverSimplicialClassifyingSpace'>Relation to the s… |
|
4302 |
|
4303 <p>\begin{prop}\label{QuillenEquivalenceToSliceOverSimplicialClassifying… |
|
4304 <div class='maruku-equation' id='eq:QuillenAdjunctionWithSliceOverSimpli… |
|
4305 |
|
4306 G_\bullet Acts(sSet)_{proj} |
|
4307 \underoverset |
|
4308 {\underset{ \big((-) \times W G\big)/G }{\longrightarrow}} |
|
4309 {\overset{ (-) \times_{\overline{W}G} W G }{\longleftarrow}} |
|
4310 {\bot} |
|
4311 sSet_{/\overline{W}G} |
|
4312 |
|
4313 </annotation></semantics></math></div> |
|
4314 <p>which constitute a <a class='existingWikiWord' href='/nlab/show/simpl… |
|
4315 |
|
4316 <p>\end{prop}</p> |
|
4317 |
|
4318 <p>(<a href='#DDK80'>DDK 80, Prop. 2.3, Prop. 2.4</a>) Here:</p> |
|
4319 |
|
4320 <ul> |
|
4321 <li> |
|
4322 <p>the <a class='existingWikiWord' href='/nlab/show/right+adjoint'>right… |
|
4323 </li> |
|
4324 |
|
4325 <li> |
|
4326 <p>the <a class='existingWikiWord' href='/nlab/show/left+adjoint'>left a… |
|
4327 </li> |
|
4328 </ul> |
|
4329 |
|
4330 <p>In fact, these are <a class='existingWikiWord' href='/nlab/show/SimpS… |
|
4331 |
|
4332 <p>This kind of relation is discussed in more detail at <em><a class='ex… |
|
4333 |
|
4334 <p>\begin{remark}\label{sSetEnrichmentOfAdjunctionToSliceOverSimpClassSp… |
|
4335 |
|
4336 <p>The remaining observation that we have a <a class='existingWikiWord' … |
|
4337 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
4338 \big[ |
|
4339 X \times_{\overline{W}G} W G, |
|
4340 \, |
|
4341 V |
|
4342 \big] |
|
4343 \;\simeq\; |
|
4344 \big[ |
|
4345 X, |
|
4346 \, |
|
4347 (V \times W G)/G |
|
4348 \big] |
|
4349 |
|
4350 </annotation></semantics></math></div> |
|
4351 <p>hence</p> |
|
4352 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
4353 Hom |
|
4354 \Big( |
|
4355 \big( X \times_{\overline{W}G} W G \big) \times \Delta[\bullet], |
|
4356 \, |
|
4357 V |
|
4358 \Big) |
|
4359 \;\simeq\; |
|
4360 Hom |
|
4361 \big( |
|
4362 X \times \Delta[\bullet], |
|
4363 \, |
|
4364 (V \times W G)/G |
|
4365 \big) |
|
4366 |
|
4367 </annotation></semantics></math></div> |
|
4368 <p>follows from the plain adjunction and the natural isomorphism</p> |
|
4369 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
4370 (X \times_{\overline{W}G} W G) \times \Delta[\bullet] |
|
4371 \;\simeq\; |
|
4372 (X \times \Delta[\bullet]) \times_{\overline{W}G} W G |
|
4373 \,, |
|
4374 |
|
4375 </annotation></semantics></math></div> |
|
4376 <p>which, in turn, follows, for instance, via the <a class='existingWiki… |
|
4377 |
|
4378 <p>\begin{tikzcd} { { (X \times_{\overline{W}G} W G) \times \Delta[k] } … |
|
4379 |
|
4380 <p>\end{remark}</p> |
|
4381 |
|
4382 <h3 id='RelationToModelStructureOnPlainSimplicialSets'>Relation to the m… |
|
4383 |
|
4384 <p>For <math class='maruku-mathml' display='inline' id='mathml_6bd24f229… |
|
4385 |
|
4386 <p>\begin{proposition}\label{CofreeAction} <strong>(underlying simplicia… |
|
4387 |
|
4388 <p>Its <a class='existingWikiWord' href='/nlab/show/right+adjoint'>right… |
|
4389 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
4390 sSet |
|
4391 \underoverset |
|
4392 {\underset{ \;\;\; [\mathcal{G},-] \;\;\; }{\longrightarrow}} |
|
4393 {\overset{ \;\;\; undrl \;\;\; }{\longleftarrow}} |
|
4394 {\bot} |
|
4395 \mathcal{G}Actions(sSet) |
|
4396 |
|
4397 </annotation></semantics></math></div> |
|
4398 <p>sends <math class='maruku-mathml' display='inline' id='mathml_6bd24f2… |
|
4399 |
|
4400 <ul> |
|
4401 <li> |
|
4402 <p>the simplicial set</p> |
|
4403 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
4404 [\mathcal{G},\mathcal{X}] |
|
4405 \;\coloneqq\; |
|
4406 Hom_{sSet}\big( \mathcal{G} \times \Delta[\bullet], \mathcal{X}\big) |
|
4407 \;\;\; |
|
4408 \in |
|
4409 sSet |
|
4410 |
|
4411 </annotation></semantics></math></div></li> |
|
4412 |
|
4413 <li> |
|
4414 <p>equipped with the <math class='maruku-mathml' display='inline' id='ma… |
|
4415 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
4416 \mathcal{G} \times [\mathcal{G},\mathcal{X}] |
|
4417 \overset{ (-) \cdot (-) }{\longrightarrow} |
|
4418 \mathcal{G} |
|
4419 |
|
4420 </annotation></semantics></math></div> |
|
4421 <p>which in degree <math class='maruku-mathml' display='inline' id='math… |
|
4422 <div class='maruku-equation' id='eq:CofreeSimplicialActionComponentFunct… |
|
4423 |
|
4424 Hom(\Delta[n], \mathcal{G}) |
|
4425 \,\times\, |
|
4426 Hom |
|
4427 \big( |
|
4428 \mathcal{G} \times \Delta[n], |
|
4429 \, |
|
4430 \mathcal{X} |
|
4431 \big) |
|
4432 \longrightarrow |
|
4433 Hom |
|
4434 \big( |
|
4435 \mathcal{G} \times \Delta[n], |
|
4436 \, |
|
4437 \mathcal{X} |
|
4438 \big) |
|
4439 |
|
4440 </annotation></semantics></math></div> |
|
4441 <p>that sends</p> |
|
4442 <div class='maruku-equation' id='eq:CofreeSimplicialActionInComponents'>… |
|
4443 |
|
4444 \begin{aligned} |
|
4445 & |
|
4446 \Big( |
|
4447 \Delta[n] \overset{g_n}{\to} \mathcal{G}, |
|
4448 \; |
|
4449 \mathcal{G}\times \Delta[n] |
|
4450 \overset{\phi}{\to} |
|
4451 \mathcal{X}, |
|
4452 \Big) |
|
4453 \\ |
|
4454 \;\;\mapsto\;\; |
|
4455 & |
|
4456 \Big( |
|
4457 \mathcal{G} \times \Delta[n] |
|
4458 \overset{id \times diag}{\longrightarrow} |
|
4459 \mathcal{G} \times \Delta[n] \times \Delta[n] |
|
4460 \overset{ id \times g_n \times id }{\longrightarrow} |
|
4461 \mathcal{G} \times \mathcal{G} \times \Delta[n] |
|
4462 \overset{(-)\cdot(-) \times id}{\to} |
|
4463 \mathcal{G} \times \Delta[n] |
|
4464 \overset{\phi}{\to} |
|
4465 \mathcal{X} |
|
4466 \Big) |
|
4467 \end{aligned} |
|
4468 |
|
4469 </annotation></semantics></math></div></li> |
|
4470 </ul> |
|
4471 |
|
4472 <p>\end{proposition}</p> |
|
4473 |
|
4474 <p>Here and in the following proof we make free use of the <a class='exi… |
|
4475 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
4476 Hom_{sSet}(\Delta[n], \mathcal{S}) \;\simeq\; \mathcal{S}_n |
|
4477 |
|
4478 </annotation></semantics></math></div> |
|
4479 <p>for any <a class='existingWikiWord' href='/nlab/show/simplicial+set'>… |
|
4480 |
|
4481 <p>\begin{proof}</p> |
|
4482 |
|
4483 <p>We already know from Def. \ref{BorelModelStructure} that <math class=… |
|
4484 |
|
4485 <p>The idea of the existence of the <a class='existingWikiWord' href='/n… |
|
4486 |
|
4487 <p>To check that <a class='maruku-eqref' href='#eq:CofreeSimplicialActio… |
|
4488 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
4489 \big\{ |
|
4490 \mathcal{P} |
|
4491 \overset{\;\;\phi_{(-)}\;\;}{\longrightarrow} |
|
4492 [\mathcal{G}, \mathcal{X}] |
|
4493 \big\} |
|
4494 \;\;\;\overset{ \;\; \widetilde{(-)} \;\; }{\leftrightarrow}\;\;\; |
|
4495 \big\{ |
|
4496 undrl(\mathcal{P}) |
|
4497 \overset{\;\; {\widetilde \phi}_{(-)} \;\; }{\longrightarrow} |
|
4498 \mathcal{X} |
|
4499 \big\} |
|
4500 \,. |
|
4501 |
|
4502 </annotation></semantics></math></div> |
|
4503 <p>So given</p> |
|
4504 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
4505 \phi_{(-)} |
|
4506 \;\colon\; |
|
4507 p_n |
|
4508 \mapsto |
|
4509 \big( |
|
4510 \phi_{p_n} |
|
4511 \;\colon\; |
|
4512 \mathcal{G} \times \Delta[n] \to \mathcal{X} |
|
4513 \big) |
|
4514 |
|
4515 </annotation></semantics></math></div> |
|
4516 <p>on the left, define</p> |
|
4517 <div class='maruku-equation' id='eq:AdjunctOfHomomorphismToCofreeSimplic… |
|
4518 |
|
4519 \widetilde \phi_{(-)} |
|
4520 \;\colon\; |
|
4521 p_n |
|
4522 \mapsto |
|
4523 \phi_{p_n}(e_n, \sigma_n) |
|
4524 \;\in\; |
|
4525 \mathcal{X}_n |
|
4526 \,, |
|
4527 |
|
4528 </annotation></semantics></math></div> |
|
4529 <p>where <math class='maruku-mathml' display='inline' id='mathml_6bd24f2… |
|
4530 |
|
4531 <p>It is clear that this is a <a class='existingWikiWord' href='/nlab/sh… |
|
4532 |
|
4533 <p>To that end, observe for any <math class='maruku-mathml' display='inl… |
|
4534 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
4535 \begin{aligned} |
|
4536 \phi_{p_n}(g_n, \sigma_n) |
|
4537 & \;=\; |
|
4538 \phi_{p_n}( e_n \cdot g_n, \sigma_n ) |
|
4539 \\ |
|
4540 & \;=\; |
|
4541 \big( |
|
4542 g_n \cdot \phi_{p_n} |
|
4543 \big) |
|
4544 ( e_n, \sigma_n ) |
|
4545 \\ |
|
4546 & \;=\; |
|
4547 \phi_{ g_n \cdot p_n } |
|
4548 (e_n, \sigma_n) |
|
4549 \\ |
|
4550 & \;=\; |
|
4551 {\widetilde \phi}_{g_n \cdot p_n} |
|
4552 \end{aligned} |
|
4553 |
|
4554 </annotation></semantics></math></div> |
|
4555 <p>Here:</p> |
|
4556 |
|
4557 <ul> |
|
4558 <li> |
|
4559 <p>the first step is the unit law in the component group <math class='ma… |
|
4560 </li> |
|
4561 |
|
4562 <li> |
|
4563 <p>the second step uses the definition <a class='maruku-eqref' href='#eq… |
|
4564 </li> |
|
4565 |
|
4566 <li> |
|
4567 <p>the third step is the assumption that <math class='maruku-mathml' dis… |
|
4568 </li> |
|
4569 |
|
4570 <li> |
|
4571 <p>the fourth step is the definition <a class='maruku-eqref' href='#eq:A… |
|
4572 </li> |
|
4573 </ul> |
|
4574 |
|
4575 <p>These identifications show that <math class='maruku-mathml' display='… |
|
4576 |
|
4577 <p>\end{proof}</p> |
|
4578 |
|
4579 <p>\begin{example}\label{BZActionOnInertiaGroupoid} <strong>(<math class… |
|
4580 |
|
4581 <ul> |
|
4582 <li> |
|
4583 <p><math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d9… |
|
4584 |
|
4585 <p>be a <a class='existingWikiWord' href='/nlab/show/discrete+group'>dis… |
|
4586 </li> |
|
4587 |
|
4588 <li> |
|
4589 <p><math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d9… |
|
4590 |
|
4591 <p>be a <math class='maruku-mathml' display='inline' id='mathml_6bd24f22… |
|
4592 </li> |
|
4593 |
|
4594 <li> |
|
4595 <p><math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d9… |
|
4596 |
|
4597 <p>the <a class='existingWikiWord' href='/nlab/show/simplicial+set'>simp… |
|
4598 </li> |
|
4599 |
|
4600 <li> |
|
4601 <p><math class='maruku-mathml' display='inline' id='mathml_6bd24f22919d9… |
|
4602 |
|
4603 <p>the <a class='existingWikiWord' href='/nlab/show/simplicial+group'>si… |
