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href="/nlab/show/category">category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> </li> </ul> </li> </ul> <h2 id="toposes">Toposes</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%280%2C1%29-topos">(0,1)-topos</a>, <a class="existingWikiWord" href="/nlab/show/Heyting+algebra">Heyting algebra</a>, <a class="existingWikiWord" href="/nlab/show/locale">locale</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pretopos">pretopos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topos">topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+topos">Grothendieck topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+presheaves">category of presheaves</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/representable+functor">representable presheaf</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+sheaves">category of sheaves</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/site">site</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sieve">sieve</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/coverage">coverage</a>, <a class="existingWikiWord" href="/nlab/show/Grothendieck+pretopology">pretopology</a>, <a class="existingWikiWord" href="/nlab/show/Grothendieck+topology">topology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sheafification">sheafification</a></p> </li> </ul> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quasitopos">quasitopos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/base+topos">base topos</a>, <a class="existingWikiWord" href="/nlab/show/indexed+topos">indexed topos</a></p> </li> </ul> <h2 id="toc_internal_logic">Internal Logic</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/categorical+semantics">categorical semantics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/internal+logic">internal logic</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/subobject+classifier">subobject classifier</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/natural+numbers+object">natural numbers object</a></p> </li> </ul> </li> </ul> <h2 id="topos_morphisms">Topos morphisms</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/logical+morphism">logical morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+morphism">geometric morphism</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/direct+image">direct image</a>/<a class="existingWikiWord" href="/nlab/show/inverse+image">inverse image</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/global+section">global sections</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+embedding">geometric embedding</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/surjective+geometric+morphism">surjective geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/essential+geometric+morphism">essential geometric morphism</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+connected+geometric+morphism">locally connected geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connected+geometric+morphism">connected geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/totally+connected+geometric+morphism">totally connected geometric morphism</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%C3%A9tale+geometric+morphism">étale geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/open+geometric+morphism">open geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/proper+geometric+morphism">proper geometric morphism</a>, <a class="existingWikiWord" href="/nlab/show/compact+topos">compact topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/separated+geometric+morphism">separated geometric morphism</a>, <a class="existingWikiWord" href="/nlab/show/Hausdorff+topos">Hausdorff topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+geometric+morphism">local geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bounded+geometric+morphism">bounded geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/base+change">base change</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/localic+geometric+morphism">localic geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hyperconnected+geometric+morphism">hyperconnected geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/atomic+geometric+morphism">atomic geometric morphism</a></p> </li> </ul> </li> </ul> <h2 id="extra_stuff_structure_properties">Extra stuff, structure, properties</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+locale">topological locale</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/localic+topos">localic topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/petit+topos">petit topos/gros topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+connected+topos">locally connected topos</a>, <a class="existingWikiWord" href="/nlab/show/connected+topos">connected topos</a>, <a class="existingWikiWord" href="/nlab/show/totally+connected+topos">totally connected topos</a>, <a class="existingWikiWord" href="/nlab/show/strongly+connected+topos">strongly connected topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+topos">local topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohesive+topos">cohesive topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/classifying+topos">classifying topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+topos">smooth topos</a></p> </li> </ul> <h2 id="cohomology_and_homotopy">Cohomology and homotopy</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+groups+in+an+%28infinity%2C1%29-topos">homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">model structure on simplicial presheaves</a></p> </li> </ul> <h2 id="in_higher_category_theory">In higher category theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+topos+theory">higher topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%280%2C1%29-topos">(0,1)-topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/%280%2C1%29-site">(0,1)-site</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-topos">2-topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-site">2-site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-sheaf">2-sheaf</a>, <a class="existingWikiWord" href="/nlab/show/stack">stack</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-site">(∞,1)-site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-sheaf">(∞,1)-sheaf</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stack">∞-stack</a>, <a class="existingWikiWord" href="/nlab/show/derived+stack">derived stack</a></p> </li> </ul> </li> </ul> <h2 id="theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Diaconescu%27s+theorem">Diaconescu's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Barr%27s+theorem">Barr's theorem</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/topos+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="yoneda_lemma">Yoneda lemma</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a></strong></p> <p><strong>Ingredients</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category">category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+presheaves">category of presheaves</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/representable+presheaf">representable presheaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yoneda+embedding">Yoneda embedding</a></p> </li> </ul> <p><strong>Incarnations</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+Yoneda+lemma">enriched Yoneda lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/co-Yoneda+lemma">co-Yoneda lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yoneda+reduction">Yoneda reduction</a></p> </li> </ul> <p><strong>Properties</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/free+cocompletion">free cocompletion</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yoneda+extension">Yoneda extension</a></p> </li> </ul> <p><strong>Universal aspects</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/representable+functor">representable functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+construction">universal construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+element">universal element</a></p> </li> </ul> <p><strong>Classification</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a>, <a class="existingWikiWord" href="/nlab/show/classifying+stack">classifying stack</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/moduli+space">moduli space</a>, <a class="existingWikiWord" href="/nlab/show/moduli+stack">moduli stack</a>, <a class="existingWikiWord" href="/nlab/show/derived+moduli+space">derived moduli space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/classifying+topos">classifying topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/subobject+classifier">subobject classifier</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+principal+bundle">universal principal bundle</a>, <a class="existingWikiWord" href="/nlab/show/universal+principal+%E2%88%9E-bundle">universal principal ∞-bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/classifying+morphism">classifying morphism</a></p> </li> </ul> <p><strong>Induced theorems</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a></li> </ul> <p>…</p> <p><strong>In higher category theory</strong></p> <p><a class="existingWikiWord" href="/nlab/show/Yoneda+lemma+for+higher+categories">Yoneda lemma for higher categories</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Yoneda+lemma+for+%28%E2%88%9E%2C1%29-categories">Yoneda lemma for (∞,1)-categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yoneda+lemma+for+bicategories">Yoneda lemma for bicategories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yoneda+lemma+for+tricategories">Yoneda lemma for tricategories</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/Yoneda+lemma+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <li><a href='#background_on_the_theory_of_theories'>Background on the theory of theories</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#GeometricMorphismsAndMorphismsOfSites'>Geometric morphisms equivalent to morphisms of sites</a></li> </ul> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#for_nothing'>For nothing</a></li> <li><a href='#for_contradictions'>For contradictions</a></li> <li><a href='#ForObjects'>For objects</a></li> <li><a href='#ForPointedObjects'>For pointed objects</a></li> <li><a href='#for_groups'>For groups</a></li> <li><a href='#for_rings'>For rings</a></li> <li><a href='#ForLinearOrders'>For (inhabited) linear orders</a></li> <li><a href='#ForIntervals'>For intervals</a></li> <li><a href='#for_abstract_circles'>For abstract circles</a></li> <li><a href='#for_local_rings'>For local rings</a></li> <ul> <li><a href='#local_rings'>Local rings</a></li> <li><a href='#StrictLocalRings'>Strict local rings</a></li> </ul> <li><a href='#PrincipalBund'>For principal bundles</a></li> <ul> <li><a href='#BareGTor'>Over bare groups</a></li> <li><a href='#BareGTorGeomTheo'>In terms of geometric theories</a></li> <li><a href='#TopGTor'>Over topological groups</a></li> <li><a href='#UniversalBundle'>The universal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-bundle topos</a></li> </ul> <li><a href='#ForLocalicGroupoids'>For general localic groupoids</a></li> <li><a href='#for_flat_functors'>For flat functors</a></li> <li><a href='#CoverPreservingFLatFunctors'>For geometric theories / cover-preserving flat functors on a site</a></li> <li><a href='#LocalAlgebras'>For local algebras</a></li> </ul> <li><a href='#as_a_generalization_of_the_notion_of_classifying_space_in_topology'>As a generalization of the notion of classifying space in topology</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#ReferencesRelationToForcing'>Relation to forcing</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p>The <em>classifying topos</em> for a given type of mathematical <a class="existingWikiWord" href="/nlab/show/structure">structure</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> — for example the structures: “<a class="existingWikiWord" href="/nlab/show/group">group</a>”, “<a class="existingWikiWord" href="/nlab/show/torsor">torsor</a>”, “<a class="existingWikiWord" href="/nlab/show/ring">ring</a>”, “<a class="existingWikiWord" href="/nlab/show/category">category</a>” etc. — is a (<a class="existingWikiWord" href="/nlab/show/Grothendieck+topos">Grothendieck</a>) <a class="existingWikiWord" href="/nlab/show/topos">topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo stretchy="false">[</mo><mi>T</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">S[T]</annotation></semantics></math> such that <a class="existingWikiWord" href="/nlab/show/geometric+morphisms">geometric morphisms</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>E</mi><mo>→</mo><mi>S</mi><mo stretchy="false">[</mo><mi>T</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">f: E \to S[T]</annotation></semantics></math> are the same as structures of this sort in the topos <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>, i.e. groups <a class="existingWikiWord" href="/nlab/show/internalization">internal to</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>, torsors internal to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>, etc. In other words, a classifying topos is a <a class="existingWikiWord" href="/nlab/show/representing+object">representing object</a> for the functor which sends a topos <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> to the category of structures of the desired sort in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>.</p> <p>In particular for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/sheaf+topos">sheaf topos</a> on a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> a (bare, i.e. discrete) <a class="existingWikiWord" href="/nlab/show/group">group</a>, a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-torsor in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. There is a classifying topos denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mi>G</mi></mrow><annotation encoding="application/x-tex">B G</annotation></semantics></math>, such that the <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mi>Bund</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G Bund(X)</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a>s over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is equivalent to geometric morphims <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sh</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mi>B</mi><mi>G</mi></mrow><annotation encoding="application/x-tex">Sh(X) \to B G</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>G</mi><mi>Bund</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>Topos</mi><mo stretchy="false">(</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><mi>B</mi><mi>G</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> G Bund(X) \simeq Topos(Sh(X), B G) \,. </annotation></semantics></math></div> <p>This is evidently analogous to the notion of <a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a> in <a class="existingWikiWord" href="/nlab/show/topology">topology</a>, which for the discrete group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℬ</mi><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathcal{B} G</annotation></semantics></math> such that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><mi>G</mi><mi>Bund</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>π</mi> <mn>0</mn></msub><mi>Top</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℬ</mi><mi>G</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \pi_0 G Bund(X) \simeq \pi_0 Top(X, \mathcal{B}G) \,. </annotation></semantics></math></div> <p>Hence one can think of classifying topoi as a grand generalization of the notion of <a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a> in topology.