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float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="topos_theory">Topos Theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/topos+theory">topos theory</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Toposes">Toposes</a></li> </ul> <h2 id="background">Background</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></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> </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="category_theory">Category Theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></strong></p> <h2 id="sidebar_concepts">Concepts</h2> <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/Cat">Cat</a></p> </li> </ul> <h2 id="sidebar_universal_constructions">Universal constructions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+construction">universal construction</a></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/adjoint+functor">adjoint functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/limit">limit</a>/<a class="existingWikiWord" href="/nlab/show/colimit">colimit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/weighted+limit">weighted limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/end">end</a>/<a class="existingWikiWord" href="/nlab/show/coend">coend</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kan+extension">Kan extension</a></p> </li> </ul> </li> </ul> <h2 id="sidebar_theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+construction">Grothendieck construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+functor+theorem">adjoint functor theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monadicity+theorem">monadicity theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+lifting+theorem">adjoint lifting theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gabriel-Ulmer+duality">Gabriel-Ulmer duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freyd-Mitchell+embedding+theorem">Freyd-Mitchell embedding theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relation+between+type+theory+and+category+theory">relation between type theory and category theory</a></p> </li> </ul> <h2 id="sidebar_extensions">Extensions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf+and+topos+theory">sheaf and topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a></p> </li> </ul> <h2 id="sidebar_applications">Applications</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/applications+of+%28higher%29+category+theory">applications of (higher) category theory</a></li> </ul> <div> <p> <a href="/nlab/edit/category+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#definition'>Definition</a></li> <ul> <li><a href='#related_2categories'>Related 2-categories</a></li> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#from_topological_spaces_to_toposes'>From topological spaces to toposes</a></li> <li><a href='#from_toposes_to_higher_toposes'>From toposes to higher toposes</a></li> <li><a href='#AdjunctionToLocallyPresentable'>From locally presentable categories to toposes</a></li> <li><a href='#Limits'>Limits and colimits</a></li> <ul> <li><a href='#Colimits'>Colimits</a></li> <li><a href='#Pullbacks'>Pullbacks</a></li> </ul> <li><a href='#free_loop_spaces'>Free loop spaces</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="definition">Definition</h2> <p>By <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Topos</mi></mrow><annotation encoding="application/x-tex">Topos</annotation></semantics></math></strong> (or <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Toposes</mi></mrow><annotation encoding="application/x-tex">Toposes</annotation></semantics></math></strong>) is denoted the <a class="existingWikiWord" href="/nlab/show/category">category</a> of <a class="existingWikiWord" href="/nlab/show/topos">topos</a>es. Usually this means:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/object">object</a>s are <a class="existingWikiWord" href="/nlab/show/topos">topos</a>es;</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/morphism">morphism</a>s are <a class="existingWikiWord" href="/nlab/show/geometric+morphism">geometric morphism</a>s of toposes.