|
4604 </li> |
|
4605 </ul> |
|
4606 |
|
4607 <p>Then the <a class='existingWikiWord' href='/nlab/show/functor+categor… |
|
4608 <div class='maruku-equation' id='eq:InertiaGroupoidAsFunctorGroupoidOutO… |
|
4609 |
|
4610 \begin{aligned} |
|
4611 \Lambda(X \!\sslash\! G) |
|
4612 & \;\coloneqq\; |
|
4613 \big[ |
|
4614 \mathbf{B}\mathbb{Z}, X \!\sslash\! G |
|
4615 \big] |
|
4616 \\ |
|
4617 & |
|
4618 \;\simeq\; |
|
4619 Func |
|
4620 \big( |
|
4621 (\mathbb{Z} \rightrightarrows \ast), |
|
4622 \, |
|
4623 (X \times G \rightrightarrows X) |
|
4624 \big) |
|
4625 \\ |
|
4626 & \;\underset{\in \mathrm{W}}{\leftarrow}\; |
|
4627 \underset{ |
|
4628 [g] \in ConjCl(G) |
|
4629 }{\coprod} |
|
4630 \Big( |
|
4631 X^{g} \!\sslash\! C_g |
|
4632 \Big) |
|
4633 \end{aligned} |
|
4634 |
|
4635 </annotation></semantics></math></div> |
|
4636 <p>is known as the <em><a class='existingWikiWord' href='/nlab/show/iner… |
|
4637 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
4638 ConjCla(G) |
|
4639 \;\coloneqq\; |
|
4640 G/_{ad} G |
|
4641 \,, |
|
4642 \;\;\;\;\;\;\;\;\;\;\; |
|
4643 C_g |
|
4644 \;\coloneqq\; |
|
4645 \big\{ |
|
4646 h \in G |
|
4647 \,\left\vert\, |
|
4648 h \cdot g = g \cdot h |
|
4649 \right. |
|
4650 \big\} |
|
4651 |
|
4652 </annotation></semantics></math></div> |
|
4653 <p>denotes, respectively, the set of <a class='existingWikiWord' href='/… |
|
4654 |
|
4655 <p>Now, by Prop. \ref{CofreeAction} the inertia groupoid <a class='maruk… |
|
4656 |
|
4657 <p>By the formula <a class='maruku-eqref' href='#eq:CofreeSimplicialActi… |
|
4658 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
4659 {\color{purple}n} |
|
4660 \;\colon\; |
|
4661 \Delta[1] |
|
4662 \longrightarrow |
|
4663 \mathbf{B}G |
|
4664 |
|
4665 </annotation></semantics></math></div> |
|
4666 <p>acts on the morphisms</p> |
|
4667 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
4668 (x,g) \overset{h}{\longrightarrow} (h\cdot x, g) |
|
4669 \;\;\; |
|
4670 \in |
|
4671 \; |
|
4672 \Lambda(X \!\sslash\! G) |
|
4673 |
|
4674 </annotation></semantics></math></div> |
|
4675 <p>of the inertia groupoid as follows (recall the nature of <a class='ex… |
|
4676 |
|
4677 <p><img src='https://ncatlab.org/nlab/files/BZActionOnInertiaGroupoid202… |
|
4678 |
|
4679 <p>\end{example}</p> |
|
4680 |
|
4681 <h3 id='relation_to_the_fine_model_structure_of_equivariant_homotopy_the… |
|
4682 |
|
4683 <p>The <a class='existingWikiWord' href='/nlab/show/identity+functor'>id… |
|
4684 |
|
4685 <p>The left adjoint is</p> |
|
4686 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
4687 L = id |
|
4688 \;\colon\; |
|
4689 G_\bullet Act_{coarse} |
|
4690 \longrightarrow |
|
4691 G_\bullet Act_{fine} |
|
4692 |
|
4693 </annotation></semantics></math></div> |
|
4694 <p>from the Borel model structure to the genuine <a class='existingWikiW… |
|
4695 |
|
4696 <p>Because:</p> |
|
4697 |
|
4698 <p>First of all, by (<a href='#Guillou'>Guillou, theorem 3.12, example 4… |
|
4699 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
4700 R = id \;\colon\; G_\bullet Act_{fine} \longrightarrow G_\bullet Act_{… |
|
4701 |
|
4702 </annotation></semantics></math></div> |
|
4703 <p>is right Quillen.</p> |
|
4704 |
|
4705 <h3 id='GeneralizationToSimplicialPresheaves'>Generalization to simplici… |
|
4706 |
|
4707 <p>Since the <a class='existingWikiWord' href='/nlab/show/simplicial+cla… |
|
4708 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
4709 SimplicialGroups |
|
4710 \xrightarrow{\;\; W \;\;} |
|
4711 SimplicialSets |
|
4712 |
|
4713 </annotation></semantics></math></div> |
|
4714 <p>with <a class='existingWikiWord' href='/nlab/show/natural+transformat… |
|
4715 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
4716 \mathcal{G} |
|
4717 \xrightarrow{\;\; i \;\;} |
|
4718 W\mathcal{G} |
|
4719 \xrightarrow{\;\; p \;\;} |
|
4720 \overline{W}\mathcal{G} |
|
4721 |
|
4722 </annotation></semantics></math></div> |
|
4723 <p>the pair of <a class='existingWikiWord' href='/nlab/show/adjoint+func… |
|
4724 |
|
4725 <p>\begin{prop} For <math class='maruku-mathml' display='inline' id='mat… |
|
4726 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
4727 sPSh(\mathcal{C}) |
|
4728 \;\coloneqq\; |
|
4729 sSetCat( \mathcal{C}^{op}, \, sSet ) |
|
4730 |
|
4731 </annotation></semantics></math></div> |
|
4732 <p>denoting its category of <a class='existingWikiWord' href='/nlab/show… |
|
4733 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
4734 \underline{\mathcal{G}} |
|
4735 \;\in\; |
|
4736 Groups |
|
4737 \big( |
|
4738 sPSh(\mathcal{C}) |
|
4739 \big) |
|
4740 |
|
4741 </annotation></semantics></math></div> |
|
4742 <p>a <a class='existingWikiWord' href='/nlab/show/group+object'>group ob… |
|
4743 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
4744 \underline{\mathcal{G}} |
|
4745 Acts |
|
4746 \big( |
|
4747 sPSh(\mathcal{C}) |
|
4748 \big) |
|
4749 |
|
4750 </annotation></semantics></math></div> |
|
4751 <p>denoting its category of <a class='existingWikiWord' href='/nlab/show… |
|
4752 |
|
4753 <p>we have an <a class='existingWikiWord' href='/nlab/show/adjoint+funct… |
|
4754 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
4755 \underline{\mathcal{G}} |
|
4756 Acts |
|
4757 \big( |
|
4758 sPSh(\mathcal{C}) |
|
4759 \big) |
|
4760 \underoverset |
|
4761 { |
|
4762 \underset{ |
|
4763 \big( |
|
4764 (-) \times W\underline{\mathcal{G}} |
|
4765 \big) |
|
4766 \big/ |
|
4767 \underline{\mathcal{G}} |
|
4768 } |
|
4769 {\longrightarrow}} |
|
4770 { |
|
4771 \overset{ |
|
4772 (-) |
|
4773 \times_{\overline{W}\underline{\mathcal{G}}} |
|
4774 W\underline{\mathcal{G}} |
|
4775 }{\longleftarrow} |
|
4776 } |
|
4777 {\bot} |
|
4778 sPSh(\mathcal{C})_{/\overline{W}\underline{\mathcal{G}}} |
|
4779 |
|
4780 </annotation></semantics></math></div> |
|
4781 <p>\end{prop} \begin{proof}</p> |
|
4782 |
|
4783 <p>The required <a href='adjoint+functor#InTermsOfHomIsomorphism'>hom-is… |
|
4784 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
4785 \begin{aligned} |
|
4786 Hom |
|
4787 \Big( |
|
4788 (\underline{X},p), |
|
4789 \, |
|
4790 \big( |
|
4791 \underline{Y} \times W\underline{\mathcal{G}} |
|
4792 \big) / \underline{\mathcal{G}} |
|
4793 \Big) |
|
4794 & |
|
4795 \;\simeq\; |
|
4796 Hom |
|
4797 \Big( |
|
4798 \underline{X}, |
|
4799 \, |
|
4800 \big( |
|
4801 \underline{Y} \times W\underline{\mathcal{G}} |
|
4802 \big) / \underline{\mathcal{G}} |
|
4803 \Big) |
|
4804 \underset{ |
|
4805 Hom |
|
4806 \Big( |
|
4807 \underline{X}, |
|
4808 \, |
|
4809 \overline{W} \underline{\mathcal{G}} |
|
4810 \Big) |
|
4811 }{\times} |
|
4812 \{p\} |
|
4813 \\ |
|
4814 & \;\simeq\; |
|
4815 \int^c |
|
4816 Hom |
|
4817 \Big( |
|
4818 \underline{X}(c), |
|
4819 \, |
|
4820 \big( |
|
4821 \underline{Y}(c) \times W\underline{\mathcal{G}(c)} |
|
4822 \big) / \underline{\mathcal{G}}(c) |
|
4823 \Big) |
|
4824 \underset{ |
|
4825 \int^c |
|
4826 Hom |
|
4827 \Big( |
|
4828 \underline{X}(c), |
|
4829 \, |
|
4830 \overline{W} \underline{\mathcal{G}}(c) |
|
4831 \Big) |
|
4832 }{\times} |
|
4833 \{p\} |
|
4834 \\ |
|
4835 & \;\simeq\; |
|
4836 \int^c |
|
4837 \left( |
|
4838 Hom |
|
4839 \Big( |
|
4840 \underline{X}(c), |
|
4841 \, |
|
4842 \big( |
|
4843 \underline{Y}(c) \times W\underline{\mathcal{G}}(c) |
|
4844 \big) / \underline{\mathcal{G}}(c) |
|
4845 \Big) |
|
4846 \underset{ |
|
4847 Hom |
|
4848 \Big( |
|
4849 \underline{X}(c), |
|
4850 \, |
|
4851 \overline{W} \underline{\mathcal{G}}(c) |
|
4852 \Big) |
|
4853 }{\times} |
|
4854 \{p(c)\} |
|
4855 \right) |
|
4856 \\ |
|
4857 & \;\simeq\; |
|
4858 \int^c |
|
4859 Hom_{/\overline{W}\underline{\mathcal{G}}(c)} |
|
4860 \Big( |
|
4861 \big( \underline{X}(c), p(c)\big), |
|
4862 \, |
|
4863 \big( |
|
4864 \underline{Y}(c) \times \overline{W} \underline{\mathcal{G}}(c) |
|
4865 \big)\big/ \mathcal{G}(c) |
|
4866 \Big) |
|
4867 \\ |
|
4868 & \;\simeq\; |
|
4869 \int^c |
|
4870 \left( |
|
4871 \underline{\mathcal{G}}(c) |
|
4872 Acts(sSet) |
|
4873 \big( |
|
4874 \underline{X}(c) |
|
4875 \underset{ \overline{W}\underline{\mathcal{G}}(c) }{\times} |
|
4876 W \underline{\mathcal{G}}(c), |
|
4877 \, |
|
4878 \underline{Y}(c) |
|
4879 \big) |
|
4880 \right) |
|
4881 \\ |
|
4882 & \;\simeq\; |
|
4883 \mathcal{G}Acts(sPSh(\mathcal{C})) |
|
4884 \big( |
|
4885 \underline{X} |
|
4886 \underset{\overline{W}\underline{\mathcal{G}}}{\times} |
|
4887 W \underline{\mathcal{G}}, |
|
4888 \, |
|
4889 \underline{Y} |
|
4890 \big) |
|
4891 \end{aligned} |
|
4892 |
|
4893 </annotation></semantics></math></div> |
|
4894 <p>Here:</p> |
|
4895 |
|
4896 <ul> |
|
4897 <li> |
|
4898 <p>the first step is the characterization of hom-sets of a <a class='exi… |
|
4899 </li> |
|
4900 |
|
4901 <li> |
|
4902 <p>the second step is the description of the hom-set of a <a class='exis… |
|
4903 </li> |
|
4904 |
|
4905 <li> |
|
4906 <p>the third step uses that <a class='existingWikiWord' href='/nlab/show… |
|
4907 </li> |
|
4908 |
|
4909 <li> |
|
4910 <p>the fourth step recognizes again, now object-wise, the hom-set in a <… |
|
4911 </li> |
|
4912 |
|
4913 <li> |
|
4914 <p>the fifth step is objectwise the <a href='adjoint+functor#InTermsOfHo… |
|
4915 </li> |
|
4916 |
|
4917 <li> |
|
4918 <p>the sixth step recognizes again the <a class='existingWikiWord' href=… |
|
4919 </li> |
|
4920 </ul> |
|
4921 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
4922 \array{ |
|
4923 \underline{\mathcal{G}}Acts |
|
4924 \big( |
|
4925 \underline{A}, \, \underline{B} |
|
4926 \big) |
|
4927 &\longrightarrow& |
|
4928 \mathcal{G}(c_1)Acts |
|
4929 \big( |
|
4930 \underline{A}(c_1), \, \underline{B}(c_1) |
|
4931 \big) |
|
4932 \\ |
|
4933 \big\downarrow |
|
4934 && |
|
4935 \big\downarrow |
|
4936 \\ |
|
4937 \mathcal{G}(c_2)Acts |
|
4938 \big( |
|
4939 \underline{A}(c_2), \, \underline{B}(c_2) |
|
4940 \big) |
|
4941 &\longrightarrow& |
|
4942 Hom |
|
4943 \big( |
|