</p> <h2 id="definition">Definition</h2> <p>In a tautological way, every <a class="existingWikiWord" href="/nlab/show/topos">topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> is the classifying topos for something, namely for the categories of <a class="existingWikiWord" href="/nlab/show/geometric+morphisms">geometric morphisms</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>→</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">E \to F</annotation></semantics></math> into it. The concept of <a class="existingWikiWord" href="/nlab/show/geometric+theory">geometric theory</a> allows one to usefully interpret these categories as <em>categories of certain structures in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math></em> :</p> <p>as decribed in <em><a href="geometric+theory#InTermsOfSheafTopoi">Geometric theories – In terms of sheaf topoi</a></em>, every <a class="existingWikiWord" href="/nlab/show/sheaf+topos">sheaf topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> is a completion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo stretchy="false">[</mo><mi>T</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">S[T]</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/syntactic+category">syntactic category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mi>T</mi></msub></mrow><annotation encoding="application/x-tex">C_T</annotation></semantics></math> of <em>some</em> <a class="existingWikiWord" href="/nlab/show/geometric+theory">geometric theory</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>≃</mo><mi>S</mi><mo stretchy="false">[</mo><mi>T</mi><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> F \simeq S[T] \,. </annotation></semantics></math></div> <p>And structures of type <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> is what geometric morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>→</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">E \to F</annotation></semantics></math> classify.</p> <p>So the <strong>classifying <a class="existingWikiWord" href="/nlab/show/topos">topos</a></strong> for the <a class="existingWikiWord" href="/nlab/show/geometric+theory">geometric theory</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/Grothendieck+topos">Grothendieck topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo stretchy="false">[</mo><mi>T</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">S[T]</annotation></semantics></math> equipped with a “universal model <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>”. This means that for any Grothendieck topos <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> together with a model <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>, there exists a unique (up to isomorphism) <a class="existingWikiWord" href="/nlab/show/geometric+morphism">geometric morphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>E</mi><mo>→</mo><mi>S</mi><mo stretchy="false">[</mo><mi>T</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">f: E \to S[T]</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">f^*</annotation></semantics></math> maps the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>-model <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> to the model <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. More precisely, for any Grothendieck topos <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>, the category of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>-models in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> is equivalent to the category of geometric morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>→</mo><mi>S</mi><mo stretchy="false">[</mo><mi>T</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">E \to S[T]</annotation></semantics></math>.</p> <p>The fact that a classifying topos is like the ambient <a class="existingWikiWord" href="/nlab/show/set+theory">set theory</a> but equipped with that universal model is essentially the notion of <em><a class="existingWikiWord" href="/nlab/show/forcing">forcing</a></em> in <a class="existingWikiWord" href="/nlab/show/logic">logic</a>: the passage to the <a class="existingWikiWord" href="/nlab/show/internal+logic">internal logic</a> of the classifying topos <em>forces</em> the universal model to exist.</p> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mi>T</mi></msub></mrow><annotation encoding="application/x-tex">C_T</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/syntactic+category">syntactic category</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>, so that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>-models are the same as <a class="existingWikiWord" href="/nlab/show/geometric+category">geometric functors</a> out of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mi>T</mi></msub></mrow><annotation encoding="application/x-tex">C_T</annotation></semantics></math>, then this universal model can be identified with a certain geometric functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>:</mo><msub><mi>C</mi> <mi>T</mi></msub><mo>→</mo><mi>S</mi><mo stretchy="false">[</mo><mi>T</mi><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> U : C_T \to S[T] \,. </annotation></semantics></math></div> <p>Its universality property means that any geometric functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>:</mo><msub><mi>C</mi> <mi>T</mi></msub><mo>→</mo><mi>E</mi></mrow><annotation encoding="application/x-tex"> X : C_T \to E </annotation></semantics></math></div> <p>factors essentially uniquely as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>:</mo><msub><mi>C</mi> <mi>T</mi></msub><mover><mo>→</mo><mi>U</mi></mover><mi>S</mi><mo stretchy="false">[</mo><mi>T</mi><mo stretchy="false">]</mo><mover><mo>→</mo><mrow><msup><mi>f</mi> <mo>*</mo></msup></mrow></mover><mi>E</mi></mrow><annotation encoding="application/x-tex"> X : C_T \stackrel{U}{\to} S[T] \stackrel{f^*}{\to} E </annotation></semantics></math></div> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> the universal model and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">f^*</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> part of a <a class="existingWikiWord" href="/nlab/show/geometric+morphism">geometric morphism</a>. More precisely, composition with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> defines an equivalence between the category of geometric morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>→</mo><mi>S</mi><mo stretchy="false">[</mo><mi>T</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">E\to S[T]</annotation></semantics></math> and the category of geometric functors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mi>T</mi></msub><mo>→</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">C_T\to E</annotation></semantics></math>.</p> <p>More specifically, for any <a class="existingWikiWord" href="/nlab/show/cartesian+theory">cartesian theory</a>, <a class="existingWikiWord" href="/nlab/show/regular+theory">regular theory</a> or <a class="existingWikiWord" href="/nlab/show/coherent+theory">coherent theory</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕋</mi></mrow><annotation encoding="application/x-tex">\mathbb{T}</annotation></semantics></math> (which in ascending order are special cases of each other and all of geometric theories), the corresponding <a class="existingWikiWord" href="/nlab/show/syntactic+category">syntactic category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mi>𝕋</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{C}_{\mathbb{T}}</annotation></semantics></math> comes equipped with the structure of a <a class="existingWikiWord" href="/nlab/show/syntactic+site">syntactic site</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mi>𝕋</mi><mo>,</mo><mi>J</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C},\mathbb{T}, J)</annotation></semantics></math> (see there) and the classifying topos for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕋</mi></mrow><annotation encoding="application/x-tex">\mathbb{T}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/sheaf+topos">sheaf topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sh</mi><mo stretchy="false">(</mo><msub><mi>𝒞</mi> <mi>𝕋</mi></msub><mo>,</mo><mi>J</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh(\mathcal{C}_{\mathbb{T}}, J)</annotation></semantics></math>.</p> <p>Classifying toposes can also be defined over any <a class="existingWikiWord" href="/nlab/show/base+topos">base topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> instead of <a class="existingWikiWord" href="/nlab/show/Set">Set</a>. In that case “Grothendieck topos” is replaced by “<a class="existingWikiWord" href="/nlab/show/bounded+topos">bounded</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-topos”. The <em>general existence of classifying toposes</em> for geometric theories for bounded <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-toposes is then intimately connected to the existence of the <a class="existingWikiWord" href="/nlab/show/classifying+topos+for+the+theory+of+objects">classifying topos for the theory of objects</a> which in turn hinges on the existence of a <a class="existingWikiWord" href="/nlab/show/natural+number+object">natural number object</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>. See <a href="#ForObjects">below</a> and, for further details, <a class="existingWikiWord" href="/nlab/show/classifying+topos+for+the+theory+of+objects">classifying topos for the theory of objects</a> or Blass (<a href="#blass">1989</a>).</p> <p>If the classifying topos of a geometric theory <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/presheaf+topos">presheaf topos</a>, one calls <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> a <em><a class="existingWikiWord" href="/nlab/show/theory+of+presheaf+type">theory of presheaf type</a></em>.</p> <h2 id="background_on_the_theory_of_theories">Background on the theory of theories</h2> <p>The notion of <em>classifying topos</em> is part of a trend, begun by <a class="existingWikiWord" href="/nlab/show/Bill+Lawvere">Lawvere</a>, of viewing a mathematical <a class="existingWikiWord" href="/nlab/show/theory">theory</a> in <a class="existingWikiWord" href="/nlab/show/logic">logic</a> as a <a class="existingWikiWord" href="/nlab/show/category">category</a> with suitable <a class="existingWikiWord" href="/nlab/show/exact+functor">exactness</a> properties and which contains a “generic model”, and a <a class="existingWikiWord" href="/nlab/show/model">model</a> of the theory as a functor which preserves those properties. This is described in more detail at <a class="existingWikiWord" href="/nlab/show/internal+logic">internal logic</a> and <a class="existingWikiWord" href="/nlab/show/type+theory">type theory</a>, but here are some simple examples to give the flavor. The original example is that of a ‘finite products theory’:</p> <ul> <li> <p><strong>Finite products theory.</strong> Roughly speaking, a ‘finite products theory’, ‘<a class="existingWikiWord" href="/nlab/show/Lawvere+theory">Lawvere theory</a>’, or ‘<a class="existingWikiWord" href="/nlab/show/algebraic+theory">algebraic theory</a>’ is a <a class="existingWikiWord" href="/nlab/show/theory">theory</a> describing some mathematical structure that can be defined in an arbitrary category with finite <a class="existingWikiWord" href="/nlab/show/product">product</a>s. An example would be the theory of <a class="existingWikiWord" href="/nlab/show/groups">groups</a>. As explained in the entry for <a class="existingWikiWord" href="/nlab/show/Lawvere+theory">Lawvere theory</a>, for each such theory <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> there is a category with finite products <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mi>fp</mi></msub><mo stretchy="false">[</mo><mi>T</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">C_{fp}[T]</annotation></semantics></math> – the <a class="existingWikiWord" href="/nlab/show/syntactic+category">syntactic category</a>, which serves as a “classifying category” for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>, in that models of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> in the category of sets correspond to product-preserving <a class="existingWikiWord" href="/nlab/show/functor">functor</a>s <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><msub><mi>C</mi> <mi>fp</mi></msub><mo stretchy="false">[</mo><mi>T</mi><mo stretchy="false">]</mo><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">f : C_{fp}[T] \to Set</annotation></semantics></math>. More generally, for any category with finite products, say <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>, models of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> correspond to product-preserving functors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><msub><mi>C</mi> <mi>fp</mi></msub><mo stretchy="false">[</mo><mi>T</mi><mo stretchy="false">]</mo><mo>→</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">f : C_{fp}[T] \to E</annotation></semantics></math>.</p> </li> <li> <p><strong>Finite limits theory.