</p> </li> </ul> <p>This is naturally a <a class="existingWikiWord" href="/nlab/show/2-category">2-category</a>, where</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/2-morphism">2-morphism</a> are <a class="existingWikiWord" href="/nlab/show/geometric+transformation">geometric transformation</a>s</li> </ul> <p>That is, a 2-morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>→</mo><mi>g</mi></mrow><annotation encoding="application/x-tex">f\to g</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a> <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><msup><mi>g</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">f^* \to g^*</annotation></semantics></math> (which is, by <a class="existingWikiWord" href="/nlab/show/mate">mate</a> calculus, equivalent to a natural transformation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mo>*</mo></msub><mo>→</mo><msub><mi>f</mi> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">g_* \to f_*</annotation></semantics></math> between <a class="existingWikiWord" href="/nlab/show/direct+image">direct image</a>s). Thus, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Toposes</mi></mrow><annotation encoding="application/x-tex">Toposes</annotation></semantics></math> is equivalent to both of</p> <ul> <li>the (non-full) <a class="existingWikiWord" href="/nlab/show/sub-2-category">sub-2-category</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Cat</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">Cat^{op}</annotation></semantics></math> on categories that are toposes and morphisms that are the inverse image parts of geometric morphisms, and</li> <li>the (non-full) sub-2-category of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Cat</mi> <mi>co</mi></msup></mrow><annotation encoding="application/x-tex">Cat^{co}</annotation></semantics></math> on categories that are toposes and morphisms that are the direct image parts of geometric morphisms.</li> </ul> <h3 id="related_2categories">Related 2-categories</h3> <ul> <li> <p>There is also the sub-2-category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ShToposes</mi><mo>=</mo><mi>GrToposes</mi></mrow><annotation encoding="application/x-tex">ShToposes = GrToposes</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/sheaf+topos">sheaf topos</a>es (i.e. Grothendieck toposes).</p> </li> <li> <p>Note that in some literature this 2-category is denoted merely <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Top</mi></mrow><annotation encoding="application/x-tex">Top</annotation></semantics></math>, but that is also commonly used to denote <a class="existingWikiWord" href="/nlab/show/Top">the category</a> of <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a>.</p> </li> <li> <p>We obtain a very different 2-category of toposes if we take the morphisms to be <a class="existingWikiWord" href="/nlab/show/logical+functors">logical functors</a>; this 2-category is sometimes denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Log</mi></mrow><annotation encoding="application/x-tex">Log</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>LogTopos</mi></mrow><annotation encoding="application/x-tex">LogTopos</annotation></semantics></math>.</p> </li> </ul> <h2 id="properties">Properties</h2> <h3 id="from_topological_spaces_to_toposes">From topological spaces to toposes</h3> <p>The operation of forming <a class="existingWikiWord" href="/nlab/show/categories+of+sheaves">categories of sheaves</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Sh</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>:</mo><mi>Top</mi><mo>→</mo><mi>ShToposes</mi></mrow><annotation encoding="application/x-tex"> Sh(-) : Top \to ShToposes </annotation></semantics></math></div> <p>embeds <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>s into toposes. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f : X \to Y</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/continuous+map">continuous map</a> we have that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sh</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh(f)</annotation></semantics></math> is the <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>Sh</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>:</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><munder><mo>→</mo><mrow><msub><mi>f</mi> <mo>*</mo></msub></mrow></munder><mover><mo>←</mo><mrow><msup><mi>f</mi> <mo>*</mo></msup></mrow></mover></mover><mi>Sh</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Sh(f) : Sh(X) \stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}} Sh(Y) </annotation></semantics></math></div> <p>with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">f_*</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/direct+image">direct image</a> 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/inverse+image">inverse image</a>.</p> <p>Strictly speaking, this functor is not an <a class="existingWikiWord" href="/nlab/show/embedding">embedding</a> if we consider <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Top</mi></mrow><annotation encoding="application/x-tex">Top</annotation></semantics></math> as a 1-category and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Toposes</mi></mrow><annotation encoding="application/x-tex">Toposes</annotation></semantics></math> as a 2-category, since it is then not <a class="existingWikiWord" href="/nlab/show/fully+faithful">fully faithful</a> in the 2-categorical sense—there can be nontrivial 2-cells between geometric morphisms between toposes of sheaves on topological spaces.