4944 \underline{A}(c_1), \, \underline{B}(c_2) |
|
4945 \big) |
|
4946 } |
|
4947 |
|
4948 </annotation></semantics></math></div> |
|
4949 <p>\end{proof}</p> |
|
4950 |
|
4951 <h2 id='references'>References</h2> |
|
4952 |
|
4953 <p>The model structure, the characterization of its cofibrations, and it… |
|
4954 |
|
4955 <ul> |
|
4956 <li id='DDK80'><a class='existingWikiWord' href='/nlab/show/Emmanuel+Dro… |
|
4957 </ul> |
|
4958 |
|
4959 <p>This Quillen equivalence also mentioned as:</p> |
|
4960 |
|
4961 <ul> |
|
4962 <li id='Dwyer2008'><a class='existingWikiWord' href='/nlab/show/William+… |
|
4963 </ul> |
|
4964 |
|
4965 <p>Discussion in relation to the “fine” model structure of <a class=… |
|
4966 |
|
4967 <ul> |
|
4968 <li id='Guillou'><a class='existingWikiWord' href='/nlab/show/Bert+Guill… |
|
4969 </ul> |
|
4970 |
|
4971 <p>Textbook account of (just) the Borel model structure:</p> |
|
4972 |
|
4973 <ul> |
|
4974 <li id='GoerssJardine09'><a class='existingWikiWord' href='/nlab/show/Pa… |
|
4975 </ul> |
|
4976 |
|
4977 <p>Discussion with the model of <a class='existingWikiWord' href='/nlab/… |
|
4978 |
|
4979 <ul> |
|
4980 <li><a class='existingWikiWord' href='/nlab/show/Matan+Prasma'>Matan Pra… |
|
4981 </ul> |
|
4982 |
|
4983 <p>Discussion of a <a class='existingWikiWord' href='/nlab/show/global+e… |
|
4984 |
|
4985 <ul> |
|
4986 <li><a class='existingWikiWord' href='/nlab/show/Yonatan+Harpaz'>Yonatan… |
|
4987 </ul> |
|
4988 |
|
4989 <p> |
|
4990 </p> </div> |
|
4991 </content> |
|
4992 </entry> |
|
4993 <entry> |
|
4994 <title type="html">Sandbox</title> |
|
4995 <link rel="alternate" type="application/xhtml+xml" href="https://nca… |
|
4996 <updated>2021-07-01T16:14:56Z</updated> |
|
4997 <published>2009-07-07T06:11:26Z</published> |
|
4998 <id>tag:ncatlab.org,2009-07-07:nLab,Sandbox</id> |
|
4999 <author> |
|
5000 <name>Urs Schreiber</name> |
|
5001 </author> |
|
5002 <content type="xhtml" xml:base="https://ncatlab.org/nlab/show/Sandbo… |
|
5003 <div xmlns="http://www.w3.org/1999/xhtml"> |
|
5004 <h3 id='GeneralizationToSimplicialPresheaves'>Generalization to simplici… |
|
5005 |
|
5006 <p>Since the <a class='existingWikiWord' href='/nlab/show/simplicial+cla… |
|
5007 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
5008 SimplicialGroups |
|
5009 \xrightarrow{\;\; W \;\;} |
|
5010 SimplicialSets |
|
5011 |
|
5012 </annotation></semantics></math></div> |
|
5013 <p>with <a class='existingWikiWord' href='/nlab/show/natural+transformat… |
|
5014 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
5015 \mathcal{G} |
|
5016 \xrightarrow{\;\; i \;\;} |
|
5017 W\mathcal{G} |
|
5018 \xrightarrow{\;\; p \;\;} |
|
5019 \overline{W}\mathcal{G} |
|
5020 |
|
5021 </annotation></semantics></math></div> |
|
5022 <p>the pair of <a class='existingWikiWord' href='/nlab/show/adjoint+func… |
|
5023 |
|
5024 <p>\begin{prop} For <math class='maruku-mathml' display='inline' id='mat… |
|
5025 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
5026 sPSh(\mathcal{C}) |
|
5027 \;\coloneqq\; |
|
5028 sSetCat( \mathcal{C}^{op}, \, sSet ) |
|
5029 |
|
5030 </annotation></semantics></math></div> |
|
5031 <p>denoting its category of <a class='existingWikiWord' href='/nlab/show… |
|
5032 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
5033 \underline{\mathcal{G}} |
|
5034 \;\in\; |
|
5035 Groups |
|
5036 \big( |
|
5037 sPSh(\mathcal{C}) |
|
5038 \big) |
|
5039 |
|
5040 </annotation></semantics></math></div> |
|
5041 <p>a <a class='existingWikiWord' href='/nlab/show/group+object'>group ob… |
|
5042 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
5043 \underline{\mathcal{G}} |
|
5044 Acts |
|
5045 \big( |
|
5046 sPSh(\mathcal{C}) |
|
5047 \big) |
|
5048 |
|
5049 </annotation></semantics></math></div> |
|
5050 <p>denoting its category of <a class='existingWikiWord' href='/nlab/show… |
|
5051 |
|
5052 <p>we have an <a class='existingWikiWord' href='/nlab/show/adjoint+funct… |
|
5053 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
5054 \underline{\mathcal{G}} |
|
5055 Acts |
|
5056 \big( |
|
5057 sPSh(\mathcal{C}) |
|
5058 \big) |
|
5059 \underoverset |
|
5060 { |
|
5061 \underset{ |
|
5062 \big( |
|
5063 (-) \times W\underline{\mathcal{G}} |
|
5064 \big) |
|
5065 \big/ |
|
5066 \underline{\mathcal{G}} |
|
5067 } |
|
5068 {\longrightarrow}} |
|
5069 { |
|
5070 \overset{ |
|
5071 (-) |
|
5072 \times_{\overline{W}\underline{\mathcal{G}}} |
|
5073 W\underline{\mathcal{G}} |
|
5074 }{\longleftarrow} |
|
5075 } |
|
5076 {\bot} |
|
5077 sPSh(\mathcal{C})_{/\overline{W}\underline{\mathcal{G}}} |
|
5078 |
|
5079 </annotation></semantics></math></div> |
|
5080 <p>\end{prop} \begin{proof}</p> |
|
5081 |
|
5082 <p>The required <a href='adjoint+functor#InTermsOfHomIsomorphism'>hom-is… |
|
5083 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
5084 \begin{aligned} |
|
5085 Hom |
|
5086 \Big( |
|
5087 (\underline{X},p), |
|
5088 \, |
|
5089 \big( |
|
5090 \underline{Y} \times W\underline{\mathcal{G}} |
|
5091 \big) / \underline{\mathcal{G}} |
|
5092 \Big) |
|
5093 & |
|
5094 \;\simeq\; |
|
5095 Hom |
|
5096 \Big( |
|
5097 \underline{X}, |
|
5098 \, |
|
5099 \big( |
|
5100 \underline{Y} \times W\underline{\mathcal{G}} |
|
5101 \big) / \underline{\mathcal{G}} |
|
5102 \Big) |
|
5103 \underset{ |
|
5104 Hom |
|
5105 \Big( |
|
5106 \underline{X}, |
|
5107 \, |
|
5108 \overline{W} \underline{\mathcal{G}} |
|
5109 \Big) |
|
5110 }{\times} |
|
5111 \{p\} |
|
5112 \\ |
|
5113 & \;\simeq\; |
|
5114 \int^c |
|
5115 Hom |
|
5116 \Big( |
|
5117 \underline{X}(c), |
|
5118 \, |
|
5119 \big( |
|
5120 \underline{Y}(c) \times W\underline{\mathcal{G}(c)} |
|
5121 \big) / \underline{\mathcal{G}}(c) |
|
5122 \Big) |
|
5123 \underset{ |
|
5124 \int^c |
|
5125 Hom |
|
5126 \Big( |
|
5127 \underline{X}(c), |
|
5128 \, |
|
5129 \overline{W} \underline{\mathcal{G}}(c) |
|
5130 \Big) |
|
5131 }{\times} |
|
5132 \{p\} |
|
5133 \\ |
|
5134 & \;\simeq\; |
|
5135 \int^c |
|
5136 \left( |
|
5137 Hom |
|
5138 \Big( |
|
5139 \underline{X}(c), |
|
5140 \, |
|
5141 \big( |
|
5142 \underline{Y}(c) \times W\underline{\mathcal{G}(c)} |
|
5143 \big) / \underline{\mathcal{G}}(c) |
|
5144 \Big) |
|
5145 \underset{ |
|
5146 Hom |
|
5147 \Big( |
|
5148 \underline{X}(c), |
|
5149 \, |
|
5150 \overline{W} \underline{\mathcal{G}}(c) |
|
5151 \Big) |
|
5152 }{\times} |
|
5153 \{p(c)\} |
|
5154 \right) |
|
5155 \\ |
|
5156 & \;\simeq\; |
|
5157 \int^c |
|
5158 Hom_{/\overline{W}\underline{\mathcal{G}}(c)} |
|
5159 \Big( |
|
5160 \big(X(c), p(c)\big), |
|
5161 \, |
|
5162 \big( |
|
5163 Y(c) \times \overline{W} \mathcal{G}(c) |
|
5164 \big)\big/ \mathcal{G}(c) |
|
5165 \Big) |
|
5166 \\ |
|
5167 & \;\simeq\; |
|
5168 \int^c |
|
5169 \left( |
|
5170 \underline{\mathcal{G}}(c) |
|
5171 Acts(sSet) |
|
5172 \big( |
|
5173 X(c) |
|
5174 \underset{ \overline{W}\underline{\mathcal{G}}(c) }{\times} |
|
5175 W \underline{\mathcal{G}}(c), |
|
5176 \, |
|
5177 Y(c) |
|
5178 \big) |
|
5179 \right) |
|
5180 \\ |
|
5181 & \;\simeq\; |
|
5182 \mathcal{G}Acts(sPSh(\mathcal{C})) |
|
5183 \big( |
|
5184 X |
|
5185 \underset{\overline{W}\underline{\mathcal{G}}}{\times} |
|
5186 W \underline{\mathcal{G}}, |
|
5187 \, |
|
5188 Y |
|
5189 \big) |
|
5190 \end{aligned} |
|
5191 |
|
5192 </annotation></semantics></math></div> |
|
5193 <p>Here:</p> |
|
5194 |
|
5195 <ul> |
|
5196 <li> |
|
5197 <p>the first step is the characterization of hom-sets of a <a class='exi… |
|
5198 </li> |
|
5199 |
|
5200 <li> |
|
5201 <p>the second step is the description of the hom-set of a <a class='exis… |
|
5202 </li> |
|
5203 |
|
5204 <li> |
|
5205 <p>the third step uses that <a class='existingWikiWord' href='/nlab/show… |
|
5206 </li> |
|
5207 |
|
5208 <li> |
|
5209 <p>the fourth step recognizes again, now object-wise, the hom-set in a <… |
|
5210 </li> |
|
5211 |
|
5212 <li> |
|
5213 <p>the fifth step is objectwise the <a href='adjoint+functor#InTermsOfHo… |
|
5214 </li> |
|
5215 |
|
5216 <li> |
|
5217 <p>the sixth step recognizes again the <a class='existingWikiWord' href=… |
|
5218 </li> |
|
5219 </ul> |
|
5220 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
5221 \array{ |
|
5222 \underline{\mathcal{G}}Acts |
|
5223 \big( |
|
5224 A, \, B |
|
5225 \big) |
|
5226 &\longrightarrow& |
|
5227 \mathcal{G}(c_1)Acts |
|
5228 \big( |
|
5229 A(c_1), \, B(c_1) |
|
5230 \big) |
|
5231 \\ |
|
5232 \big\downarrow |
|
5233 && |
|
5234 \big\downarrow |
|
5235 \\ |
|
5236 \mathcal{G}(c_2)Acts |
|
5237 \big( |
|
5238 A(c_2), \, B(c_2) |
|
5239 \big) |
|
5240 &\longrightarrow& |
|
5241 Hom |
|
5242 \big( |
|
5243 A(c_1), \, B(c_2) |
|
5244 \big) |
|
5245 } |
|
5246 |
|
5247 </annotation></semantics></math></div> |
|
5248 <p>\end{proof}</p> |
|
5249 |
|
5250 <p>\linebreak</p> |
|
5251 |
|
5252 <p>\linebreak</p> |
|
5253 |
|
5254 <p>added the observation (<a href='https://ncatlab.org/nlab/show/Borel+m… |
|
5255 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
5256 \mathcal{G} Acts(sSet) |
|
5257 \underoverset |
|
5258 {\underset{ \big((-) \times W \mathcal{G}\big)/\mathcal{G} }{\lo… |
|
5259 {\overset{ (-) \times_{\overline{W}\mathcal{G}} W \mathcal{G} }… |
|
5260 {\bot} |
|
5261 sSet_{/\overline{W}\mathcal{G}} |
|
5262 |
|
5263 </annotation></semantics></math></div> |
|
5264 <p>generalizes to one for presheaves of simplicial groups</p> |
|
5265 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
5266 \underline{\mathcal{G}} |
|
5267 Acts |
|
5268 \big( |
|
5269 sPSh(\mathcal{C}) |
|
5270 \big) |
|
5271 \underoverset |
|
5272 { |
|
5273 \underset{ |
|
5274 \big( |
|
5275 (-) \times W\underline{\mathcal{G}} |
|
5276 \big) |
|
5277 \big/ |
|
5278 \underline{\mathcal{G}} |
|
5279 } |
|
5280 {\longrightarrow}} |
|
5281 { |
|
5282 \overset{ |
|
5283 (-) |
|
5284 \times_{\overline{W}\underline{\mathcal{G}}} |
|
5285 W\underline{\mathcal{G}} |
|
5286 }{\longleftarrow} |
|
5287 } |
|
5288 {\bot} |
|
5289 sPSh(\mathcal{C})_{/\overline{W}\underline{\mathcal{G}}} |