</strong> Next up the line is the notion of ‘finite limits theory’, sometimes called an <a class="existingWikiWord" href="/nlab/show/essentially+algebraic+theory">essentially algebraic theory</a>. This is roughly a theory describing some structure that can be defined in an arbitrary category with <a class="existingWikiWord" href="/nlab/show/finite+limit">finite limit</a>s (also called a <a class="existingWikiWord" href="/nlab/show/finitely+complete+category">finitely complete category</a>). An example of a finite limits theory would be the theory of categories. (The notion of ‘category’ requires finite limits, while the notion of ‘group’ does not, because categories but not groups involve a <em>partially defined</em> operation, namely composition of morphisms.) Every finite limits theory <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> admits a <a class="existingWikiWord" href="/nlab/show/syntactic+category">classifying category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mi>fl</mi></msub><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C_{fl}(T)</annotation></semantics></math>: a finitely complete category such that models of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> in a category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> with finite limits correspond to functors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><msub><mi>C</mi> <mi>fl</mi></msub><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo><mo>→</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">f: C_{fl}(T) \to E</annotation></semantics></math> that preserve finite limits. (Such functors are called <a class="existingWikiWord" href="/nlab/show/left+exact">left exact</a>, or ‘lex’ for short.)</p> </li> <li> <p><strong>Geometric theory.</strong> Further up the line, a <a class="existingWikiWord" href="/nlab/show/geometric+theory">geometric theory</a> is roughly a theory which can be formulated in that fragment of first-order logic that deals in finite limits and arbitrary (small) colimits, plus certain <a class="existingWikiWord" href="/nlab/show/familial+regularity+and+exactness">exactness</a> properties the details of which need not concern us. The point is that a category with finite limits, small colimits, and appropriate exactness is just a Grothendieck topos, and a functor preserving finite limits and small colimits is just the inverse image part of a geometric morphism. Just as in the previous two cases, any ‘geometric theory’ has a classifying category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo stretchy="false">[</mo><mi>T</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">S[T]</annotation></semantics></math> (which is now a Grothendieck topos) which possesses a “generic object” for that theory, and T-models in any other Grothendieck topos E can be identified with geometric morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>E</mi><mo>→</mo><mi>S</mi><mo stretchy="false">[</mo><mi>T</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">f\colon E\to S[T]</annotation></semantics></math>, or specifically with their inverse image parts.</p> </li> </ul> <p>Each type of theory may be considered a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-theory, or <a class="existingWikiWord" href="/nlab/show/doctrine">doctrine</a>. Furthermore, each type of theory can be promoted to a theory “further up the line”, by <a class="existingWikiWord" href="/nlab/show/completion">freely adding</a> the missing structure to the classifying category. This can always be done purely formally, but in a few cases this promotion also has other, more explicit descriptions.</p> <p>For instance, to go from a finite products theory <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> to the corresponding finite limits theory, we can take the opposite of the category of <a class="existingWikiWord" href="/nlab/show/finitely+presentable+object">finitely presentable models</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Set</mi></mrow><annotation encoding="application/x-tex">Set</annotation></semantics></math>, thanks to <a class="existingWikiWord" href="/nlab/show/Gabriel-Ulmer+duality">Gabriel-Ulmer duality</a>. Similarly, to go from a finite limits theory to the classifying topos of the corresponding geometric theory, we can take the category of presheaves on the classifying category of the finite limits theory.</p> <h2 id="properties">Properties</h2> <h3 id="GeometricMorphismsAndMorphismsOfSites">Geometric morphisms equivalent to morphisms of sites</h3> <p>The fact that classifying toposes are what they are all comes down, if spelled out explicitly, to the fact that a <a class="existingWikiWord" href="/nlab/show/geometric+morphism">geometric morphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>ℰ</mi><mo>→</mo><mi>ℱ</mi></mrow><annotation encoding="application/x-tex">f : \mathcal{E} \to \mathcal{F}</annotation></semantics></math> of toposes can be identified with a certain morphism of <a class="existingWikiWord" href="/nlab/show/site">site</a>s <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mi>ℰ</mi></msub></mrow><annotation encoding="application/x-tex">C_{\mathcal{E}}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mi>ℱ</mi></msub></mrow><annotation encoding="application/x-tex">C_{\mathcal{F}}</annotation></semantics></math> for these toposes, going the other way round, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mi>ℰ</mi></msub><mo>←</mo><msub><mi>C</mi> <mi>ℱ</mi></msub></mrow><annotation encoding="application/x-tex">C_\mathcal{E} \leftarrow C_{\mathcal{F}}</annotation></semantics></math>, and having certain properties. If here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mi>ℱ</mi></msub></mrow><annotation encoding="application/x-tex">C_\mathcal{F}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/syntactic+site">syntactic site</a> of some <a class="existingWikiWord" href="/nlab/show/theory">theory</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕋</mi></mrow><annotation encoding="application/x-tex">\mathbb{T}</annotation></semantics></math> and we choose <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mi>ℰ</mi></msub></mrow><annotation encoding="application/x-tex">C_{\mathcal{E}}</annotation></semantics></math> to be the <a class="existingWikiWord" href="/nlab/show/canonical+site">canonical site</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℰ</mi></mrow><annotation encoding="application/x-tex">\mathcal{E}</annotation></semantics></math> (itself equipped with the <a class="existingWikiWord" href="/nlab/show/canonical+coverage">canonical coverage</a>) this makes manifest why the geometric morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℱ</mi></mrow><annotation encoding="application/x-tex">\mathcal{F}</annotation></semantics></math> correspond to <a class="existingWikiWord" href="/nlab/show/model">model</a>s of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕋</mi></mrow><annotation encoding="application/x-tex">\mathbb{T}</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℰ</mi></mrow><annotation encoding="application/x-tex">\mathcal{E}</annotation></semantics></math>.</p> <p>We now say this in precise manner. In the following a <em><a class="existingWikiWord" href="/nlab/show/cartesian+site">cartesian site</a></em> means a <a class="existingWikiWord" href="/nlab/show/site">site</a> whose underlying category is <a class="existingWikiWord" href="/nlab/show/finitely+complete+category">finitely complete</a>.</p> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mi>J</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C}, J)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒟</mi><mo>,</mo><mi>K</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{D}, K)</annotation></semantics></math> be <a class="existingWikiWord" href="/nlab/show/cartesian+site">cartesian site</a>s such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/small+category">small category</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">\mathcal{D}</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/essentially+small+category">essentially small category</a> and the <a class="existingWikiWord" href="/nlab/show/coverage">coverage</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/subcanonical+coverage">subcanonical</a>.</p> <p>Then a <a class="existingWikiWord" href="/nlab/show/geometric+morphism">geometric morphism</a> of the corresponding <a class="existingWikiWord" href="/nlab/show/sheaf+toposes">sheaf toposes</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>𝒟</mi><mo>,</mo><mi>K</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mi>J</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> f : Sh(\mathcal{D}, K) \to Sh(\mathcal{C}, J) </annotation></semantics></math></div> <p>is induced by a <a class="existingWikiWord" href="/nlab/show/morphism+of+sites">morphism of sites</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒟</mi><mo>,</mo><mi>K</mi><mo stretchy="false">)</mo><mo>←</mo><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mi>J</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (\mathcal{D}, K) \leftarrow (\mathcal{C}, J) </annotation></semantics></math></div> <p>precisely if the <a class="existingWikiWord" href="/nlab/show/inverse+image">inverse image</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> respects the <a class="existingWikiWord" href="/nlab/show/Yoneda+embedding">Yoneda embedding</a>s <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math> as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>𝒟</mi></mtd> <mtd><mo>←</mo></mtd> <mtd><mi>𝒞</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>j</mi> <mi>𝒟</mi></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>j</mi> <mi>𝒞</mi></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Sh</mi><mo stretchy="false">(</mo><mi>𝒟</mi><mo>,</mo><mi>K</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>←</mo><mrow><msup><mi>f</mi> <mo>*</mo></msup></mrow></mover></mtd> <mtd><mi>Sh</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mi>J</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \mathcal{D } &\leftarrow& \mathcal{C} \\ {}^{\mathllap{j_{\mathcal{D}}}}\downarrow && \downarrow^{\mathrlap{j_{\mathcal{C}}}} \\ Sh(\mathcal{D}, K) &\stackrel{f^*}{\leftarrow}& Sh(\mathcal{C}, J) } \,. </annotation></semantics></math></div></div> <p>This appears as (<a href="#Johnstone">Johnstone, lemma C2.3.8</a>).</p> <div class="proof"> <h6 id="proof">Proof</h6> <p>It suffices to observe that the factorization, if it exists, is a morphism of sites.</p> </div> <div class="num_cor" id="SheafToposesAreClassifyingForTheirTheoryOfLocalAlgegras"> <h6 id="corollary">Corollary</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mi>J</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C},J)</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/small+category">small</a> <a class="existingWikiWord" href="/nlab/show/cartesian+site">cartesian site</a> and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℰ</mi></mrow><annotation encoding="application/x-tex">\mathcal{E}</annotation></semantics></math> be any <a class="existingWikiWord" href="/nlab/show/sheaf+topos">sheaf topos</a>. Then we have an <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence of categories</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Topos</mi><mo stretchy="false">(</mo><mi>ℰ</mi><mo>,</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mi>J</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≃</mo><mi>Site</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mi>J</mi><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">(</mo><mi>ℰ</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Topos(\mathcal{E}, Sh(\mathcal{C}, J)) \simeq Site((\mathcal{C}, J), (\mathcal{E}, C)) </annotation></semantics></math></div> <p>between the <a class="existingWikiWord" href="/nlab/show/geometric+morphism">geometric morphism</a>s from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℰ</mi></mrow><annotation encoding="application/x-tex">\mathcal{E}</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sh</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mi>J</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh(\mathcal{C}, J)</annotation></semantics></math> and the morphisms of <a class="existingWikiWord" href="/nlab/show/sites">sites</a> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mi>J</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C}, J)</annotation></semantics></math> to the <a class="existingWikiWord" href="/nlab/show/big+site">big site</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>ℰ</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{E}, C)</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/canonical+coverage">canonical coverage</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℰ</mi></mrow><annotation encoding="application/x-tex">\mathcal{E}</annotation></semantics></math>.</p> </div> <p>This appears as (<a href="#Johnstone">Johnstone, cor. C2.3.9</a>).</p> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>This means that a sheaf topos <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sh</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mi>J</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh(\mathcal{C},J)</annotation></semantics></math> is the classifying topos for the theory of <a class="existingWikiWord" href="/nlab/show/local+algebras">local algebras</a> determined by the <a class="existingWikiWord" href="/nlab/show/site">site</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mi>J</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C},J)</annotation></semantics></math>.</p> </div> <h2 id="examples">Examples</h2> <p>We list and discuss explicit examples of classifying toposes.</p> <h3 id="for_nothing">For nothing</h3> <p>Since the <a class="existingWikiWord" href="/nlab/show/empty+theory">empty geometric theory</a> has a unique model in any Grothendieck topos, its classifying topos is the terminal Grothendieck topos, namely <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Set</mi></mrow><annotation encoding="application/x-tex">Set</annotation></semantics></math>.</p> <p>Note that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Set</mi></mrow><annotation encoding="application/x-tex">Set</annotation></semantics></math> has no non-trivial <a class="existingWikiWord" href="/nlab/show/subtopos">subtoposes</a>. Thus relative to the empty signature, the empty theory is complete: either a sequent <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math> follows from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝕋</mi> <mrow><msub><mi>Σ</mi> <mi>∅</mi></msub></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbb{T}_{\Sigma_\emptyset}</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>σ</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{\sigma\}</annotation></semantics></math> is inconsistent. In other words, the only toposes classifying theories over the empty signature are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Set</mi></mrow><annotation encoding="application/x-tex">Set</annotation></semantics></math> and the inconsistent topos <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mn>1</mn></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{1}</annotation></semantics></math>.