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Classifying topos
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<h2 id="examples">Examples</h2> <ul><li>The classifying topos for objects of a topos is the topos of <a href="/facts/Presheaf/MM3hcRw4">presheaves</a> over the opposite of the category of finite sets.</li> <li>The classifying topos for rings of a topos is the topos of presheaves over the opposite of the category of finitely presented rings.</li> <li>The classifying topos for local rings of a topos is the topos of sheaves over the opposite of the category of finitely presented rings with the <a href="/facts/Zariski_topology/uolkKBpM">Zariski topology</a>.</li> <li>The classifying topos for <a href="/facts/Linear_order/xnTtmuYJ">linear orders</a> with distinct largest and smallest elements of a topos is the topos of <a href="/facts/Simplicial_set/6DaqXMLT">simplicial sets</a>.</li> <li>If <i>G</i> is a <a href="/facts/Discrete_group/HIjkpJhS">discrete group</a>, the classifying topos for <i>G</i>-<a href="/facts/Torsor/N27maCdL">torsors</a> over a topos is the topos <i>BG</i> of <i>G</i>-sets.</li> <li>The <a href="/facts/Classifying_space/gGIAmhtL">classifying space</a> of <a href="/facts/Topological_group/cAgNMMRU">topological groups</a> in <a href="/facts/Homotopy_theory/MJwYyRny">homotopy theory</a>.</li></ul> <ul><li><a href="/facts/Olivia_Caramello/qaRmmtjk">Caramello, Olivia</a> (2017), <i>Theories, Sites, Toposes: Relating and studying mathematical theories through topos-theoretic 'bridges'</i>, Oxford University Press, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1093%2Foso%2F9780198758914.001.0001">10.1093/oso/9780198758914.001.0001</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9780198758914</li> <li>Mac Lane, Saunders; Moerdijk, Ieke (1992), <i>Sheaves in geometry and logic. A first introduction to topos theory</i>, Universitext, New York: Springer-Verlag, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-387-97710-4, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1300636">1300636</a></li> <li>Moerdijk, I. (1995), <i>Classifying spaces and classifying topoi</i>, Lecture Notes in Mathematics, vol. 1616, Berlin: Springer-Verlag, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBFb0094441">10.1007/BFb0094441</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-60319-0, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1440857">1440857</a></li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="https://ncatlab.org/nlab/show/classifying+topos">Classifying topos</a> at the <a href="/facts/NLab/lpPvRmgo"><i>n</i>Lab</a></li></ul>