|
5290 |
|
5291 </annotation></semantics></math></div> </div> |
|
5292 </content> |
|
5293 </entry> |
|
5294 <entry> |
|
5295 <title type="html">simplicial presheaf</title> |
|
5296 <link rel="alternate" type="application/xhtml+xml" href="https://nca… |
|
5297 <updated>2021-07-01T14:31:25Z</updated> |
|
5298 <published>2009-01-29T18:34:04Z</published> |
|
5299 <id>tag:ncatlab.org,2009-01-29:nLab,simplicial+presheaf</id> |
|
5300 <author> |
|
5301 <name>Urs Schreiber</name> |
|
5302 </author> |
|
5303 <content type="xhtml" xml:base="https://ncatlab.org/nlab/show/simpli… |
|
5304 <div xmlns="http://www.w3.org/1999/xhtml"> |
|
5305 <div class='rightHandSide'> |
|
5306 <div class='toc clickDown' tabindex='0'> |
|
5307 <h3 id='context'>Context</h3> |
|
5308 |
|
5309 <h4 id='homotopy_theory'>Homotopy theory</h4> |
|
5310 |
|
5311 <div class='hide'> |
|
5312 <p><strong><a class='existingWikiWord' href='/nlab/show/homotopy+theory'… |
|
5313 |
|
5314 <p>flavors: <a class='existingWikiWord' href='/nlab/show/stable+homotopy… |
|
5315 |
|
5316 <p>models: <a class='existingWikiWord' href='/nlab/show/topological+homo… |
|
5317 |
|
5318 <p>see also <strong><a class='existingWikiWord' href='/nlab/show/algebra… |
|
5319 |
|
5320 <p><strong>Introductions</strong></p> |
|
5321 |
|
5322 <ul> |
|
5323 <li> |
|
5324 <p><a class='existingWikiWord' href='/nlab/show/Introduction+to+Topology… |
|
5325 </li> |
|
5326 |
|
5327 <li> |
|
5328 <p><a class='existingWikiWord' href='/nlab/show/Introduction+to+Homotopy… |
|
5329 </li> |
|
5330 |
|
5331 <li> |
|
5332 <p><a class='existingWikiWord' href='/nlab/show/geometry+of+physics+--+h… |
|
5333 </li> |
|
5334 </ul> |
|
5335 |
|
5336 <p><strong>Definitions</strong></p> |
|
5337 |
|
5338 <ul> |
|
5339 <li> |
|
5340 <p><a class='existingWikiWord' href='/nlab/show/homotopy'>homotopy</a>, … |
|
5341 </li> |
|
5342 |
|
5343 <li> |
|
5344 <p><a class='existingWikiWord' href='/nlab/show/homotopy+type'>homotopy … |
|
5345 </li> |
|
5346 |
|
5347 <li> |
|
5348 <p><a class='existingWikiWord' href='/nlab/show/Pi-algebra'>Pi-algebra</… |
|
5349 </li> |
|
5350 |
|
5351 <li> |
|
5352 <p><a class='existingWikiWord' href='/nlab/show/homotopy+coherent+catego… |
|
5353 |
|
5354 <ul> |
|
5355 <li> |
|
5356 <p><a class='existingWikiWord' href='/nlab/show/homotopical+category'>ho… |
|
5357 |
|
5358 <ul> |
|
5359 <li> |
|
5360 <p><a class='existingWikiWord' href='/nlab/show/model+category'>model ca… |
|
5361 </li> |
|
5362 |
|
5363 <li> |
|
5364 <p><a class='existingWikiWord' href='/nlab/show/category+of+fibrant+obje… |
|
5365 </li> |
|
5366 |
|
5367 <li> |
|
5368 <p><a class='existingWikiWord' href='/nlab/show/Waldhausen+category'>Wal… |
|
5369 </li> |
|
5370 </ul> |
|
5371 </li> |
|
5372 |
|
5373 <li> |
|
5374 <p><a class='existingWikiWord' href='/nlab/show/homotopy+category'>homot… |
|
5375 |
|
5376 <ul> |
|
5377 <li><a class='existingWikiWord' href='/nlab/show/Ho%28Top%29'>Ho(Top)</a… |
|
5378 </ul> |
|
5379 </li> |
|
5380 </ul> |
|
5381 </li> |
|
5382 |
|
5383 <li> |
|
5384 <p><a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-categ… |
|
5385 |
|
5386 <ul> |
|
5387 <li><a class='existingWikiWord' href='/nlab/show/homotopy+category+of+an… |
|
5388 </ul> |
|
5389 </li> |
|
5390 </ul> |
|
5391 |
|
5392 <p><strong>Paths and cylinders</strong></p> |
|
5393 |
|
5394 <ul> |
|
5395 <li> |
|
5396 <p><a class='existingWikiWord' href='/nlab/show/homotopy'>left homotopy<… |
|
5397 |
|
5398 <ul> |
|
5399 <li> |
|
5400 <p><a class='existingWikiWord' href='/nlab/show/cylinder+object'>cylinde… |
|
5401 </li> |
|
5402 |
|
5403 <li> |
|
5404 <p><a class='existingWikiWord' href='/nlab/show/mapping+cone'>mapping co… |
|
5405 </li> |
|
5406 </ul> |
|
5407 </li> |
|
5408 |
|
5409 <li> |
|
5410 <p><a class='existingWikiWord' href='/nlab/show/homotopy'>right homotopy… |
|
5411 |
|
5412 <ul> |
|
5413 <li> |
|
5414 <p><a class='existingWikiWord' href='/nlab/show/path+space+object'>path … |
|
5415 </li> |
|
5416 |
|
5417 <li> |
|
5418 <p><a class='existingWikiWord' href='/nlab/show/mapping+cocone'>mapping … |
|
5419 </li> |
|
5420 |
|
5421 <li> |
|
5422 <p><a class='existingWikiWord' href='/nlab/show/generalized+universal+bu… |
|
5423 </li> |
|
5424 </ul> |
|
5425 </li> |
|
5426 |
|
5427 <li> |
|
5428 <p><a class='existingWikiWord' href='/nlab/show/interval+object'>interva… |
|
5429 |
|
5430 <ul> |
|
5431 <li> |
|
5432 <p><a class='existingWikiWord' href='/nlab/show/localization+at+geometri… |
|
5433 </li> |
|
5434 |
|
5435 <li> |
|
5436 <p><a class='existingWikiWord' href='/nlab/show/infinitesimal+interval+o… |
|
5437 </li> |
|
5438 </ul> |
|
5439 </li> |
|
5440 </ul> |
|
5441 |
|
5442 <p><strong>Homotopy groups</strong></p> |
|
5443 |
|
5444 <ul> |
|
5445 <li> |
|
5446 <p><a class='existingWikiWord' href='/nlab/show/homotopy+group'>homotopy… |
|
5447 |
|
5448 <ul> |
|
5449 <li> |
|
5450 <p><a class='existingWikiWord' href='/nlab/show/fundamental+group'>funda… |
|
5451 |
|
5452 <ul> |
|
5453 <li><a class='existingWikiWord' href='/nlab/show/fundamental+group+of+a+… |
|
5454 </ul> |
|
5455 </li> |
|
5456 |
|
5457 <li> |
|
5458 <p><a class='existingWikiWord' href='/nlab/show/Brown-Grossman+homotopy+… |
|
5459 </li> |
|
5460 |
|
5461 <li> |
|
5462 <p><a class='existingWikiWord' href='/nlab/show/categorical+homotopy+gro… |
|
5463 </li> |
|
5464 |
|
5465 <li> |
|
5466 <p><a class='existingWikiWord' href='/nlab/show/geometric+homotopy+group… |
|
5467 </li> |
|
5468 </ul> |
|
5469 </li> |
|
5470 |
|
5471 <li> |
|
5472 <p><a class='existingWikiWord' href='/nlab/show/fundamental+infinity-gro… |
|
5473 |
|
5474 <ul> |
|
5475 <li> |
|
5476 <p><a class='existingWikiWord' href='/nlab/show/fundamental+groupoid'>fu… |
|
5477 |
|
5478 <ul> |
|
5479 <li><a class='existingWikiWord' href='/nlab/show/path+groupoid'>path gro… |
|
5480 </ul> |
|
5481 </li> |
|
5482 |
|
5483 <li> |
|
5484 <p><a class='existingWikiWord' href='/nlab/show/fundamental+infinity-gro… |
|
5485 </li> |
|
5486 |
|
5487 <li> |
|
5488 <p><a class='existingWikiWord' href='/nlab/show/fundamental+infinity-gro… |
|
5489 </li> |
|
5490 </ul> |
|
5491 </li> |
|
5492 |
|
5493 <li> |
|
5494 <p><a class='existingWikiWord' href='/nlab/show/fundamental+%28infinity%… |
|
5495 |
|
5496 <ul> |
|
5497 <li><a class='existingWikiWord' href='/nlab/show/fundamental+category'>f… |
|
5498 </ul> |
|
5499 </li> |
|
5500 </ul> |
|
5501 |
|
5502 <p><strong>Basic facts</strong></p> |
|
5503 |
|
5504 <ul> |
|
5505 <li><a class='existingWikiWord' href='/nlab/show/fundamental+group+of+th… |
|
5506 </ul> |
|
5507 |
|
5508 <p><strong>Theorems</strong></p> |
|
5509 |
|
5510 <ul> |
|
5511 <li> |
|
5512 <p><a class='existingWikiWord' href='/nlab/show/fundamental+theorem+of+c… |
|
5513 </li> |
|
5514 |
|
5515 <li> |
|
5516 <p><a class='existingWikiWord' href='/nlab/show/Freudenthal+suspension+t… |
|
5517 </li> |
|
5518 |
|
5519 <li> |
|
5520 <p><a class='existingWikiWord' href='/nlab/show/Blakers-Massey+theorem'>… |
|
5521 </li> |
|
5522 |
|
5523 <li> |
|
5524 <p><a class='existingWikiWord' href='/nlab/show/higher+homotopy+van+Kamp… |
|
5525 </li> |
|
5526 |
|
5527 <li> |
|
5528 <p><a class='existingWikiWord' href='/nlab/show/nerve+theorem'>nerve the… |
|
5529 </li> |
|
5530 |
|
5531 <li> |
|
5532 <p><a class='existingWikiWord' href='/nlab/show/Whitehead+theorem'>White… |
|
5533 </li> |
|
5534 |
|
5535 <li> |
|
5536 <p><a class='existingWikiWord' href='/nlab/show/Hurewicz+theorem'>Hurewi… |
|
5537 </li> |
|
5538 |
|
5539 <li> |
|
5540 <p><a class='existingWikiWord' href='/nlab/show/Galois+theory'>Galois th… |
|
5541 </li> |
|
5542 |
|
5543 <li> |
|
5544 <p><a class='existingWikiWord' href='/nlab/show/homotopy+hypothesis'>hom… |
|
5545 </li> |
|
5546 </ul> |
|
5547 </div> |
|
5548 |
|
5549 <h4 id='topos_theory'><math class='maruku-mathml' display='inline' id='m… |
|
5550 |
|
5551 <div class='hide'> |
|
5552 <p><strong><a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%… |
|
5553 |
|
5554 <h2 id='background'>Background</h2> |
|
5555 |
|
5556 <ul> |
|
5557 <li> |
|
5558 <p><a class='existingWikiWord' href='/nlab/show/sheaf+and+topos+theory'>… |
|
5559 </li> |
|
5560 |
|
5561 <li> |
|
5562 <p><a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-categ… |
|
5563 </li> |
|
5564 |
|
5565 <li> |
|
5566 <p><a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-funct… |
|
5567 </li> |
|
5568 |
|
5569 <li> |
|
5570 <p><a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-presh… |
|
5571 </li> |
|
5572 |
|
5573 <li> |
|
5574 <p><a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-categ… |
|
5575 </li> |
|
5576 </ul> |
|
5577 |
|
5578 <h2 id='definitions'>Definitions</h2> |
|
5579 |
|
5580 <ul> |
|
5581 <li> |
|
5582 <p><a class='existingWikiWord' href='/nlab/show/elementary+%28infinity%2… |
|
5583 </li> |
|
5584 |
|
5585 <li> |
|
5586 <p><a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-site'… |
|
5587 </li> |
|
5588 |
|
5589 <li> |
|
5590 <p><a class='existingWikiWord' href='/nlab/show/reflective+sub-%28infini… |
|
5591 |
|
5592 <ul> |
|
5593 <li> |
|
5594 <p><a class='existingWikiWord' href='/nlab/show/localization+of+an+%28in… |
|
5595 </li> |
|
5596 |
|
5597 <li> |
|
5598 <p><a class='existingWikiWord' href='/nlab/show/topological+localization… |
|
5599 </li> |
|
5600 |
|
5601 <li> |
|
5602 <p><a class='existingWikiWord' href='/nlab/show/hypercompletion'>hyperco… |
|
5603 </li> |
|
5604 </ul> |
|
5605 </li> |
|
5606 |
|
5607 <li> |
|
5608 <p><a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-categ… |
|
5609 |
|
5610 <ul> |
|
5611 <li><a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-shea… |
|
5612 </ul> |
|
5613 </li> |
|
5614 |
|
5615 <li> |
|
5616 <p><a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-topos… |
|
5617 </li> |
|
5618 |
|
5619 <li> |
|
5620 <p><a class='existingWikiWord' href='/nlab/show/%28n%2C1%29-topos'>(n,1)… |
|
5621 |
|
5622 <ul> |
|
5623 <li> |
|