</p> <p>The empty theory is not the only theory classified by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Set</mi></mrow><annotation encoding="application/x-tex">Set</annotation></semantics></math>: any theory that has a unique model in any Grothendieck topos will do. For instance, the theories of initial objects, of terminal objects, and of natural numbers objects are all classified by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Set</mi></mrow><annotation encoding="application/x-tex">Set</annotation></semantics></math>. Note that these theories have nonempty signatures, e.g. to axiomatize initial objects one has to add the sequent <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊤</mo><msub><mo>⊢</mo> <mi>x</mi></msub><mo>⊥</mo></mrow><annotation encoding="application/x-tex">\top\vdash_x\bot</annotation></semantics></math> to the <a class="existingWikiWord" href="/nlab/show/theory+of+objects">theory of objects</a> below, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> is the unique sort.</p> <h3 id="for_contradictions">For contradictions</h3> <p>The contradictory theory <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mo>⊤</mo><mo>⊢</mo><mo>⊥</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{\top\vdash\bot\}</annotation></semantics></math> has no models in any nontrivial Grothendieck topos. Thus its classifying topos is the initial Grothendieck topos <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mn>1</mn></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{1}</annotation></semantics></math> (which is a <a class="existingWikiWord" href="/nlab/show/strict+initial+object">strict initial object</a>).</p> <p>More generally, any theory that has no models in any nontrivial Grothendieck topos is classified by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mn>1</mn></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{1}</annotation></semantics></math>, such as the theory of <a class="existingWikiWord" href="/nlab/show/zero+objects">zero objects</a>.</p> <h3 id="ForObjects">For objects</h3> <p>The <a class="existingWikiWord" href="/nlab/show/presheaf+topos">presheaf topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>FinSet</mi><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[FinSet, Set]</annotation></semantics></math> on the <a class="existingWikiWord" href="/nlab/show/opposite+category">opposite category</a> of <a class="existingWikiWord" href="/nlab/show/FinSet">FinSet</a> is the classifying topos for the <a class="existingWikiWord" href="/nlab/show/theory+of+objects">theory of objects</a>, sometimes called the “object classifier”. This is not to be confused with the notion of an <a class="existingWikiWord" href="/nlab/show/object+classifier">object classifier</a> in an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> and maybe better called in full the <em><a class="existingWikiWord" href="/nlab/show/classifying+topos+for+the+theory+of+objects">classifying topos for the theory of objects</a></em>.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> any <a class="existingWikiWord" href="/nlab/show/topos">topos</a>, a <a class="existingWikiWord" href="/nlab/show/geometric+morphism">geometric morphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>→</mo><mo stretchy="false">[</mo><mi>FinSet</mi><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">E \to [FinSet,Set]</annotation></semantics></math> is equivalently just an <a class="existingWikiWord" href="/nlab/show/object">object</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>, given by the inverse image of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>FinSet</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><mo>*</mo><mo stretchy="false">}</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">FinSet(\{ * \}, -)</annotation></semantics></math>.</p> <h3 id="ForPointedObjects">For pointed objects</h3> <p>Similarly, the presheaf topos <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msub><mi>FinSet</mi> <mo>*</mo></msub><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[FinSet_*, Set]</annotation></semantics></math> (where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>FinSet</mi> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">FinSet_*</annotation></semantics></math> is the category of finite <a class="existingWikiWord" href="/nlab/show/pointed+sets">pointed sets</a>) classifies <a class="existingWikiWord" href="/nlab/show/pointed+objects">pointed objects</a>; cf. <a href="http://mathoverflow.net/questions/85600/what-do-gamma-sets-classify">this question</a> and answer. This is the topos of “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi></mrow><annotation encoding="application/x-tex">\Gamma</annotation></semantics></math>-sets”; see <a class="existingWikiWord" href="/nlab/show/Gamma-space">Gamma-space</a>.</p> <h3 id="for_groups">For groups</h3> <p>We discuss the <a class="existingWikiWord" href="/nlab/show/finite+product+theory">finite product theory</a> of <a class="existingWikiWord" href="/nlab/show/group">group</a>s. This theory has a classifying category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mi>fp</mi></msub><mo stretchy="false">(</mo><mi>Grp</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C_{fp}(Grp)</annotation></semantics></math>. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mi>fp</mi></msub><mo stretchy="false">(</mo><mi>Grp</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C_{fp}(Grp)</annotation></semantics></math> is a category with finite products equipped with an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>, the “<a class="existingWikiWord" href="/nlab/show/walking">walking</a> group”, a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi><mo>:</mo><mi>G</mi><mo>×</mo><mi>G</mi><mo>→</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">m: G \times G \to G</annotation></semantics></math> describing multiplication, a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>inv</mi><mo>:</mo><mi>G</mi><mo>→</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">inv : G \to G</annotation></semantics></math> describing inverses, and a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>:</mo><mn>1</mn><mo>→</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">i: 1 \to G</annotation></semantics></math> describing the identity element of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>, obeying the usual group axioms. For any category with finite products, say <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>, a finite-product-preserving functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><msub><mi>C</mi> <mi>fp</mi></msub><mo stretchy="false">(</mo><mi>Grp</mi><mo stretchy="false">)</mo><mo>→</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">f: C_{fp}(Grp) \to E</annotation></semantics></math> is the same as a <a class="existingWikiWord" href="/nlab/show/group+object">group object</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>. For more details, see <a class="existingWikiWord" href="/nlab/show/Lawvere+theory">Lawvere theory</a>.</p> <p>We can promote <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mi>fp</mi></msub><mo stretchy="false">(</mo><mi>Grp</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C_{fp}(Grp)</annotation></semantics></math> to a category with finite limits, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mi>fl</mi></msub><mo stretchy="false">(</mo><mi>Grp</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C_{fl}(Grp)</annotation></semantics></math>, by adjoining all finite limits. As mentioned above, one way to do this is to take the category of models of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mi>fp</mi></msub><mo stretchy="false">(</mo><mi>Grp</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C_{fp}(Grp)</annotation></semantics></math> in Set, which is simply <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Grp</mi></mrow><annotation encoding="application/x-tex">Grp</annotation></semantics></math>, and then take the full subcategory of <a class="existingWikiWord" href="/nlab/show/finitely+presentable+object">finitely presentable</a> groups. By <a class="existingWikiWord" href="/nlab/show/Gabriel-Ulmer+duality">Gabriel-Ulmer duality</a>, the opposite of this is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mi>fl</mi></msub><mo stretchy="false">(</mo><mi>Grp</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C_{fl}(Grp)</annotation></semantics></math>. For any category with finite products, say <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>, a left exact functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><msub><mi>C</mi> <mi>fl</mi></msub><mo stretchy="false">(</mo><mi>Grp</mi><mo stretchy="false">)</mo><mo>→</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">f: C_{fl}(Grp) \to E</annotation></semantics></math> is the same as a <a class="existingWikiWord" href="/nlab/show/group+object">group object</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>.</p> <p>We can further promote <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mi>fl</mi></msub><mo stretchy="false">(</mo><mi>Grp</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C_{fl}(Grp)</annotation></semantics></math> to a <a class="existingWikiWord" href="/nlab/show/Grothendieck+topos">Grothendieck topos</a> by taking the category of <a class="existingWikiWord" href="/nlab/show/presheaves">presheaves</a>. This gives the classifying topos for groups:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo stretchy="false">[</mo><mi>Grp</mi><mo stretchy="false">]</mo><mo>=</mo><msup><mi>Set</mi> <mrow><msub><mi>C</mi> <mi>fl</mi></msub><mo stretchy="false">(</mo><mi>Grp</mi><msup><mo stretchy="false">)</mo> <mi>op</mi></msup></mrow></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> S[Grp] = Set^{C_{fl}(Grp)^{op}} \, . </annotation></semantics></math></div> <p>By invoking <a class="existingWikiWord" href="/nlab/show/Diaconescu%27s+theorem">Diaconescu's theorem</a>, for any Grothendieck topos, say <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>, a left exact left adjoint functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup><mo>:</mo><mi>S</mi><mo stretchy="false">[</mo><mi>Grp</mi><mo stretchy="false">]</mo><mo>→</mo><mi>E</mi></mrow><annotation encoding="application/x-tex"> f^*: S[Grp] \to E </annotation></semantics></math> is the same as a <a class="existingWikiWord" href="/nlab/show/group+object">group object</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>.</p> <h3 id="for_rings">For rings</h3> <p>The discussion above for groups can be repeated verbatim for rings, since they too are described by a finite products theory.</p> <h3 id="ForLinearOrders">For (inhabited) linear orders</h3> <div class="num_prop"> <h6 id="proposition_2">Proposition</h6> <p>The category of <a class="existingWikiWord" href="/nlab/show/cosimplicial+sets">cosimplicial sets</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>Δ</mi><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\Delta, Set]</annotation></semantics></math> – hence the <a class="existingWikiWord" href="/nlab/show/presheaf+topos">presheaf topos</a> over the <a class="existingWikiWord" href="/nlab/show/opposite+category">opposite category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Δ</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">\Delta^{op}</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/simplex+category">simplex category</a> – is the classifying topos for <a class="existingWikiWord" href="/nlab/show/inhabited+set">inhabited</a> <a class="existingWikiWord" href="/nlab/show/linear+orders">linear orders</a>.</p> </div> <p>This appears as (<a href="#Moerdijk95">Moerdijk 95, prop. 5.4</a>).</p> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>For ease of notation we discuss this in <a class="existingWikiWord" href="/nlab/show/Set">Set</a>, hence we show that <a class="existingWikiWord" href="/nlab/show/geometric+morphisms">geometric morphisms</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Set</mi><mo>→</mo><mi>PSh</mi><mo stretchy="false">(</mo><msup><mi>Δ</mi> <mi>op</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Set \to PSh(\Delta^{op})</annotation></semantics></math> are equivalently <a class="existingWikiWord" href="/nlab/show/linear+orders">linear orders</a>. Or, by <a class="existingWikiWord" href="/nlab/show/Diaconescu%27s+theorem">Diaconescu's theorem</a>, that <a class="existingWikiWord" href="/nlab/show/flat+functors">flat functors</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>:</mo><msup><mi>Δ</mi> <mi>op</mi></msup><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex"> X : \Delta^{op} \to Set </annotation></semantics></math></div> <p>are equivalently linear orders. Evidently, such a functor is in particular a <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a> and we will show that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> being flat is equivalent to this simplicial set being the <a class="existingWikiWord" href="/nlab/show/nerve">nerve</a> of an <a class="existingWikiWord" href="/nlab/show/inhabited+set">inhabited</a> linear order regarded as a <a class="existingWikiWord" href="/nlab/show/category">category</a> (a <a class="existingWikiWord" href="/nlab/show/%280%2C1%29-category">(0,1)-category</a>).</p> <p>First assume that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/flat+functor">flat functor</a>. Since (by the discussion there) this preserves all <a class="existingWikiWord" href="/nlab/show/finite+limits">finite limits</a> that exist in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Δ</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">\Delta^{op}</annotation></semantics></math>, equivalently that it sends the finite <a class="existingWikiWord" href="/nlab/show/colimits">colimits</a> that exist in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi></mrow><annotation encoding="application/x-tex">\Delta</annotation></semantics></math> to limits in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Set</mi></mrow><annotation encoding="application/x-tex">Set</annotation></semantics></math>, it in particular sends the gluings of intervals</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mtd> <mtd><mo>≃</mo><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mrow><mo stretchy="false">[</mo><mn>0</mn><mo stretchy="false">]</mo></mrow></munder><mo stretchy="false">[</mo><mi>l</mi><mo stretchy="false">]</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><mi>n</mi><mo>=</mo><mi>k</mi><mo>+</mo><mi>l</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mrow><mo stretchy="false">[</mo><mn>0</mn><mo stretchy="false">]</mo></mrow></munder><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mrow><mo stretchy="false">[</mo><mn>0</mn><mo stretchy="false">]</mo></mrow></munder><mi>⋯</mi><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mrow><mo stretchy="false">[</mo><mn>0</mn><mo stretchy="false">]</mo></mrow></munder><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} [n] & \simeq [k] \coprod_{[0]} [l] \;\;\;\; (n = k + l) \\ & \simeq [1] \coprod_{[0]} [1] \coprod_{[0]} \cdots \coprod_{[0]} [1] \end{aligned} </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi></mrow><annotation encoding="application/x-tex">\Delta</annotation></semantics></math> to <a class="existingWikiWord" href="/nlab/show/isomorphisms">isomorphisms</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mi>X</mi> <mi>n</mi></msub></mtd> <mtd><mo>≃</mo><msub><mi>X</mi> <mi>k</mi></msub><msub><mo>×</mo> <mrow><msub><mi>X</mi> <mn>0</mn></msub></mrow></msub><msub><mi>X</mi> <mi>l</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><msub><mi>X</mi> <mn>1</mn></msub><msub><mo>×</mo> <mrow><msub><mi>X</mi> <mn>0</mn></msub></mrow></msub><mi>⋯</mi><msub><mo>×</mo> <mrow><msub><mi>X</mi> <mn>0</mn></msub></mrow></msub><msub><mi>X</mi> <mn>1</mn></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} X_n & \simeq X_k \times_{X_0} X_l \\ & \simeq X_1 \times_{X_0} \cdots \times_{X_0} X_1 \end{aligned} \,. </annotation></semantics></math></div> <p>This are the <a class="existingWikiWord" href="/nlab/show/Segal+space">Segal relations</a> that say that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/nerve">nerve</a> of a <a class="existingWikiWord" href="/nlab/show/category">category</a>.