</p> <p>However, if we regard <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Top</mi></mrow><annotation encoding="application/x-tex">Top</annotation></semantics></math> as a <a class="existingWikiWord" href="/nlab/show/%281%2C2%29-category">(1,2)-category</a> where the 2-cells are inequalities in the <a class="existingWikiWord" href="/nlab/show/specialization+ordering">specialization ordering</a>, then this functor does become a 2-categorically full embedding (i.e. an equivalence on hom-categories) if we restrict to the full subcategory <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SobTop</mi></mrow><annotation encoding="application/x-tex">SobTop</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/sober+spaces">sober spaces</a>. This embedding can also be extended from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SobTop</mi></mrow><annotation encoding="application/x-tex">SobTop</annotation></semantics></math> to the entire category of <a class="existingWikiWord" href="/nlab/show/locales">locales</a> (which can be viewed as “Grothendieck 0-toposes”).</p> <h3 id="from_toposes_to_higher_toposes">From toposes to higher toposes</h3> <p>There are similar full embeddings <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ShTopos</mi><mo>↪</mo><mi>Sh</mi><mn>2</mn><mi>Topos</mi></mrow><annotation encoding="application/x-tex">ShTopos \hookrightarrow Sh 2 Topos</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ShTopos</mi><mo>↪</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mi>Topos</mi></mrow><annotation encoding="application/x-tex">ShTopos \hookrightarrow Sh(n,1)Topos</annotation></semantics></math> of sheaf (1-)toposes into <a class="existingWikiWord" href="/nlab/show/2-sheaf">2-sheaf</a> <a class="existingWikiWord" href="/nlab/show/2-topos">2-topos</a>es and sheaf <a class="existingWikiWord" href="/nlab/show/%28n%2C1%29-topos">(n,1)-topos</a>es for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><mo>≤</mo><mi>n</mi><mo>≤</mo><mn>∞</mn></mrow><annotation encoding="application/x-tex">2\le n\le \infty</annotation></semantics></math>. Note that these embeddings are <em>not</em> the identity functor on underlying categories: a 1-topos is not itself an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-topos, instead we have to take <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-sheaves on a suitable generating <a class="existingWikiWord" href="/nlab/show/site">site</a> for it.</p> <h3 id="AdjunctionToLocallyPresentable">From locally presentable categories to toposes</h3> <p>There is a canonical <a class="existingWikiWord" href="/nlab/show/forgetful+functor">forgetful functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>:</mo><mi>Topos</mi><mo>→</mo></mrow><annotation encoding="application/x-tex"> U : Topos \to</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a> that lands, by definition, in the sub-2-category of <a class="existingWikiWord" href="/nlab/show/locally+presentable+categories">locally presentable categories</a> and <a class="existingWikiWord" href="/nlab/show/functors">functors</a> which preserve all limits / are <a class="existingWikiWord" href="/nlab/show/right+adjoints">right adjoints</a>.</p> <p>This <a class="existingWikiWord" href="/nlab/show/2-functor">2-functor</a> has a <a class="existingWikiWord" href="/nlab/show/2-adjunction">right 2-adjoint</a> (<a href="#BungeCarboni">Bunge-Carboni</a>).</p> <h3 id="Limits">Limits and colimits</h3> <p>The 2-category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Topos</mi></mrow><annotation encoding="application/x-tex">Topos</annotation></semantics></math> is not all that well-endowed with <a class="existingWikiWord" href="/nlab/show/limits">limits</a>, but its <a class="existingWikiWord" href="/nlab/show/slice+categories">slice categories</a> are finitely complete <a class="existingWikiWord" href="/nlab/show/2-limit">as 2-categories</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ShTopos</mi></mrow><annotation encoding="application/x-tex">ShTopos</annotation></semantics></math> is closed under finite limits in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Topos</mi><mo stretchy="false">/</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">Topos/Set</annotation></semantics></math>. In particular, the <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ShToposes</mi></mrow><annotation encoding="application/x-tex">ShToposes</annotation></semantics></math> is the topos <a class="existingWikiWord" href="/nlab/show/Set">Set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≃</mo><mi>Sh</mi><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\simeq Sh(*)</annotation></semantics></math>.