5624 <p><a class='existingWikiWord' href='/nlab/show/truncated+object'>n-trun… |
|
5625 </li> |
|
5626 |
|
5627 <li> |
|
5628 <p><a class='existingWikiWord' href='/nlab/show/connected+object'>n-conn… |
|
5629 </li> |
|
5630 |
|
5631 <li> |
|
5632 <p><a class='existingWikiWord' href='/nlab/show/topos'>(1,1)-topos</a></… |
|
5633 |
|
5634 <ul> |
|
5635 <li> |
|
5636 <p><a class='existingWikiWord' href='/nlab/show/presheaf'>presheaf</a></… |
|
5637 </li> |
|
5638 |
|
5639 <li> |
|
5640 <p><a class='existingWikiWord' href='/nlab/show/sheaf'>sheaf</a></p> |
|
5641 </li> |
|
5642 </ul> |
|
5643 </li> |
|
5644 |
|
5645 <li> |
|
5646 <p><a class='existingWikiWord' href='/nlab/show/2-topos'>(2,1)-topos</a>… |
|
5647 |
|
5648 <ul> |
|
5649 <li><a class='existingWikiWord' href='/nlab/show/%282%2C1%29-presheaf'>(… |
|
5650 </ul> |
|
5651 </li> |
|
5652 </ul> |
|
5653 </li> |
|
5654 |
|
5655 <li> |
|
5656 <p><a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-quasi… |
|
5657 |
|
5658 <ul> |
|
5659 <li> |
|
5660 <p><a class='existingWikiWord' href='/nlab/show/separated+%28infinity%2C… |
|
5661 </li> |
|
5662 |
|
5663 <li> |
|
5664 <p><a class='existingWikiWord' href='/nlab/show/quasitopos'>quasitopos</… |
|
5665 |
|
5666 <ul> |
|
5667 <li><a class='existingWikiWord' href='/nlab/show/separated+presheaf'>sep… |
|
5668 </ul> |
|
5669 </li> |
|
5670 |
|
5671 <li> |
|
5672 <p><span class='newWikiWord'>(2,1)-quasitopos<a href='/nlab/new/%282%2C1… |
|
5673 |
|
5674 <ul> |
|
5675 <li><a class='existingWikiWord' href='/nlab/show/separated+%282%2C1%29-p… |
|
5676 </ul> |
|
5677 </li> |
|
5678 </ul> |
|
5679 </li> |
|
5680 |
|
5681 <li> |
|
5682 <p><a class='existingWikiWord' href='/nlab/show/%28infinity%2C2%29-topos… |
|
5683 </li> |
|
5684 |
|
5685 <li> |
|
5686 <p><a class='existingWikiWord' href='/nlab/show/%28infinity%2Cn%29-topos… |
|
5687 </li> |
|
5688 </ul> |
|
5689 |
|
5690 <h2 id='characterization'>Characterization</h2> |
|
5691 |
|
5692 <ul> |
|
5693 <li> |
|
5694 <p><a class='existingWikiWord' href='/nlab/show/pullback-stable+colimit'… |
|
5695 </li> |
|
5696 |
|
5697 <li> |
|
5698 <p><a class='existingWikiWord' href='/nlab/show/%28sub%29object+classifi… |
|
5699 </li> |
|
5700 |
|
5701 <li> |
|
5702 <p><a class='existingWikiWord' href='/nlab/show/groupoid+object+in+an+%2… |
|
5703 |
|
5704 <ul> |
|
5705 <li><a class='existingWikiWord' href='/nlab/show/effective+epimorphism'>… |
|
5706 </ul> |
|
5707 </li> |
|
5708 </ul> |
|
5709 |
|
5710 <h2 id='morphisms'>Morphisms</h2> |
|
5711 |
|
5712 <ul> |
|
5713 <li> |
|
5714 <p><a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-geome… |
|
5715 </li> |
|
5716 |
|
5717 <li> |
|
5718 <p><a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29Topos'… |
|
5719 </li> |
|
5720 |
|
5721 <li> |
|
5722 <p><a class='existingWikiWord' href='/nlab/show/Lawvere+distribution'>La… |
|
5723 </li> |
|
5724 </ul> |
|
5725 |
|
5726 <h2 id='extra_stuff_structure_and_property'>Extra stuff, structure and p… |
|
5727 |
|
5728 <ul> |
|
5729 <li> |
|
5730 <p><a class='existingWikiWord' href='/nlab/show/hypercomplete+%28infinit… |
|
5731 |
|
5732 <ul> |
|
5733 <li> |
|
5734 <p><a class='existingWikiWord' href='/nlab/show/hypercomplete+object'>hy… |
|
5735 </li> |
|
5736 |
|
5737 <li> |
|
5738 <p><a class='existingWikiWord' href='/nlab/show/Whitehead+theorem'>White… |
|
5739 </li> |
|
5740 </ul> |
|
5741 </li> |
|
5742 |
|
5743 <li> |
|
5744 <p><a class='existingWikiWord' href='/nlab/show/over-%28infinity%2C1%29-… |
|
5745 </li> |
|
5746 |
|
5747 <li> |
|
5748 <p><a class='existingWikiWord' href='/nlab/show/n-localic+%28infinity%2C… |
|
5749 </li> |
|
5750 |
|
5751 <li> |
|
5752 <p><a class='existingWikiWord' href='/nlab/show/locally+n-connected+%28n… |
|
5753 </li> |
|
5754 |
|
5755 <li> |
|
5756 <p><a class='existingWikiWord' href='/nlab/show/structured+%28infinity%2… |
|
5757 |
|
5758 <ul> |
|
5759 <li><a class='existingWikiWord' href='/nlab/show/geometry+%28for+structu… |
|
5760 </ul> |
|
5761 </li> |
|
5762 |
|
5763 <li> |
|
5764 <p><a class='existingWikiWord' href='/nlab/show/locally+n-connected+%28n… |
|
5765 </li> |
|
5766 |
|
5767 <li> |
|
5768 <p><a class='existingWikiWord' href='/nlab/show/%28%E2%88%9E%2C1%29-loca… |
|
5769 |
|
5770 <ul> |
|
5771 <li><a class='existingWikiWord' href='/nlab/show/concrete+%28infinity%2C… |
|
5772 </ul> |
|
5773 </li> |
|
5774 |
|
5775 <li> |
|
5776 <p><a class='existingWikiWord' href='/nlab/show/cohesive+%28infinity%2C1… |
|
5777 </li> |
|
5778 </ul> |
|
5779 |
|
5780 <h2 id='models'>Models</h2> |
|
5781 |
|
5782 <ul> |
|
5783 <li> |
|
5784 <p><a class='existingWikiWord' href='/nlab/show/presentations+of+%28infi… |
|
5785 |
|
5786 <ul> |
|
5787 <li> |
|
5788 <p><a class='existingWikiWord' href='/nlab/show/model+category'>model ca… |
|
5789 </li> |
|
5790 |
|
5791 <li> |
|
5792 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+funct… |
|
5793 </li> |
|
5794 |
|
5795 <li> |
|
5796 <p><a class='existingWikiWord' href='/nlab/show/model+site'>model site</… |
|
5797 </li> |
|
5798 |
|
5799 <li> |
|
5800 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+simpl… |
|
5801 </li> |
|
5802 |
|
5803 <li> |
|
5804 <p><a class='existingWikiWord' href='/nlab/show/descent+for+simplicial+p… |
|
5805 </li> |
|
5806 |
|
5807 <li> |
|
5808 <p><a class='existingWikiWord' href='/nlab/show/Verity+on+descent+for+st… |
|
5809 </li> |
|
5810 </ul> |
|
5811 </li> |
|
5812 </ul> |
|
5813 |
|
5814 <h2 id='constructions'>Constructions</h2> |
|
5815 |
|
5816 <p><strong>structures in a <a class='existingWikiWord' href='/nlab/show/… |
|
5817 |
|
5818 <ul> |
|
5819 <li> |
|
5820 <p><a class='existingWikiWord' href='/nlab/show/shape+of+an+%28infinity%… |
|
5821 </li> |
|
5822 |
|
5823 <li> |
|
5824 <p><a class='existingWikiWord' href='/nlab/show/cohomology'>cohomology</… |
|
5825 </li> |
|
5826 |
|
5827 <li> |
|
5828 <p><a class='existingWikiWord' href='/nlab/show/homotopy+groups+in+an+%2… |
|
5829 |
|
5830 <ul> |
|
5831 <li> |
|
5832 <p><a class='existingWikiWord' href='/nlab/show/fundamental+infinity-gro… |
|
5833 </li> |
|
5834 |
|
5835 <li> |
|
5836 <p><a class='existingWikiWord' href='/nlab/show/categorical+homotopy+gro… |
|
5837 </li> |
|
5838 |
|
5839 <li> |
|
5840 <p><a class='existingWikiWord' href='/nlab/show/Postnikov+tower+in+an+%2… |
|
5841 </li> |
|
5842 |
|
5843 <li> |
|
5844 <p><a class='existingWikiWord' href='/nlab/show/Whitehead+tower+in+an+%2… |
|
5845 </li> |
|
5846 </ul> |
|
5847 </li> |
|
5848 |
|
5849 <li> |
|
5850 <p><a class='existingWikiWord' href='/nlab/show/function+algebras+on+inf… |
|
5851 </li> |
|
5852 |
|
5853 <li> |
|
5854 <p><a class='existingWikiWord' href='/nlab/show/dimension'>dimension</a>… |
|
5855 |
|
5856 <ul> |
|
5857 <li> |
|
5858 <p><a class='existingWikiWord' href='/nlab/show/homotopy+dimension'>homo… |
|
5859 </li> |
|
5860 |
|
5861 <li> |
|
5862 <p><a class='existingWikiWord' href='/nlab/show/cohomological+dimension'… |
|
5863 </li> |
|
5864 |
|
5865 <li> |
|
5866 <p><a class='existingWikiWord' href='/nlab/show/covering+dimension'>cove… |
|
5867 </li> |
|
5868 |
|
5869 <li> |
|
5870 <p><a class='existingWikiWord' href='/nlab/show/Heyting+dimension'>Heyti… |
|
5871 </li> |
|
5872 </ul> |
|
5873 </li> |
|
5874 </ul> |
|
5875 <div> |
|
5876 <p> |
|
5877 <a href='/nlab/edit/%28infinity%2C1%29-topos+-+contents'>Edit this sid… |
|
5878 </p> |
|
5879 </div></div> |
|
5880 </div> |
|
5881 </div> |
|
5882 |
|
5883 <h1 id='contents'>Contents</h1> |
|
5884 <div class='maruku_toc'><ul><li><a href='#definition'>Definition</a></li… |
|
5885 <h2 id='definition'>Definition</h2> |
|
5886 |
|
5887 <p><em>Simplicial presheaves</em> over some <a class='existingWikiWord' … |
|
5888 |
|
5889 <ul> |
|
5890 <li><a class='existingWikiWord' href='/nlab/show/presheaf'>Presheaves</a… |
|
5891 </ul> |
|
5892 |
|
5893 <p>or equivalently, using the Hom-<a class='existingWikiWord' href='/nla… |
|
5894 |
|
5895 <ul> |
|
5896 <li>simplicial objects in the category of presheaves, i.e. functors <mat… |
|
5897 </ul> |
|
5898 |
|
5899 <h2 id='interpretation_as_stacks'>Interpretation as <math class='maruku-… |
|
5900 |
|
5901 <p>Regarding <math class='maruku-mathml' display='inline' id='mathml_ebe… |
|
5902 |
|
5903 <p>In more illustrative language this means that a simplicial presheaf o… |
|
5904 |
|
5905 <h2 id='examples'>Examples</h2> |
|
5906 |
|
5907 <ul> |
|
5908 <li> |
|
5909 <p>Notice that most definitions of <math class='maruku-mathml' display='… |
|
5910 </li> |
|
5911 |
|
5912 <li> |
|
5913 <p>In particular, regarding a <a class='existingWikiWord' href='/nlab/sh… |
|
5914 </li> |
|
5915 </ul> |
|
5916 |
|
5917 <h2 id='remarks'>Remarks</h2> |
|
5918 |
|
5919 <ul> |
|
5920 <li>There are various useful <a class='existingWikiWord' href='/nlab/sho… |
|
5921 </ul> |
|
5922 |
|
5923 <h2 id='properties'>Properties</h2> |
|
5924 |
|
5925 <p>Here are some basic but useful facts about simplicial presheaves.</p> |
|
5926 |
|
5927 <div class='un_prop'> |
|
5928 <h6 id='proposition'>Proposition</h6> |
|
5929 |
|
5930 <p>Every simplicial presheaf <math class='maruku-mathml' display='inline… |
|
5931 |
|
5932 <p>More precisely, for <math class='maruku-mathml' display='inline' id='… |
|
5933 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
5934 hocolim_{[n] \in \Delta} D_X([n]) \stackrel{\simeq}{\to} X |
|
5935 \,. |
|
5936 |
|
5937 </annotation></semantics></math></div></div> |
|
5938 |
|
5939 <div class='proof'> |
|
5940 <h6 id='proof'>Proof</h6> |
|
5941 |
|
5942 <p>See for instance <a href='http://www.math.uiuc.edu/K-theory/0563/spre… |
|
5943 |
|
5944 <ul> |
|
5945 <li><a class='existingWikiWord' href='/nlab/show/Daniel+Dugger'>Daniel D… |
|
5946 </ul> |
|
5947 |
|
5948 <p>(which is otherwise about <a class='existingWikiWord' href='/nlab/sho… |
|
5949 </div> |
|
5950 |
|
5951 <div class='un_cor'> |
|
5952 <h6 id='corollary'>Corollary</h6> |
|
5953 |
|
5954 <p>Let <math class='maruku-mathml' display='inline' id='mathml_ebe0afceb… |