</p> <p>Moreover, since <a class="existingWikiWord" href="/nlab/show/monomorphisms">monomorphisms</a> are characterized by <a class="existingWikiWord" href="/nlab/show/pullbacks">pullbacks</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> being flat means that it sends jointly epimorphic families of morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi></mrow><annotation encoding="application/x-tex">\Delta</annotation></semantics></math> to monomorphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Set</mi></mrow><annotation encoding="application/x-tex">Set</annotation></semantics></math>. In particular, the epimorphic family <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mo>∂</mo> <mn>0</mn></msub><mo>:</mo><mo stretchy="false">[</mo><mn>0</mn><mo stretchy="false">]</mo><mo>→</mo><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo>,</mo><msub><mo>∂</mo> <mn>1</mn></msub><mo>:</mo><mo stretchy="false">[</mo><mn>0</mn><mo stretchy="false">]</mo><mo>→</mo><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{\partial_0 : [0] \to [1], \partial_1 : [0] \to [1]\}</annotation></semantics></math> is sent to an injection</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>d</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>d</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>:</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>↪</mo><msub><mi>X</mi> <mn>0</mn></msub><mo>×</mo><msub><mi>X</mi> <mn>0</mn></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (d_0, d_1) : X_1 \hookrightarrow X_0 \times X_0 \,. </annotation></semantics></math></div> <p>Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">X_1</annotation></semantics></math> is the set of <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> of the category that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is the nerve of, this means that there is at most one morphism in this category from any one object to any other. Hence this category is a <a class="existingWikiWord" href="/nlab/show/poset">poset</a>.</p> <p>Finally to show that this poset is an inhabited linear order, we use the fact that a functor is flat precisely if its <a class="existingWikiWord" href="/nlab/show/category+of+elements">category of elements</a> <a class="existingWikiWord" href="/nlab/show/filtered+category">cofiltered</a>.</p> <p>This means</p> <ol> <li> <p>The category of elements is inhabited, hence the poset of which <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is the nerve is inhabited.</p> </li> <li> <p>For every two elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>,</mo><mi>z</mi><mo>∈</mo><msub><mi>X</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">y, z \in X_0</annotation></semantics></math> there exist morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>,</mo><mi>β</mi><mo>:</mo><mo stretchy="false">[</mo><mn>0</mn><mo stretchy="false">]</mo><mo>→</mo><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\alpha, \beta : [0] \to [k]</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi></mrow><annotation encoding="application/x-tex">\Delta</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>w</mi><mo>∈</mo><msub><mi>X</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">w \in X_k</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo><mo>:</mo><mi>w</mi><mo>↦</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">X(\alpha) : w \mapsto y</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo stretchy="false">(</mo><mi>β</mi><mo stretchy="false">)</mo><mo>:</mo><mi>w</mi><mo>↦</mo><mi>z</mi></mrow><annotation encoding="application/x-tex">X(\beta) : w \mapsto z</annotation></semantics></math>. Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is the nerve of a poset, this means that there is a totally ordered set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>w</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mi>w</mi> <mn>0</mn></msub><mo>≤</mo><mi>⋯</mi><mo>≤</mo><msub><mi>w</mi> <mi>k</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">w = (w_0 \leq \cdots \leq w_k)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>z</mi></mrow><annotation encoding="application/x-tex">z</annotation></semantics></math> are among its elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>=</mo><msub><mi>w</mi> <mrow><mi>α</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">y = w_{\alpha(0)}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>z</mi><mo>=</mo><msub><mi>w</mi> <mrow><mi>β</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">z = w_{\beta(0)}</annotation></semantics></math>. Accordingly we have either <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>≤</mo><mi>z</mi></mrow><annotation encoding="application/x-tex">y \leq z</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>z</mi><mo>≤</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">z \leq y</annotation></semantics></math> and hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is in fact the nerve of a <a class="existingWikiWord" href="/nlab/show/total+order">total order</a>.</p> </li> <li> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>,</mo><mi>z</mi></mrow><annotation encoding="application/x-tex">y,z</annotation></semantics></math> are elements in the total order with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>≤</mo><mi>z</mi></mrow><annotation encoding="application/x-tex">y \leq z</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>z</mi><mo>≤</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">z \leq y</annotation></semantics></math>, this means that in the nerve we have elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo><mo>∈</mo><msub><mi>X</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">(y,z) \in X_1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>∈</mo><msub><mi>X</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">(z,y) \in X_1</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>d</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">d_0(y,z) = d_1(z,y)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>d</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">d_1(y,z) = d_1(z,y)</annotation></semantics></math>.</p> <p>By co-filtering, there exists a <a class="existingWikiWord" href="/nlab/show/cone">cone</a> over this diagram in the category of elements, hence morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>,</mo><mi>β</mi><mo>:</mo><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo>→</mo><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\alpha, \beta : [1] \to [k]</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi></mrow><annotation encoding="application/x-tex">\Delta</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>w</mi><mo>∈</mo><msub><mi>X</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">w \in X_k</annotation></semantics></math> such that</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo><mo>:</mo><mi>w</mi><mo>↦</mo><mo stretchy="false">(</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X(\alpha) : w \mapsto (y,z)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo stretchy="false">(</mo><mi>β</mi><mo stretchy="false">)</mo><mo>:</mo><mi>w</mi><mo>↦</mo><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X(\beta) : w \mapsto (z,y)</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∂</mo> <mn>0</mn></msub><mo>∘</mo><mi>α</mi><mo>=</mo><msub><mo>∂</mo> <mn>1</mn></msub><mo>∘</mo><mi>β</mi></mrow><annotation encoding="application/x-tex">\partial_0 \circ \alpha = \partial_1 \circ \beta</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∂</mo> <mn>1</mn></msub><mo>∘</mo><mi>α</mi><mo>=</mo><msub><mo>∂</mo> <mn>0</mn></msub><mo>∘</mo><mi>β</mi></mrow><annotation encoding="application/x-tex">\partial_1 \circ \alpha = \partial_0 \circ \beta</annotation></semantics></math>.</p> </li> </ol> <p>Here the last condition in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi></mrow><annotation encoding="application/x-tex">\Delta</annotation></semantics></math> can only hold if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>=</mo><mi>β</mi><mo>=</mo><msub><mi>const</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\alpha = \beta = const_{i}</annotation></semantics></math>, hence if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>=</mo><mi>z</mi></mrow><annotation encoding="application/x-tex">y = z</annotation></semantics></math>.</p> </li> </ol> <p>Conversely, assume that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is the nerve of a linear order. We show that then it is a flat functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>:</mo><msup><mi>Δ</mi> <mi>op</mi></msup><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">X : \Delta^{op} \to Set</annotation></semantics></math>.</p> <p>(…)</p> </div> <h3 id="ForIntervals">For intervals</h3> <p><a class="existingWikiWord" href="/nlab/show/Andre+Joyal">Andre Joyal</a> showed that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Set</mi> <mrow><msup><mi>Δ</mi> <mi>op</mi></msup></mrow></msup></mrow><annotation encoding="application/x-tex">Set^{{\Delta}^{op}}</annotation></semantics></math>, the category of <a class="existingWikiWord" href="/nlab/show/simplicial+sets">simplicial sets</a>, is the classifying topos for <a class="existingWikiWord" href="/nlab/show/linear+intervals">linear intervals</a>.</p> <p>Specifically a <a class="existingWikiWord" href="/nlab/show/geometric+morphism">geometric morphism</a> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Set</mi></mrow><annotation encoding="application/x-tex">Set</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Set</mi> <mrow><msup><mi>Δ</mi> <mi>op</mi></msup></mrow></msup></mrow><annotation encoding="application/x-tex">Set^{{\Delta}^{op}}</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/linear+interval">linear interval</a> in <a class="existingWikiWord" href="/nlab/show/Set">Set</a>, meaning a <a class="existingWikiWord" href="/nlab/show/totally+ordered+set">totally ordered set</a> with distinct <a class="existingWikiWord" href="/nlab/show/top">top</a> and <a class="existingWikiWord" href="/nlab/show/bottom">bottom</a> elements. In general, a linear interval is a model for the one-sorted <a class="existingWikiWord" href="/nlab/show/geometric+theory">geometric theory</a> whose <a class="existingWikiWord" href="/nlab/show/signature">signature</a> consists of a binary <a class="existingWikiWord" href="/nlab/show/relation">relation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≤</mo></mrow><annotation encoding="application/x-tex">\leq</annotation></semantics></math> and two <span class="newWikiWord">constants<a href="/nlab/new/constant">?</a></span> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math>, subject to the following <a class="existingWikiWord" href="/nlab/show/axiom">axioms</a>:</p> <ul> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊢</mo><mo stretchy="false">(</mo><mi>x</mi><mo>≤</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\vdash (x \leq x)</annotation></semantics></math></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∃</mo> <mi>y</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>≤</mo><mi>y</mi><mo stretchy="false">)</mo><mo>∧</mo><mo stretchy="false">(</mo><mi>y</mi><mo>≤</mo><mi>z</mi><mo stretchy="false">)</mo><mo>⊢</mo><mo stretchy="false">(</mo><mi>x</mi><mo>≤</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\exists_y (x \leq y) \wedge (y \leq z) \vdash (x \leq z)</annotation></semantics></math></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>≤</mo><mi>y</mi><mo stretchy="false">)</mo><mo>∧</mo><mo stretchy="false">(</mo><mi>y</mi><mo>≤</mo><mi>x</mi><mo stretchy="false">)</mo><mo>⊢</mo><mo stretchy="false">(</mo><mi>x</mi><mo>=</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x \leq y) \wedge (y \leq x) \vdash (x = y)</annotation></semantics></math></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊢</mo><mo stretchy="false">(</mo><mi>x</mi><mo>≤</mo><mi>y</mi><mo stretchy="false">)</mo><mo>∨</mo><mo stretchy="false">(</mo><mi>y</mi><mo>≤</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\vdash (x \leq y) \vee (y \leq x)</annotation></semantics></math></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊢</mo><mo stretchy="false">(</mo><mn>0</mn><mo>≤</mo><mi>x</mi><mo stretchy="false">)</mo><mo>∧</mo><mo stretchy="false">(</mo><mi>x</mi><mo>≤</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\vdash (0 \leq x) \wedge (x \leq 1)</annotation></semantics></math></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>=</mo><mn>1</mn><mo stretchy="false">)</mo><mo>⊢</mo><mi>false</mi></mrow><annotation encoding="application/x-tex">(0 = 1) \vdash false</annotation></semantics></math></li> </ul> <p>(Joyal calls this a <strong>strict</strong> linear interval; by removing the hypothesis of distinct top and bottom, one arrives at a weaker notion he calls “linear interval”. Linear intervals in this sense are classified by the topos <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Set</mi> <mrow><msubsup><mi>Δ</mi> <mi>a</mi> <mi>op</mi></msubsup></mrow></msup></mrow><annotation encoding="application/x-tex">Set^{\Delta_{a}^{op}}</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>a</mi></msub></mrow><annotation encoding="application/x-tex">\Delta_a</annotation></semantics></math>, sometimes called the algebraist’s Delta or the augmented <a class="existingWikiWord" href="/nlab/show/simplex+category">simplex category</a>, is the category of all finite ordinals including the empty one.)</p> <p>The generic such interval is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Δ</mi> <mn>1</mn></msup><mo>∈</mo><msup><mi>Set</mi> <mrow><msup><mi>Δ</mi> <mi>op</mi></msup></mrow></msup></mrow><annotation encoding="application/x-tex">\Delta^1 \in Set^{{\Delta}^{op}}</annotation></semantics></math>; see <a class="existingWikiWord" href="/nlab/show/generic+interval">generic interval</a> for more details and references.</p> <h3 id="for_abstract_circles">For abstract circles</h3> <p>The category of <a class="existingWikiWord" href="/nlab/show/cyclic+sets">cyclic sets</a> is the classifying topos for <a class="existingWikiWord" href="/nlab/show/abstract+circles">abstract circles</a> (<a href="#Moerdijk96">Moerdijk 96</a>).</p> <h3 id="for_local_rings">For local rings</h3> <h4 id="local_rings">Local rings</h4> <p>The classifying topos for <a class="existingWikiWord" href="/nlab/show/local+ring">local rings</a> is the <a class="existingWikiWord" href="/nlab/show/big+Zariski+topos">big Zariski topos</a> of the <a class="existingWikiWord" href="/nlab/show/scheme">scheme</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spec</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spec(\mathbb{Z})</annotation></semantics></math>. A <strong>local ring</strong> is a model of the geometric theory of commutative unital rings subject to the extra axioms</p> <ul> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>=</mo><mn>1</mn><mo stretchy="false">)</mo><mo>⊢</mo><mi>false</mi></mrow><annotation encoding="application/x-tex">(0 = 1) \vdash false</annotation></semantics></math></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>+</mo><mi>y</mi><mo>=</mo><mn>1</mn><mo>⊢</mo><msub><mo>∃</mo> <mi>z</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mi>z</mi><mo>=</mo><mn>1</mn><mo stretchy="false">)</mo><mo>∨</mo><msub><mo>∃</mo> <mi>z</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mi>z</mi><mo>=</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x + y = 1 \vdash \exists_z (x z = 1) \vee \exists_z (y z = 1)</annotation></semantics></math></li> </ul> <p>In a topos of sheaves over a <a class="existingWikiWord" href="/nlab/show/sober+space">sober space</a>, a local ring is precisely what algebraic geometers usually call a “sheaf of local rings”: namely, a sheaf of rings all of whose <a class="existingWikiWord" href="/nlab/show/stalk">stalks</a> are local. See <a class="existingWikiWord" href="/nlab/show/locally+ringed+topos">locally ringed topos</a>. This is a special case of the case of <a href="#CoverPreservingFLatFunctors">Cover-preserving flat functors</a> below.</p> <h4 id="StrictLocalRings">Strict local rings</h4> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spec</mi><mi>R</mi></mrow><annotation encoding="application/x-tex">Spec R</annotation></semantics></math> an <a class="existingWikiWord" href="/nlab/show/affine+scheme">affine scheme</a>, the <a class="existingWikiWord" href="/nlab/show/%C3%A9tale+topos">étale topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sh</mi><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>et</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh(X_{et})</annotation></semantics></math> classifies “<a class="existingWikiWord" href="/nlab/show/strict+local+ring">strict local R-algebras</a>”. The <a class="existingWikiWord" href="/nlab/show/point+of+a+topos">points of this topos</a> are <em><span class="newWikiWord">strict Henselian R-algebras<a href="/nlab/new/strict+Henselian+ring">?</a></span></em> (<a href="#Hakim">Hakim, III.2-4</a>) and (<a href="#Wraith">Wraith</a>).</p> <p>See also <a href="http://mathoverflow.net/questions/48690/what-does-an-etale-topos-classify">this MO discussion</a></p> <h3 id="PrincipalBund">For principal bundles</h3> <p>Essentially every topos may be regarded as a classifying topos for certain <a class="existingWikiWord" href="/nlab/show/torsor">torsor</a>s/<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a>s.</p> <h4 id="BareGTor">Over bare groups</h4> <p>For any (bare / discrete) <a class="existingWikiWord" href="/nlab/show/group">group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>, write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math> for its <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> groupoid, the groupoid with a single object and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> as its endomorphisms. The presheaf topos</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>G</mi><mi>Set</mi><mo>:</mo><mo>=</mo><mi>PSh</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> G Set := PSh(\mathbf{B}G) </annotation></semantics></math></div> <p>of <a class="existingWikiWord" href="/nlab/show/permutation+representation">permutation representation</a>s (objects are <a class="existingWikiWord" href="/nlab/show/set">set</a>s equipped with a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/action">action</a>, morphisms are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-equivariant maps between these) is the classifying topos for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/torsor">torsor</a>s.</p> <p>For example, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, geometric morphisms from the <a class="existingWikiWord" href="/nlab/show/sheaf+topos">sheaf topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sh</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh(X)</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/sheaves">sheaves</a> on (the <a class="existingWikiWord" href="/nlab/show/category+of+open+subsets">category of open subsets</a> of) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mi>Set</mi></mrow><annotation encoding="application/x-tex">G Set</annotation></semantics></math> are the same as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a>s over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>G</mi><mi>Bund</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>Topos</mi><mo stretchy="false">(</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><mi>G</mi><mi>Set</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> G Bund(X) \simeq Topos(Sh(X), G Set) \,. </annotation></semantics></math></div> <p>This follows via <a class="existingWikiWord" href="/nlab/show/Diaconescu%27s+theorem">Diaconescu's theorem</a>, which asserts that <a class="existingWikiWord" href="/nlab/show/geometric+morphism">geometric morphism</a>s <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sh</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Sh</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh(X) \to Sh(\mathbf{B}G)</annotation></semantics></math> are equivalent to <a class="existingWikiWord" href="/nlab/show/flat+functor">flat functor</a>s</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>→</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{B}G \to Sh(X) \,. </annotation></semantics></math></div> <p>Such a flat functor picks a single sheaf on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and encodes a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-action on this sheaf such that this sheaf is the sheaf of <a class="existingWikiWord" href="/nlab/show/section">section</a>s of a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> <div class="num_theorem"> <h6 id="theorem">Theorem</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> be a (bare, discrete) group, write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℬ</mi><mi>G</mi><mo>∈</mo></mrow><annotation encoding="application/x-tex">\mathcal{B}G \in </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Top">Top</a> for the ordinary <a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>∈</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}G \in </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Grpd">Grpd</a> the one-object groupoid version of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>. There is a canonical <a class="existingWikiWord" href="/nlab/show/geometric+morphism">geometric morphism</a>s</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>PSh</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>ℬ</mi><mi>G</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> PSh(\mathbf{B}G) \to Sh(\mathcal{B}G) \,. </annotation></semantics></math></div> <p>This is a <em>weak homotopy equivalence</em> of toposes, in that it induces isomorphisms on <a class="existingWikiWord" href="/nlab/show/geometric+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">geometric homotopy groups</a> of the terminal object.</p> </div> <p>This is (<a href="#Moerdijk95">Moerdijk 95, theorem 1.1, proven in chapter IV</a>).</p> <h4 id="BareGTorGeomTheo">In terms of geometric theories</h4> <p>A <a class="existingWikiWord" href="/nlab/show/geometric+theory">geometric theory</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> whose <a class="existingWikiWord" href="/nlab/show/model">model</a>s are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-torsors can be described as follows. It has one sort, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, and one unary operation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">g:X\to X</annotation></semantics></math> for every element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>∈</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">g\in G</annotation></semantics></math>. It has algebraic axioms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊤</mo><msub><mo>⊢</mo> <mi>x</mi></msub><mspace width="thickmathspace"></mspace><mn>1</mn><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">\top\vdash_x \;1(x) = x</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊤</mo><msub><mo>⊢</mo> <mi>x</mi></msub><mspace width="thickmathspace"></mspace><mi>g</mi><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>g</mi><mi>h</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\top\vdash_x \;g(h(x)) = (g h)(x)</annotation></semantics></math>, which make <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> into a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-set, and geometric axioms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊤</mo><mo>⊢</mo><mspace width="thickmathspace"></mspace><mo>∃</mo><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\top \vdash\; \exists x \in X</annotation></semantics></math> (inhabited-ness), <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>x</mi><mspace width="thickmathspace"></mspace><msub><mo>⊢</mo> <mi>x</mi></msub><mspace width="thickmathspace"></mspace><mo>⊥</mo></mrow><annotation encoding="application/x-tex">g(x) = x \;\vdash_x \;\bot</annotation></semantics></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>≠</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">g\neq 1</annotation></semantics></math> (freeness), and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊤</mo><msub><mo>⊢</mo> <mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></msub><mspace width="thickmathspace"></mspace><msub><mo lspace="thinmathspace" rspace="thinmathspace">⋁</mo> <mrow><mi>g</mi><mo>∈</mo><mi>G</mi></mrow></msub><mspace width="thickmathspace"></mspace><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">\top\vdash_{x,y}\; \bigvee_{g\in G}\; g(x) = y</annotation></semantics></math> (transitivity).</p> <h4 id="TopGTor">Over topological groups</h4> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> is a general <a class="existingWikiWord" href="/nlab/show/topological+group">topological group</a> we have a <a class="existingWikiWord" href="/nlab/show/simplicial+object">simplicial</a> <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>G</mi> <mrow><mo>×</mo><mo>•</mo></mrow></msup></mrow><annotation encoding="application/x-tex">G^{\times \bullet}</annotation></semantics></math>. The category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sh</mi><mo stretchy="false">(</mo><msup><mi>G</mi> <mrow><mo>×</mo><mo>•</mo></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh(G^{\times \bullet})</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/sheaves+on+a+simplicial+topological+space">sheaves on this simplicial space</a> is a topos.</p> <p>This is such that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a topological space, geometric morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sh</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Sh</mi><mo stretchy="false">(</mo><msup><mi>G</mi> <mrow><mo>×</mo><mo>•</mo></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh(X) \to Sh(G^{\times \bullet})</annotation></semantics></math> classifies topological <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-principal bundles on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> <p>This idea admits generalizations to <a class="existingWikiWord" href="/nlab/show/localic+groups">localic groups</a> — and even to <a class="existingWikiWord" href="/nlab/show/localic+groupoids">localic groupoids</a>. For more details, see <a class="existingWikiWord" href="/nlab/show/classifying+topos+of+a+localic+groupoid">classifying topos of a localic groupoid</a> .</p> <h4 id="UniversalBundle">The universal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-bundle topos</h4> <p>At <a class="existingWikiWord" href="/nlab/show/generalized+universal+bundle">generalized universal bundle</a> and <a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a> it is discussed that the principal bundle classified by a morphims into a classifying object is its <a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</a>, and how the universal bundle is a replacement of the point such that its ordinary pullback models that <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a>.</p> <p>Concretely, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/group">group</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>=</mo><mo stretchy="false">{</mo><mo>•</mo><mover><mo>→</mo><mrow><mi>g</mi><mo>∈</mo><mi>G</mi></mrow></mover><mo>•</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}G = \{\bullet \stackrel{g \in G}{\to} \bullet\}</annotation></semantics></math> in <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a> its <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a>, the universal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-bundle is really just the point inclusion</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ * \\ \downarrow \\ \mathbf{B}G } </annotation></semantics></math></div> <p>in that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">X \to \mathbf{B}G</annotation></semantics></math> a morphism, the corresponding <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a> in <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a> is the <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>P</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><msup><mrow></mrow> <mo>≃</mo></msup><mo>⇙</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ P &\to& * \\ \downarrow &{}^{\simeq}\swArrow& \downarrow \\ X &\to& \mathbf{B}G } \,. </annotation></semantics></math></div> <p>We can send this morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo>*</mo><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(* \to \mathbf{B}G)</annotation></semantics></math> in <a class="existingWikiWord" href="/nlab/show/Grpd">Grpd</a> with</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>PSh</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>:</mo><mi>Grpd</mi><mo>→</mo><mi>Toposes</mi></mrow><annotation encoding="application/x-tex"> PSh(-) : Grpd \to Toposes </annotation></semantics></math></div> <p>to the <span class="newWikiWord">2-category of toposes<a href="/nlab/new/2-category+of+toposes">?