|
5955 |
|
5956 <p>It follows in particular from the above that every such <a class='exi… |
|
5957 </div> |
|
5958 |
|
5959 <div class='proof'> |
|
5960 <h6 id='proof_2'>Proof</h6> |
|
5961 |
|
5962 <p>First the above yields</p> |
|
5963 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
5964 \begin{aligned} |
|
5965 [X, A ] & \simeq [ hocolim_{[n] \in \Delta} X_n , A ] |
|
5966 \\ |
|
5967 & holim_{[n] \in \Delta} [X_n, A] |
|
5968 \end{aligned} |
|
5969 \,. |
|
5970 |
|
5971 </annotation></semantics></math></div> |
|
5972 <p>Next from the <a class='existingWikiWord' href='/nlab/show/co-Yoneda+… |
|
5973 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
5974 \begin{aligned} |
|
5975 \cdots & \simeq |
|
5976 holim_{[n] \in \Delta} |
|
5977 [ colim_i U_{i}, A] |
|
5978 \\ |
|
5979 & \simeq |
|
5980 holim_{[n] \in \Delta} lim_i |
|
5981 [ U_{i}, A] |
|
5982 \end{aligned} |
|
5983 \,. |
|
5984 |
|
5985 </annotation></semantics></math></div> |
|
5986 <p>And finally the <a class='existingWikiWord' href='/nlab/show/Yoneda+l… |
|
5987 <div class='maruku-equation'><math class='maruku-mathml' display='block'… |
|
5988 \begin{aligned} |
|
5989 \cdots |
|
5990 & |
|
5991 holim_{[n] \in \Delta} lim_i |
|
5992 A(U_i) |
|
5993 \end{aligned} |
|
5994 \,. |
|
5995 |
|
5996 </annotation></semantics></math></div></div> |
|
5997 |
|
5998 <p>Notice that these kinds of computations are in particular often used … |
|
5999 |
|
6000 <h2 id='related_entries'>Related entries</h2> |
|
6001 |
|
6002 <ul> |
|
6003 <li> |
|
6004 <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+simpl… |
|
6005 </li> |
|
6006 |
|
6007 <li> |
|
6008 <p><a class='existingWikiWord' href='/nlab/show/descent+for+simplicial+p… |
|
6009 </li> |
|
6010 |
|
6011 <li> |
|
6012 <p><a class='existingWikiWord' href='/nlab/show/sheaf+of+spectra'>preshe… |
|
6013 </li> |
|
6014 </ul> |
|
6015 |
|
6016 <p>Applications appear for instance at</p> |
|
6017 |
|
6018 <ul> |
|
6019 <li><a class='existingWikiWord' href='/nlab/show/geometric+infinity-func… |
|
6020 </ul> |
|
6021 |
|
6022 <h2 id='references'>References</h2> |
|
6023 |
|
6024 <p>The original articles are</p> |
|
6025 |
|
6026 <ul> |
|
6027 <li> |
|
6028 <p><a class='existingWikiWord' href='/nlab/show/Kenneth+Brown'>Kenneth S… |
|
6029 </li> |
|
6030 |
|
6031 <li> |
|
6032 <p><a class='existingWikiWord' href='/nlab/show/Kenneth+Brown'>Kenneth S… |
|
6033 </li> |
|
6034 |
|
6035 <li> |
|
6036 <p><a class='existingWikiWord' href='/nlab/show/John+Frederick+Jardine'>… |
|
6037 </li> |
|
6038 |
|
6039 <li> |
|
6040 <p><a class='existingWikiWord' href='/nlab/show/John+Frederick+Jardine'>… |
|
6041 </li> |
|
6042 </ul> |
|
6043 |
|
6044 <p>A modern expository account is</p> |
|
6045 |
|
6046 <ul> |
|
6047 <li><a class='existingWikiWord' href='/nlab/show/John+Frederick+Jardine'… |
|
6048 </ul> |
|
6049 |
|
6050 <p>Further articles:</p> |
|
6051 |
|
6052 <ul> |
|
6053 <li> |
|
6054 <p><a class='existingWikiWord' href='/nlab/show/John+Frederick+Jardine'>… |
|
6055 </li> |
|
6056 |
|
6057 <li> |
|
6058 <p><a class='existingWikiWord' href='/nlab/show/John+Frederick+Jardine'>… |
|
6059 </li> |
|
6060 </ul> |
|
6061 |
|
6062 <p>For their interpretation in the more general context of <a class='exi… |
|
6063 |
|
6064 <ul> |
|
6065 <li><a class='existingWikiWord' href='/nlab/show/Jacob+Lurie'>Jacob Luri… |
|
6066 </ul> |
|
6067 |
|
6068 <p> |
|
6069 </p> |
|
6070 |
|
6071 <p> |
|
6072 |
|
6073 </p> |
|
6074 |
|
6075 <p> |
|
6076 </p> </div> |
|
6077 </content> |
|
6078 </entry> |
|
6079 <entry> |
|
6080 <title type="html">Milky Way</title> |
|
6081 <link rel="alternate" type="application/xhtml+xml" href="https://nca… |
|
6082 <updated>2021-07-01T10:22:24Z</updated> |
|
6083 <published>2019-04-10T13:55:53Z</published> |
|
6084 <id>tag:ncatlab.org,2019-04-10:nLab,Milky+Way</id> |
|
6085 <author> |
|
6086 <name>Urs Schreiber</name> |
|
6087 </author> |
|
6088 <content type="xhtml" xml:base="https://ncatlab.org/nlab/show/Milky+… |
|
6089 <div xmlns="http://www.w3.org/1999/xhtml"> |
|
6090 <div class='rightHandSide'> |
|
6091 <div class='toc clickDown' tabindex='0'> |
|
6092 <h3 id='context'>Context</h3> |
|
6093 |
|
6094 <h4 id='physics'>Physics</h4> |
|
6095 |
|
6096 <div class='hide'> |
|
6097 <p><strong><a class='existingWikiWord' href='/nlab/show/physics'>physics… |
|
6098 |
|
6099 <h2 id='surveys_textbooks_and_lecture_notes'>Surveys, textbooks and lect… |
|
6100 |
|
6101 <ul> |
|
6102 <li> |
|
6103 <p><em><a class='existingWikiWord' href='/nlab/show/higher+category+theo… |
|
6104 </li> |
|
6105 |
|
6106 <li> |
|
6107 <p><em><a class='existingWikiWord' href='/nlab/show/geometry+of+physics'… |
|
6108 </li> |
|
6109 |
|
6110 <li> |
|
6111 <p><a class='existingWikiWord' href='/nlab/show/books+and+reviews+in+mat… |
|
6112 </li> |
|
6113 </ul> |
|
6114 <hr/> |
|
6115 <p><a class='existingWikiWord' href='/nlab/show/theory+%28physics%29'>th… |
|
6116 |
|
6117 <p><a class='existingWikiWord' href='/nlab/show/experimental+observation… |
|
6118 |
|
6119 <ul> |
|
6120 <li> |
|
6121 <p><strong><a class='existingWikiWord' href='/nlab/show/mechanics'>mecha… |
|
6122 |
|
6123 <ul> |
|
6124 <li> |
|
6125 <p><a class='existingWikiWord' href='/nlab/show/mass'>mass</a>, <a class… |
|
6126 </li> |
|
6127 |
|
6128 <li> |
|
6129 <p><a class='existingWikiWord' href='/nlab/show/Hamiltonian+dynamics+on+… |
|
6130 |
|
6131 <ul> |
|
6132 <li><a class='existingWikiWord' href='/nlab/show/rigid+body+dynamics'>ri… |
|
6133 </ul> |
|
6134 </li> |
|
6135 </ul> |
|
6136 </li> |
|
6137 |
|
6138 <li> |
|
6139 <p><a class='existingWikiWord' href='/nlab/show/field+%28physics%29'>fie… |
|
6140 |
|
6141 <ul> |
|
6142 <li> |
|
6143 <p><a class='existingWikiWord' href='/nlab/show/Lagrangian+density'>Lagr… |
|
6144 |
|
6145 <ul> |
|
6146 <li> |
|
6147 <p><a class='existingWikiWord' href='/nlab/show/configuration+space'>con… |
|
6148 </li> |
|
6149 |
|
6150 <li> |
|
6151 <p><a class='existingWikiWord' href='/nlab/show/action+functional'>actio… |
|
6152 </li> |
|
6153 |
|
6154 <li> |
|
6155 <p><a class='existingWikiWord' href='/nlab/show/phase+space'>covariant p… |
|
6156 </li> |
|
6157 </ul> |
|
6158 </li> |
|
6159 |
|
6160 <li> |
|
6161 <p><a class='existingWikiWord' href='/nlab/show/Hamiltonian+mechanics'>H… |
|
6162 |
|
6163 <ul> |
|
6164 <li> |
|
6165 <p><a class='existingWikiWord' href='/nlab/show/phase+space'>phase space… |
|
6166 </li> |
|
6167 |
|
6168 <li> |
|
6169 <p><a class='existingWikiWord' href='/nlab/show/symplectic+geometry'>sym… |
|
6170 |
|
6171 <ul> |
|
6172 <li> |
|
6173 <p><a class='existingWikiWord' href='/nlab/show/Poisson+manifold'>Poisso… |
|
6174 </li> |
|
6175 |
|
6176 <li> |
|
6177 <p><a class='existingWikiWord' href='/nlab/show/symplectic+manifold'>sym… |
|
6178 </li> |
|
6179 |
|
6180 <li> |
|
6181 <p><a class='existingWikiWord' href='/nlab/show/symplectic+groupoid'>sym… |
|
6182 </li> |
|
6183 </ul> |
|
6184 </li> |
|
6185 |
|
6186 <li> |
|
6187 <p><a class='existingWikiWord' href='/nlab/show/multisymplectic+geometry… |
|
6188 |
|
6189 <ul> |
|
6190 <li><a class='existingWikiWord' href='/nlab/show/symplectic+Lie+n-algebr… |
|
6191 </ul> |
|
6192 </li> |
|
6193 </ul> |
|
6194 </li> |
|
6195 |
|
6196 <li> |
|
6197 <p><a class='existingWikiWord' href='/nlab/show/spacetime'>spacetime</a>… |
|
6198 |
|
6199 <ul> |
|
6200 <li> |
|
6201 <p><a class='existingWikiWord' href='/nlab/show/smooth+Lorentzian+space'… |
|
6202 </li> |
|
6203 |
|
6204 <li> |
|
6205 <p><a class='existingWikiWord' href='/nlab/show/special+relativity'>spec… |
|
6206 </li> |
|
6207 |
|
6208 <li> |
|
6209 <p><a class='existingWikiWord' href='/nlab/show/general+relativity'>gene… |
|
6210 </li> |
|
6211 |
|
6212 <li> |
|
6213 <p><a class='existingWikiWord' href='/nlab/show/gravity'>gravity</a></p> |
|
6214 |
|
6215 <ul> |
|
6216 <li> |
|
6217 <p><a class='existingWikiWord' href='/nlab/show/supergravity'>supergravi… |
|
6218 </li> |
|
6219 |
|
6220 <li> |
|
6221 <p><a class='existingWikiWord' href='/nlab/show/black+hole'>black hole</… |
|
6222 </li> |
|
6223 </ul> |
|
6224 </li> |
|
6225 </ul> |
|
6226 </li> |
|
6227 </ul> |
|
6228 </li> |
|
6229 |
|
6230 <li> |
|
6231 <p><strong><a class='existingWikiWord' href='/nlab/show/classical+field+… |
|
6232 |
|
6233 <ul> |
|
6234 <li> |
|
6235 <p><a class='existingWikiWord' href='/nlab/show/classical+physics'>class… |
|
6236 |
|
6237 <ul> |
|
6238 <li><a class='existingWikiWord' href='/nlab/show/classical+mechanics'>cl… |
|
6239 |
|
6240 <li><a class='existingWikiWord' href='/nlab/show/wave'>waves</a> and <a … |
|
6241 |
|
6242 <li><a class='existingWikiWord' href='/nlab/show/thermodynamics'>thermod… |
|
6243 </ul> |
|
6244 </li> |
|
6245 </ul> |
|
6246 </li> |
|
6247 |
|
6248 <li> |
|
6249 <p><strong><a class='existingWikiWord' href='/nlab/show/quantum+mechanic… |
|
6250 |
|
6251 <ul> |
|
6252 <li> |
|
6253 <p><a class='existingWikiWord' href='/nlab/show/finite+quantum+mechanics… |
|
6254 </li> |
|
6255 |
|
6256 <li> |
|
6257 <p><a class='existingWikiWord' href='/nlab/show/quantum+information'>qua… |
|
6258 </li> |
|
6259 |
|
6260 <li> |
|
6261 <p><a class='existingWikiWord' href='/nlab/show/Hamiltonian'>Hamiltonian… |
|
6262 </li> |
|
6263 |
|
6264 <li> |
|
6265 <p><a class='existingWikiWord' href='/nlab/show/density+matrix'>density … |
|
6266 </li> |
|
6267 |
|
6268 <li> |
|
6269 <p><a class='existingWikiWord' href='/nlab/show/Kochen-Specker+theorem'>… |
|
6270 </li> |
|
6271 |
|
6272 <li> |
|
6273 <p><a class='existingWikiWord' href='/nlab/show/Bell%27s+theorem'>Bell&#… |
|
6274 </li> |
|
6275 |
|
6276 <li> |
|
6277 <p><a class='existingWikiWord' href='/nlab/show/Gleason%27s+theorem'>Gle… |
|
6278 </li> |
|
6279 </ul> |
|
6280 </li> |
|
6281 |
|
6282 <li> |
|
6283 <p><strong><a class='existingWikiWord' href='/nlab/show/quantization'>Qu… |
|
6284 |
|
6285 <ul> |
|
6286 <li> |
|
6287 <p><a class='existingWikiWord' href='/nlab/show/geometric+quantization'>… |
|
6288 </li> |
|
6289 |
|
6290 <li> |
|
6291 <p><a class='existingWikiWord' href='/nlab/show/deformation+quantization… |
|
6292 </li> |
|
6293 |
|
6294 <li> |
|
6295 <p><a class='existingWikiWord' href='/nlab/show/path+integral'>path inte… |
|
6296 </li> |
|
6297 |
|
6298 <li> |
|
6299 <p><a class='existingWikiWord' href='/nlab/show/semiclassical+approximat… |
|
6300 </li> |
|
6301 </ul> |
|
6302 </li> |
|
6303 |
|
6304 <li> |
|
6305 <p><strong><a class='existingWikiWord' href='/nlab/show/quantum+field+th… |
|
6306 |
|
6307 <ul> |
|
6308 <li> |
|
6309 <p>Axiomatizations</p> |
|
6310 |
|
6311 <ul> |
|
6312 <li> |
|
6313 <p><a class='existingWikiWord' href='/nlab/show/AQFT'>algebraic QFT</a><… |
|
6314 |
|
6315 <ul> |
|
6316 <li> |
|
6317 <p><a class='existingWikiWord' href='/nlab/show/Wightman+axioms'>Wightma… |
|
6318 </li> |
|
6319 |
|
6320 <li> |
|
6321 <p><a class='existingWikiWord' href='/nlab/show/Haag-Kastler+axioms'>Haa… |
|
6322 |
|
6323 <ul> |
|
6324 <li> |
|
6325 <p><a class='existingWikiWord' href='/nlab/show/operator+algebra'>operat… |
|
6326 </li> |
|
6327 |
|
6328 <li> |
|
6329 <p><a class='existingWikiWord' href='/nlab/show/causally+local+net+of+ob… |
|
6330 </li> |
|
6331 </ul> |
|
6332 </li> |
|
6333 |
|
6334 <li> |
|
6335 <p><a class='existingWikiWord' href='/nlab/show/conformal+net'>conformal… |
|
6336 </li> |
|
6337 |
|
6338 <li> |
|
6339 <p><a class='existingWikiWord' href='/nlab/show/Reeh-Schlieder+theorem'>… |
|
6340 </li> |
|
6341 |
|
6342 <li> |
|
6343 <p><a class='existingWikiWord' href='/nlab/show/Osterwalder-Schrader+the… |
|
6344 </li> |
|
6345 |
|
6346 <li> |
|
6347 <p><a class='existingWikiWord' href='/nlab/show/PCT+theorem'>PCT theorem… |
|
6348 </li> |
|
6349 |
|
6350 <li> |
|
6351 <p><a class='existingWikiWord' href='/nlab/show/Bisognano-Wichmann+theor… |
|
6352 |
|
6353 <ul> |
|
6354 <li><a class='existingWikiWord' href='/nlab/show/modular+theory'>modular… |
|
6355 </ul> |
|
6356 </li> |
|
6357 |
|
6358 <li> |
|
6359 <p><a class='existingWikiWord' href='/nlab/show/spin-statistics+theorem'… |
|
6360 |
|
6361 <ul> |
|
6362 <li><a class='existingWikiWord' href='/nlab/show/boson'>boson</a>, <a cl… |
|
6363 </ul> |
|
6364 </li> |
|
6365 </ul> |
|
6366 </li> |
|
6367 |
|
6368 <li> |
|
6369 <p><a class='existingWikiWord' href='/nlab/show/functorial+field+theory'… |
|
6370 |
|
6371 <ul> |
|
6372 <li> |
|
6373 <p><a class='existingWikiWord' href='/nlab/show/cobordism'>cobordism</a>… |
|
6374 </li> |
|
6375 |
|
6376 <li> |
|
6377 <p><a class='existingWikiWord' href='/nlab/show/%28infinity%2Cn%29-categ… |
|
6378 </li> |
|
6379 |
|
6380 <li> |
|
6381 <p><a class='existingWikiWord' href='/nlab/show/cobordism+hypothesis'>co… |
|
6382 </li> |
|
6383 |
|
6384 <li> |
|
6385 <p><a class='existingWikiWord' href='/nlab/show/extended+topological+qua… |
|
6386 </li> |
|
6387 </ul> |
|
6388 </li> |
|
6389 </ul> |
|
6390 </li> |
|
6391 |
|
6392 <li> |
|
6393 <p>Tools</p> |
|
6394 |
|
6395 <ul> |
|
6396 <li> |
|
6397 <p><a class='existingWikiWord' href='/nlab/show/perturbative+quantum+fie… |
|
6398 </li> |
|
6399 |
|
6400 <li> |
|
6401 <p><a class='existingWikiWord' href='/nlab/show/effective+quantum+field+… |
|
6402 </li> |
|
6403 |
|
6404 <li> |
|
6405 <p><a class='existingWikiWord' href='/nlab/show/renormalization'>renorma… |
|
6406 </li> |
|
6407 |
|
6408 <li> |
|
6409 <p><a class='existingWikiWord' href='/nlab/show/BV-BRST+formalism'>BV-BR… |
|
6410 </li> |
|
6411 |
|
6412 <li> |
|
6413 <p><a class='existingWikiWord' href='/nlab/show/geometric+infinity-funct… |
|
6414 </li> |
|
6415 </ul> |
|
6416 </li> |
|
6417 |
|
6418 <li> |
|
6419 <p><a class='existingWikiWord' href='/nlab/show/particle+physics'>partic… |
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6420 |
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6421 <ul> |
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6422 <li> |
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6423 <p><a class='existingWikiWord' href='/nlab/show/phenomenology'>phenomeno… |
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6424 </li> |
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6425 |
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6426 <li> |
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6427 <p><a class='existingWikiWord' href='/nlab/show/model+%28in+theoretical+… |
|
6428 |
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6429 <ul> |
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6430 <li> |
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6431 <p><a class='existingWikiWord' href='/nlab/show/standard+model+of+partic… |
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6432 </li> |
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6433 |
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6434 <li> |
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6435 <p><a class='existingWikiWord' href='/nlab/show/fields+and+quanta+-+tabl… |
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6436 </li> |
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6437 |
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6438 <li> |
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6439 <p><a class='existingWikiWord' href='/nlab/show/GUT'>Grand Unified Theor… |
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6440 </li> |
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6441 </ul> |
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6442 </li> |
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6443 |
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6444 <li> |
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6445 <p><a class='existingWikiWord' href='/nlab/show/scattering+amplitude'>sc… |
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6446 |
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6447 <ul> |
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6448 <li><a class='existingWikiWord' href='/nlab/show/on-shell+recursion'>on-… |
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6449 </ul> |
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6450 </li> |
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6451 </ul> |
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6452 </li> |
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6453 |
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6454 <li> |
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6455 <p>Structural phenomena</p> |
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6456 |
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6457 <ul> |
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6458 <li> |
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6459 <p><a class='existingWikiWord' href='/nlab/show/universality+class'>univ… |
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6460 </li> |
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6461 |
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6462 <li> |
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6463 <p><a class='existingWikiWord' href='/nlab/show/quantum+anomaly'>quantum… |
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6464 |
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6465 <ul> |
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6466 <li><a class='existingWikiWord' href='/nlab/show/Green-Schwarz+mechanism… |
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6467 </ul> |
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6468 </li> |
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6469 |
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6470 <li> |
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6471 <p><a class='existingWikiWord' href='/nlab/show/instanton'>instanton</a>… |
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6472 </li> |
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6473 |
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6474 <li> |
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6475 <p><a class='existingWikiWord' href='/nlab/show/spontaneously+broken+sym… |
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6476 </li> |
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6477 |
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6478 <li> |
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6479 <p><a class='existingWikiWord' href='/nlab/show/Kaluza-Klein+mechanism'>… |
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6480 </li> |
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6481 |
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6482 <li> |
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6483 <p><a class='existingWikiWord' href='/nlab/show/integrable+system'>integ… |
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6484 </li> |
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6485 |
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6486 <li> |
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6487 <p><a class='existingWikiWord' href='/nlab/show/holonomic+quantum+field'… |
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6488 </li> |
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6489 </ul> |
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6490 </li> |
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6491 |
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6492 <li> |
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6493 <p>Types of quantum field thories</p> |
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6494 |
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6495 <ul> |
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6496 <li> |
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6497 <p><a class='existingWikiWord' href='/nlab/show/topological+quantum+fiel… |
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6498 |
|
6499 <ul> |