</a></span> to get a <a class="existingWikiWord" href="/nlab/show/geometric+morphism">geometric morphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>PSh</mi><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo><mo>=</mo><mi>Set</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>p</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>PSh</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>Set</mi> <mi>G</mi></msup></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ PSh(*) = Set \\ \downarrow^{\mathrlap{p}} \\ PSh(\mathbf{B}G) = Set^G } \,. </annotation></semantics></math></div> <p>By the rules of morphisms of <a class="existingWikiWord" href="/nlab/show/site">site</a>s we have that the <a class="existingWikiWord" href="/nlab/show/inverse+image">inverse image</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>p</mi> <mo>*</mo></msup><mo>:</mo><mi>PSh</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">p^* : PSh(\mathbf{B}G) \to Set</annotation></semantics></math> is precomposition with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><mo>*</mo><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">p : * \to \mathbf{B}G</annotation></semantics></math>, i.e. the functor that just forgets the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-action on a set.</p> <p>Its <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mo>!</mo></msub><mo>:</mo><mi>Set</mi><mo>→</mo><mi>PSh</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p_! : Set \to PSh(\mathbf{B}G)</annotation></semantics></math> is the functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mo>!</mo></msub><mo>:</mo><mi>S</mi><mo>↦</mo><mi>S</mi><mo>×</mo><mi>G</mi></mrow><annotation encoding="application/x-tex"> p_! : S \mapsto S \times G </annotation></semantics></math></div> <p>which sends a set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> to the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>×</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">S \times G</annotation></semantics></math> equipped with the evident <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-action induced by that of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> on itself.</p> <p>Because for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mi>ρ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(V,\rho)</annotation></semantics></math> any set with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-action <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math> we have naturally</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mi>Set</mi></msub><mo stretchy="false">(</mo><mi>S</mi><mo>,</mo><mi>V</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>Hom</mi> <mrow><msup><mi>Set</mi> <mi>G</mi></msup></mrow></msub><mo stretchy="false">(</mo><mi>S</mi><mo>×</mo><mi>G</mi><mo>,</mo><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mi>ρ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Hom_{Set}(S,V) \simeq Hom_{Set^G}(S \times G, (V,\rho)) \,. </annotation></semantics></math></div> <p>The object</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mo>!</mo></msub><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo><mo>=</mo><mi>G</mi><mo>∈</mo><mi>PSh</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> p_!(*) = G \in PSh(\mathbf{B}G) </annotation></semantics></math></div> <p>singled out in this way is the universal object in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Set</mi> <mi>G</mi></msup></mrow><annotation encoding="application/x-tex">Set^G</annotation></semantics></math>, namely <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> equipped with the canonical <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-action on itself.</p> <p>It ought to be true that the topos-incarnation of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-principal bundle on a topological space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> classified by a <a class="existingWikiWord" href="/nlab/show/geometric+morphism">geometric morphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sh</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mi>PSh</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh(X) \to PSh(\mathbf{B}G)</annotation></semantics></math> is the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(2,1)</annotation></semantics></math>-pullback</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>𝒫</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>Set</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><msup><mrow></mrow> <mo>≃</mo></msup><mo>⇙</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>Sh</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>PSh</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \mathcal{P} &\to& Set \\ \downarrow &{}^{\simeq}\swArrow& \downarrow \\ Sh(X) &\to& PSh(\mathbf{B}G) } \,. </annotation></semantics></math></div> <blockquote> <p>needs more discussion…</p> </blockquote> <h3 id="ForLocalicGroupoids">For general localic groupoids</h3> <p>In fact, <em>any</em> <a class="existingWikiWord" href="/nlab/show/Grothendieck+topos">Grothendieck topos</a> can be thought of as a classifying topos for some <a class="existingWikiWord" href="/nlab/show/localic+groupoid">localic groupoid</a>. This is related to the discussion above, since Joyal and Tierney showed that any Grothendieck topos is equivalent to the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mi>G</mi></mrow><annotation encoding="application/x-tex">B G</annotation></semantics></math> for some <a class="existingWikiWord" href="/nlab/show/localic+groupoid">localic groupoid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>. A useful discussion of this idea starts <a href="http://golem.ph.utexas.edu/category/2007/10/geometric_representation_theor_2.html#c012724">here</a>.</p> <h3 id="for_flat_functors">For flat functors</h3> <p>As a special case of the above, any <a class="existingWikiWord" href="/nlab/show/presheaf+topos">presheaf topos</a>, i.e. any topos of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Set</mi> <mrow><msup><mi>C</mi> <mi>op</mi></msup></mrow></msup></mrow><annotation encoding="application/x-tex">Set^{C^{op}}</annotation></semantics></math>, is the classifying topos for <a class="existingWikiWord" href="/nlab/show/flat+functors">flat functors</a> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> (sometimes also called “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/torsor">torsor</a>s”). In other words, geometric morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>→</mo><msup><mi>Set</mi> <mrow><msup><mi>C</mi> <mi>op</mi></msup></mrow></msup></mrow><annotation encoding="application/x-tex">E \to Set^{C^{op}}</annotation></semantics></math> are the same as <a class="existingWikiWord" href="/nlab/show/flat+functor">flat functor</a>s <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>→</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">C \to E</annotation></semantics></math>. This is <a class="existingWikiWord" href="/nlab/show/Diaconescu%27s+theorem">Diaconescu's theorem</a>. If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> has finite limits, then a flat functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>→</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">C \to E</annotation></semantics></math> is the same as a functor that preserves finite limits.</p> <h3 id="CoverPreservingFLatFunctors">For geometric theories / cover-preserving flat functors on a site</h3> <p>Another way, apart from that <a href="#ForLocalicGroupoids">above</a>, of viewing any <a class="existingWikiWord" href="/nlab/show/Grothendieck+topos">Grothendieck topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> as a classifying topos is to start with a small <a class="existingWikiWord" href="/nlab/show/site">site</a> of definition for it. Any such site gives rise to a <a class="existingWikiWord" href="/nlab/show/geometric+theory">geometric theory</a> called the theory of <a class="existingWikiWord" href="/nlab/show/cover-preserving+functor">cover-preserving</a> flat functors on that site (also called the <a class="existingWikiWord" href="/nlab/show/theory+of+flat+functors"> theory of J-continuous flat functors</a>, for syntactic details see there!). The classifying topos of this theory is again <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>.</p> <p>Moreover, for any object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>, there is a small site of definition for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> which includes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, and thus for which <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is (part of) the universal object.</p> <p>We have:</p> <div class="num_prop" id="EveryToposIsAClassifyingToposForLocalAlgebras"> <h6 id="proposition_3">Proposition</h6> <p>Every <a class="existingWikiWord" href="/nlab/show/sheaf+topos">sheaf topos</a> has a <a class="existingWikiWord" href="/nlab/show/cartesian+site">cartesian site</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mi>J</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C}, J)</annotation></semantics></math> of definition.</p> <p>This <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sh</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mi>J</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh(\mathcal{C}, J)</annotation></semantics></math> is the classifying topos for cover-preserving <a class="existingWikiWord" href="/nlab/show/flat+functor">flat functor</a>s out of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>.</p> <p>Every category of such functors is the category of <a class="existingWikiWord" href="/nlab/show/model">model</a>s of some geometric theory, and for every geometric theory there is such a cartesian site.</p> </div> <p>This appears as (<a href="#Johnstone">Johnstone, remark D3.1.13</a>).</p> <h3 id="LocalAlgebras">For local algebras</h3> <p>As a special case or rather re-interpretation of the <a href="#CoverPreservingFLatFunctors">above</a>, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒯</mi></mrow><annotation encoding="application/x-tex">\mathcal{T}</annotation></semantics></math> be any <a class="existingWikiWord" href="/nlab/show/essentially+algebraic+theory">essentially algebraic theory</a> and equip its <a class="existingWikiWord" href="/nlab/show/syntactic+category">syntactic category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mi>𝕋</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{C}_{\mathbb{T}}</annotation></semantics></math> with some <a class="existingWikiWord" href="/nlab/show/coverage">coverage</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math>. Then the <a class="existingWikiWord" href="/nlab/show/sheaf+topos">sheaf topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sh</mi><mo stretchy="false">(</mo><msub><mi>𝒞</mi> <mi>𝕋</mi></msub><mo>,</mo><mi>J</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh(\mathcal{C}_{\mathbb{T}}, J)</annotation></semantics></math> is the classifying topos for <em>local <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕋</mi></mrow><annotation encoding="application/x-tex">\mathbb{T}</annotation></semantics></math>-algebras</em> :</p> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sh</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh(X)</annotation></semantics></math> any <a class="existingWikiWord" href="/nlab/show/sheaf+topos">sheaf topos</a> a <a class="existingWikiWord" href="/nlab/show/geometric+morphism">geometric morphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒪</mi><mo>:</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Sh</mi><mo stretchy="false">(</mo><msub><mi>𝒞</mi> <mi>𝕋</mi></msub><mo>,</mo><mi>J</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathcal{O} : Sh(X) \to Sh(\mathcal{C}_{\mathbb{T}}, J) </annotation></semantics></math></div> <p>is</p> <ol> <li> <p>a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕋</mi></mrow><annotation encoding="application/x-tex">\mathbb{T}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/algebra+over+an+algebraic+theory">algebra</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sh</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh(X)</annotation></semantics></math>, hence a <a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕋</mi></mrow><annotation encoding="application/x-tex">\mathbb{T}</annotation></semantics></math>-algebras over the site <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>;</p> </li> <li> <p>such that this sheaf of algebras is local as seen by the respective topologies.</p> </li> </ol> <p>See <a class="existingWikiWord" href="/nlab/show/locally+algebra-ed+topos">locally algebra-ed topos</a> for more on this.</p> <p>By prop. <a class="maruku-ref" href="#EveryToposIsAClassifyingToposForLocalAlgebras"></a> we have that every <a class="existingWikiWord" href="/nlab/show/sheaf+topos">sheaf topos</a> is the classifying topos of <em>some</em> theory of local algebras.</p> <p>The <a class="existingWikiWord" href="/nlab/show/vertical+categorification">vertical categorification</a> of this situation to the context of <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> theory is the notion of <a class="existingWikiWord" href="/nlab/show/structured+%28%E2%88%9E%2C1%29-topos">structured (∞,1)-topos</a> and of <a class="existingWikiWord" href="/nlab/show/geometry+%28for+structured+%28%E2%88%9E%2C1%29-toposes%29">geometry (for structured (∞,1)-toposes)</a>:</p> <p>The geometry <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒢</mi></mrow><annotation encoding="application/x-tex">\mathcal{G}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> that plays role of the syntactic theory. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒳</mi></mrow><annotation encoding="application/x-tex">\mathcal{X}</annotation></semantics></math> an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a>, a model of this theory is a limits and covering-preserving <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒢</mi><mo>→</mo><mi>𝒳</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathcal{G} \to \mathcal{X} \,. </annotation></semantics></math></div> <p>The <a class="existingWikiWord" href="/nlab/show/Yoneda+embedding">Yoneda embedding</a> followed by <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stackification">∞-stackification</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒢</mi><mover><mo>→</mo><mi>Y</mi></mover><msub><mi>PSh</mi> <mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>𝒢</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><mover><mo stretchy="false">(</mo><mo stretchy="false">¯</mo></mover><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></mover><msub><mi>Sh</mi> <mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>𝒢</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathcal{G} \stackrel{Y}{\to} PSh_{(\infty,1)}(\mathcal{G}) \stackrel{\bar(-)}{\to} Sh_{(\infty,1)}(\mathcal{G}) </annotation></semantics></math></div> <p>constitutes a model of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒢</mi></mrow><annotation encoding="application/x-tex">\mathcal{G}</annotation></semantics></math> in the (Cech) <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stack">∞-stack</a> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Sh</mi> <mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>𝒢</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh_{(\infty,1)}(\mathcal{G})</annotation></semantics></math> and exhibits it as the classifying topos for such models (geometries):</p> <p>This is <em><a class="existingWikiWord" href="/nlab/show/Structured+Spaces">Structured Spaces</a></em> <a href="http://arxiv.org/PS_cache/arxiv/pdf/0905/0905.0459v1.pdf#page=26">prop 1.4.2</a>.</p> <h2 id="as_a_generalization_of_the_notion_of_classifying_space_in_topology">As a generalization of the notion of classifying space in topology</h2> <p>In view of the analogy between the classifying topos denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mi>G</mi></mrow><annotation encoding="application/x-tex">B G</annotation></semantics></math>, such that the <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mi>Bund</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G Bund(X)</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a>s over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is equivalent to geometric morphims <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sh</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mi>B</mi><mi>G</mi></mrow><annotation encoding="application/x-tex">Sh(X) \to B G</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>G</mi><mi>Bund</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>Topos</mi><mo stretchy="false">(</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><mi>B</mi><mi>G</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex"> G Bund(X) \simeq Topos(Sh(X), B G) \, </annotation></semantics></math></div> <p>and the notion of <a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a> in <a class="existingWikiWord" href="/nlab/show/topology">topology</a>, which for the discrete group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℬ</mi><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathcal{B} G</annotation></semantics></math> such that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><mi>G</mi><mi>Bund</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>π</mi> <mn>0</mn></msub><mi>Top</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℬ</mi><mi>G</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex"> \pi_0 G Bund(X) \simeq \pi_0 Top(X, \mathcal{B}G) \, </annotation></semantics></math></div> <p>we should expect there to be a topos analog of the total space, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mi>G</mi></mrow><annotation encoding="application/x-tex">E G</annotation></semantics></math>, for the classifying space. This analog is the <em>generic G-torsor</em>, which is an internal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-torsor in the topos <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Set</mi> <mi>G</mi></msup></mrow><annotation encoding="application/x-tex">Set^G</annotation></semantics></math>. The important aspect of the space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mi>G</mi></mrow><annotation encoding="application/x-tex">E G</annotation></semantics></math> is that as a principal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-bundle over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℬ</mi><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathcal{B} G</annotation></semantics></math>, it is a <em>universal element</em>, i.e. the natural transformation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Hom</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℬ</mi><mi>G</mi><mo stretchy="false">)</mo><mo>→</mo><mi>G</mi><mi>Bdl</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Hom(X, \mathcal{B}G) \to G Bdl(X)</annotation></semantics></math> that it induces (by the <a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a>) is the isomorphism which exhibits <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℬ</mi><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathcal{B}G</annotation></semantics></math> as the object representing the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>↦</mo><mi>G</mi><mi>Bdl</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \mapsto G Bdl(X)</annotation></semantics></math>. For the same Yoneda reasons, the classifying topos <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sh</mi><mo stretchy="false">(</mo><msub><mi>C</mi> <mi>T</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh(C_T)</annotation></semantics></math> of any geometric theory <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> comes with a <em>generic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>-model</em>, which is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>-model in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sh</mi><mo stretchy="false">(</mo><msub><mi>C</mi> <mi>T</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh(C_T)</annotation></semantics></math> which represents the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>↦</mo><mi>T</mi><mi>Mod</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E \mapsto T Mod(E)</annotation></semantics></math> in the same way. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> = the theory of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-torsors, this generic model is the generic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-torsor.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/theory+of+flat+functors">theory of cover-preserving flat functors</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/representable+functor">representable functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/classifying+topos+for+the+theory+of+objects">classifying topos for the theory of objects</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a>, <a class="existingWikiWord" href="/nlab/show/classifying+stack">classifying stack</a>, <a class="existingWikiWord" href="/nlab/show/moduli+space">moduli space</a>, <a class="existingWikiWord" href="/nlab/show/moduli+stack">moduli stack</a>, <a class="existingWikiWord" href="/nlab/show/derived+moduli+space">derived moduli space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+principal+bundle">universal principal bundle</a>, <a class="existingWikiWord" href="/nlab/show/universal+principal+infinity-bundle">universal principal infinity-bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/classifying+morphism">classifying morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/classifying+%28infinity%2C1%29-topos">classifying (infinity,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/natural+number+object">natural number object</a></p> </li> </ul> <h2 id="references">References</h2> <h3 id="general">General</h3> <p>Early references containing some remarks on the formation of the concept are</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Myles+Tierney">Myles Tierney</a>, <em>Forcing Topologies and Classifying Toposes</em> , pp.211-219 in Heller, Tierney (eds.), <em>Algebra, Topology and Category Theory</em> , Academic Press New York 1976.</p> </li> <li id="J77"> <p><a class="existingWikiWord" href="/nlab/show/Peter+Johnstone">Peter Johnstone</a>, <em>Topos Theory</em> , Academic Press New York 1977. (Also available as Dover reprint Mineola 2014)</p> </li> </ul> <p>Standard textbook references for classifying topoi of theories</p> <ul> <li id="Johnstone"> <p><a class="existingWikiWord" href="/nlab/show/Peter+Johnstone">Peter Johnstone</a>, <em><a class="existingWikiWord" href="/nlab/show/Sketches+of+an+Elephant">Sketches of an Elephant</a></em> , Oxford UP 2002. (In particular, sections B4.2 pp.424-432, D3.2 pp.901-910)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Francis+Borceux">Francis Borceux</a>, <em>Handbook of categorical algebra</em>, (in series Enc. Math. Appl.) vol. 3, Categories and sheaves, Cambridge Univ. Press 1994, Ch.4 Classifying toposes</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Saunders+Mac+Lane">Saunders Mac Lane</a>, <a class="existingWikiWord" href="/nlab/show/Ieke+Moerdijk">Ieke Moerdijk</a>, <em><a class="existingWikiWord" href="/nlab/show/Sheaves+in+Geometry+and+Logic">Sheaves in Geometry and Logic</a></em>, Springer Heidelberg 1994. (chap. VIII)</p> </li> </ul> <p>A more advanced reference containing several developments of the general theory, especially in relation with the view of toposes as ‘bridges’, is the monograph</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Olivia+Caramello">Olivia Caramello</a>, <a href="https://global.oup.com/academic/product/theories-sites-toposes-9780198758914"><em>Theories, Sites, Toposes: Relating and studying mathematical theories through topos-theoretic ‘bridges’</em></a>, Oxford University Press, 2017.</li> </ul> <p>The relation between the existence of natural number objects and classifying toposes is discussed in</p> <ul> <li id="blass"> <p><a class="existingWikiWord" href="/nlab/show/Andreas+Blass">Andreas Blass</a>, <em>Classifying topoi and the axiom of infinity</em>, Algebra Universalis <strong>26</strong> (1989) 341-345 [<a href="https://doi.org/10.1007/BF01211840">doi:10.1007/BF01211840</a>]</p> <blockquote> <p>(in relation to the <a class="existingWikiWord" href="/nlab/show/axiom+of+infinity">axiom of infinity</a>)</p> </blockquote> </li> </ul> <p>The study of classifying spaces of topological categories is described in the monograph</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ieke+Moerdijk">Ieke Moerdijk</a>, <em>Classifying spaces, classifying topoi</em>, <p>Lec. Notes Math. 1616, Springer Verlag 1995</p> </li> </ul> <p>The original theory for a general algebraic theory is developed in</p> <ul> <li>M. Makkai, G. Reyes, <em>First-order categorical logic</em>, Lecture Notes in Mathematics 611, Springer 1977.</li> </ul> <p>The results for the continuous groupoids include</p> <ul> <li id="Moerdijk88"> <p><a class="existingWikiWord" href="/nlab/show/Ieke+Moerdijk">Ieke Moerdijk</a>, The classifying topos of a continuous groupoid I, <em>Trans. A.M.S.</em> <strong>310</strong> (1988), 629-668.</p> </li> <li id="Moerdijk90"> <p><a class="existingWikiWord" href="/nlab/show/Ieke+Moerdijk">Ieke Moerdijk</a>, <em>The classifying topos of a continuous groupoid II</em>, Cahiers de Topologie et Géométrie Différentielle Catégoriques, <strong>31</strong> no. 2 (1990), 137-168. (<a href="http://www.numdam.org/numdam-bin/fitem?id=CTGDC_1990__31_2_137_0">web</a>)</p> </li> <li id="Moerdijk95"> <p><a class="existingWikiWord" href="/nlab/show/Ieke+Moerdijk">Ieke Moerdijk</a>, <em>Classifying spaces and classifying topoi</em>, Lecture Notes in Math. <strong>1616</strong>, Springer-Verlag, New York, 1995.</p> </li> <li id="Moerdijk96"> <p><a class="existingWikiWord" href="/nlab/show/Ieke+Moerdijk">Ieke Moerdijk</a>, <em>Cyclic sets as a classifying topos</em>, 1996 (<a class="existingWikiWord" href="/nlab/files/MoerdijkCyclic.pdf" title="pdf">pdf</a>)</p> </li> </ul> <p>Classifying toposes as <a class="existingWikiWord" href="/nlab/show/locally+algebra-ed+%28infinity%2C1%29-toposes">locally algebra-ed (infinity,1)-toposes</a> are discussed in section 1.4 of</p> <ul id="Lurie"> <li><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, <em><a class="existingWikiWord" href="/nlab/show/Structured+Spaces">Structured Spaces</a></em></li> </ul> <p>The <a class="existingWikiWord" href="/nlab/show/%C3%A9tale+topos">étale topos</a> as a classifying topos for <a class="existingWikiWord" href="/nlab/show/strict+local+rings">strict local rings</a> is discussed in</p> <ul> <li id="Hakim"><a class="existingWikiWord" href="/nlab/show/Monique+Hakim">Monique Hakim</a>, <em>Topos annelés et schémas relatifs</em>, Sec. III.2-4</li> <li id="Wraith"><a class="existingWikiWord" href="/nlab/show/Gavin+Wraith">Gavin Wraith</a>, <em>Generic Galois theory of local rings</em></li> </ul> <p><a class="existingWikiWord" href="/nlab/show/Nikolai+Durov">Nikolai Durov</a> has introduced somewhat a generalization of topos called <a class="existingWikiWord" href="/nlab/show/vectoid">vectoid</a> and quite flexible notion of a classifying vectoid in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Nikolai+Durov">Nikolai Durov</a>, <em>Classifying vectoids and generalisations of operads</em>, <a href="http://arxiv.org/abs/1105.3114">arxiv/1105.3114</a>, the translation of “Классифицирующие вектоиды и классы операд”, Trudy MIAN, vol. 273</li> </ul> <h3 id="ReferencesRelationToForcing">Relation to forcing</h3> <p>Reviews of the interpretation of <em><a class="existingWikiWord" href="/nlab/show/forcing">forcing</a></em> as the passge to classifying toposes include</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Andreas+Blass">Andreas Blass</a>, <a class="existingWikiWord" href="/nlab/show/Andrej+%C5%A0%C4%8Dedrov">Andrej Ščedrov</a>, <em>Classifying topoi and finite forcing</em>, Journal of Pure and Applied Algebra <strong>28</strong> (1983) 111-140 [<a href="https://doi.org/10.1016/0022-4049(83)90085-3">doi:10.1016/0022-4049(83)90085-3</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Andrej+%C5%A0%C4%8Dedrov">Andrej Ščedrov</a>, <em>Forcing and classifying topoi</em>, Memoirs of the American Mathematical Society (1984) [<a href="https://bookstore.ams.org/memo-48-295/5">AMS:memo-48-295</a>]</p> </li> </ul> <p>For more see</p> <ul> <li id="Roberts15"><a class="existingWikiWord" href="/nlab/show/David+Roberts">David Roberts</a>, <em>Class forcing and topos theory</em>, talk at <em><a href="https://indico.math.cnrs.fr/event/747/">Topos at l’IHES</a></em> 2015 (<a href="https://www.dropbox.com/s/vk1efw7lsvta80p/Roberts_IHES.pdf?dl=0">talk notes pdf</a>, <a href="https://youtu.be/4AaSySq8-GQ">video recording</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on October 27, 2022 at 12:00:24. 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