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6500 <li> |
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6501 <p><a class='existingWikiWord' href='/nlab/show/2d+TQFT'>2d TQFT</a></p> |
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6502 </li> |
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6503 |
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6504 <li> |
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6505 <p><a class='existingWikiWord' href='/nlab/show/Dijkgraaf-Witten+theory'… |
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6506 </li> |
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6507 |
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6508 <li> |
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6509 <p><a class='existingWikiWord' href='/nlab/show/Chern-Simons+theory'>Che… |
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6510 </li> |
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6511 </ul> |
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6512 </li> |
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6513 |
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6514 <li> |
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6515 <p><a class='existingWikiWord' href='/nlab/show/TCFT'>TCFT</a></p> |
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6516 |
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6517 <ul> |
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6518 <li> |
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6519 <p><a class='existingWikiWord' href='/nlab/show/A-model'>A-model</a>, <a… |
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6520 </li> |
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6521 |
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6522 <li> |
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6523 <p><a class='existingWikiWord' href='/nlab/show/mirror+symmetry'>homolog… |
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6524 </li> |
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6525 </ul> |
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6526 </li> |
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6527 |
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6528 <li> |
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6529 <p><a class='existingWikiWord' href='/nlab/show/QFT+with+defects'>QFT wi… |
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6530 </li> |
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6531 |
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6532 <li> |
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6533 <p><a class='existingWikiWord' href='/nlab/show/conformal+field+theory'>… |
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6534 </li> |
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6535 |
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6536 <li> |
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6537 <p><a class='existingWikiWord' href='/nlab/show/%281%2C1%29-dimensional+… |
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6538 </li> |
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6539 |
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6540 <li> |
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6541 <p><a class='existingWikiWord' href='/nlab/show/%282%2C1%29-dimensional+… |
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6542 </li> |
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6543 |
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6544 <li> |
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6545 <p><a class='existingWikiWord' href='/nlab/show/conformal+field+theory'>… |
|
6546 |
|
6547 <ul> |
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6548 <li> |
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6549 <p><a class='existingWikiWord' href='/nlab/show/Wess-Zumino-Witten+model… |
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6550 </li> |
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6551 |
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6552 <li> |
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6553 <p><a class='existingWikiWord' href='/nlab/show/D%3D6+N%3D%282%2C0%29+SC… |
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6554 </li> |
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6555 </ul> |
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6556 </li> |
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6557 |
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6558 <li> |
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6559 <p><a class='existingWikiWord' href='/nlab/show/gauge+theory'>gauge theo… |
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6560 |
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6561 <ul> |
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6562 <li> |
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6563 <p><a class='existingWikiWord' href='/nlab/show/field+strength'>field st… |
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6564 </li> |
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6565 |
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6566 <li> |
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6567 <p><a class='existingWikiWord' href='/nlab/show/gauge+group'>gauge group… |
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6568 </li> |
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6569 |
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6570 <li> |
|
6571 <p>examples</p> |
|
6572 |
|
6573 <ul> |
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6574 <li><a class='existingWikiWord' href='/nlab/show/electromagnetic+field'>… |
|
6575 </ul> |
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6576 </li> |
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6577 |
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6578 <li> |
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6579 <p><a class='existingWikiWord' href='/nlab/show/electric+charge'>electri… |
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6580 </li> |
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6581 |
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6582 <li> |
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6583 <p><a class='existingWikiWord' href='/nlab/show/magnetic+charge'>magneti… |
|
6584 |
|
6585 <ul> |
|
6586 <li><a class='existingWikiWord' href='/nlab/show/Yang-Mills+field'>Yang-… |
|
6587 </ul> |
|
6588 </li> |
|
6589 |
|
6590 <li> |
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6591 <p><a class='existingWikiWord' href='/nlab/show/Yang-Mills+theory'>Yang-… |
|
6592 </li> |
|
6593 |
|
6594 <li> |
|
6595 <p><a class='existingWikiWord' href='/nlab/show/The+Dirac+Electron'>spin… |
|
6596 </li> |
|
6597 |
|
6598 <li> |
|
6599 <p><a class='existingWikiWord' href='/nlab/show/topological+Yang-Mills+t… |
|
6600 |
|
6601 <ul> |
|
6602 <li><a class='existingWikiWord' href='/nlab/show/Kalb-Ramond+field'>Kalb… |
|
6603 |
|
6604 <li><a class='existingWikiWord' href='/nlab/show/supergravity+C-field'>s… |
|
6605 |
|
6606 <li><a class='existingWikiWord' href='/nlab/show/RR+field'>RR field</a><… |
|
6607 |
|
6608 <li><a class='existingWikiWord' href='/nlab/show/first-order+formulation… |
|
6609 </ul> |
|
6610 </li> |
|
6611 |
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6612 <li> |
|
6613 <p><a class='existingWikiWord' href='/nlab/show/general+covariance'>gene… |
|
6614 </li> |
|
6615 |
|
6616 <li> |
|
6617 <p><a class='existingWikiWord' href='/nlab/show/supergravity'>supergravi… |
|
6618 </li> |
|
6619 |
|
6620 <li> |
|
6621 <p><a class='existingWikiWord' href='/nlab/show/D%27Auria-Fre+formulatio… |
|
6622 </li> |
|
6623 |
|
6624 <li> |
|
6625 <p><a class='existingWikiWord' href='/nlab/show/gravity+as+a+BF+theory'>… |
|
6626 </li> |
|
6627 </ul> |
|
6628 </li> |
|
6629 |
|
6630 <li> |
|
6631 <p><a class='existingWikiWord' href='/nlab/show/sigma-model'>sigma-model… |
|
6632 |
|
6633 <ul> |
|
6634 <li> |
|
6635 <p><a class='existingWikiWord' href='/nlab/show/particle'>particle</a>, … |
|
6636 </li> |
|
6637 |
|
6638 <li> |
|
6639 <p><a class='existingWikiWord' href='/nlab/show/string'>string</a>, <a c… |
|
6640 </li> |
|
6641 |
|
6642 <li> |
|
6643 <p><a class='existingWikiWord' href='/nlab/show/membrane'>membrane</a></… |
|
6644 </li> |
|
6645 |
|
6646 <li> |
|
6647 <p><a class='existingWikiWord' href='/nlab/show/AKSZ+sigma-model'>AKSZ t… |
|
6648 </li> |
|
6649 </ul> |
|
6650 </li> |
|
6651 </ul> |
|
6652 </li> |
|
6653 </ul> |
|
6654 </li> |
|
6655 |
|
6656 <li> |
|
6657 <p><a class='existingWikiWord' href='/nlab/show/string+theory'>String Th… |
|
6658 |
|
6659 <ul> |
|
6660 <li><a class='existingWikiWord' href='/nlab/show/string+theory+results+a… |
|
6661 </ul> |
|
6662 </li> |
|
6663 |
|
6664 <li> |
|
6665 <p><a class='existingWikiWord' href='/nlab/show/number+theory+and+physic… |
|
6666 |
|
6667 <ul> |
|
6668 <li><a class='existingWikiWord' href='/nlab/show/Riemann+hypothesis+and+… |
|
6669 </ul> |
|
6670 </li> |
|
6671 </ul> |
|
6672 <div> |
|
6673 <p> |
|
6674 <a href='/nlab/edit/physicscontents'>Edit this sidebar</a> |
|
6675 </p> |
|
6676 </div></div> |
|
6677 </div> |
|
6678 </div> |
|
6679 |
|
6680 <h1 id='contents'>Contents</h1> |
|
6681 <div class='maruku_toc'><ul><li><a href='#idea'>Idea</a></li><li><a href… |
|
6682 <h2 id='idea'>Idea</h2> |
|
6683 |
|
6684 <p>Our <a class='existingWikiWord' href='/nlab/show/galaxy'>galaxy</a>.<… |
|
6685 |
|
6686 <h2 id='references'>References</h2> |
|
6687 |
|
6688 <ul> |
|
6689 <li>Susan Gardner, Samuel D. McDermott, Brian Yanny, <em>The Milky Way, … |
|
6690 </ul> |
|
6691 |
|
6692 <p>See also</p> |
|
6693 |
|
6694 <ul> |
|
6695 <li>Wikipedia, <em><a href='https://en.wikipedia.org/wiki/Milky_Way'>Mil… |
|
6696 </ul> </div> |
|
6697 </content> |
|
6698 </entry> |
|
6699 </feed> |