</p> <h4 id="Colimits">Colimits</h4> <p>The supply with colimits is better:</p> <div class="num_prop" id="ColimitsByInverseImageLimits"> <h6 id="proposition">Proposition</h6> <p>All small (indexed) <a class="existingWikiWord" href="/nlab/show/2-colimit">2-colimit</a>s in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ShTopos</mi></mrow><annotation encoding="application/x-tex">ShTopos</annotation></semantics></math> exists and are computed as (indexed) <a class="existingWikiWord" href="/nlab/show/2-limit">2-limit</a>s in <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a> of the underlying <a class="existingWikiWord" href="/nlab/show/inverse+image">inverse image</a> functors.</p> </div> <p>This appears as (<a href="#Moerdijk">Moerdijk, theorem 2.5</a>)</p> <div class="num_prop"> <h6 id="proposition_2">Proposition</h6> <p>Let</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>ℱ</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>p</mi> <mn>2</mn></msub></mrow></mover></mtd> <mtd><msub><mi>ℰ</mi> <mn>2</mn></msub></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>p</mi> <mn>1</mn></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><mo>⇙</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>f</mi> <mn>2</mn></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>ℰ</mi> <mn>1</mn></msub></mtd> <mtd><munder><mo>→</mo><mrow><msub><mi>f</mi> <mn>1</mn></msub></mrow></munder></mtd> <mtd><mi>ℰ</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathcal{F} &\stackrel{p_2}{\to}& \mathcal{E}_2 \\ {}^{\mathllap{p_1}}\downarrow &\swArrow& \downarrow^{\mathrlap{f_2}} \\ \mathcal{E}_1 &\underset{f_1}{\to}& \mathcal{E} } </annotation></semantics></math></div> <p>be a <a class="existingWikiWord" href="/nlab/show/2-pullback">2-pullback</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Topos</mi></mrow><annotation encoding="application/x-tex">Topos</annotation></semantics></math> such that</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>f</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">f_1, f_2</annotation></semantics></math> are both <a class="existingWikiWord" href="/nlab/show/pseudomonic+morphism">pseudomonic morphism</a>s</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℰ</mi> <mn>1</mn></msub><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><msub><mi>ℰ</mi> <mn>2</mn></msub><mo>→</mo><mi>ℰ</mi></mrow><annotation encoding="application/x-tex">\mathcal{E}_1 \coprod \mathcal{E}_2 \to \mathcal{E}</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/effective+epimorphism">effective epimorphism</a>;</p> </li> </ul> <p>then the diagram of <a class="existingWikiWord" href="/nlab/show/inverse+image">inverse image</a> functors</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>ℱ</mi></mtd> <mtd><mover><mo>←</mo><mrow><msubsup><mi>p</mi> <mn>2</mn> <mo>*</mo></msubsup></mrow></mover></mtd> <mtd><msub><mi>ℰ</mi> <mn>2</mn></msub></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msubsup><mi>p</mi> <mn>1</mn> <mo>*</mo></msubsup></mrow></mpadded></msup><mo stretchy="false">↑</mo></mtd> <mtd><mo>⇙</mo></mtd> <mtd><msup><mo stretchy="false">↑</mo> <mpadded width="0"><mrow><msubsup><mi>f</mi> <mn>2</mn> <mo>*</mo></msubsup></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>ℰ</mi> <mn>1</mn></msub></mtd> <mtd><munder><mo>←</mo><mrow><msubsup><mi>f</mi> <mn>1</mn> <mo>*</mo></msubsup></mrow></munder></mtd> <mtd><mi>ℰ</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathcal{F} &\stackrel{p_2^*}{\leftarrow}& \mathcal{E}_2 \\ {}^{\mathllap{p_1^*}}\uparrow &\swArrow& \uparrow^{\mathrlap{f_2^*}} \\ \mathcal{E}_1 &\underset{f_1^*}{\leftarrow}& \mathcal{E} } </annotation></semantics></math></div> <p>is a 2-pullback in <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a> and so by the <a href="#ColimitsByInverseImageLimits">above</a> the original square is also a 2-pushout.</p> </div> <p>This appears as theorem 5.1 in (<a href="#BungeLack">BungeLack</a>)</p> <div class="num_prop"> <h6 id="proposition_3">Proposition</h6> <p>The 2-category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Topos</mi></mrow><annotation encoding="application/x-tex">Topos</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/extensive+category">extensive category</a>. Same for toposes bounded over a base.</p> </div> <p>This is in (<a href="#BungeLack">BungeLack, proposition 4.3</a>).</p> <h4 id="Pullbacks">Pullbacks</h4> <div class="num_prop"> <h6 id="proposition_4">Proposition</h6> <p>Let</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></mtd> <mtd></mtd> <mtd><mi>𝒳</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo stretchy="false">(</mo><msup><mi>g</mi> <mo>*</mo></msup><mo>⊣</mo><msub><mi>g</mi> <mo>*</mo></msub><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>𝒴</mi></mtd> <mtd><mover><mo>→</mo><mrow><mo stretchy="false">(</mo><msup><mi>f</mi> <mo>*</mo></msup><mo>⊣</mo><msub><mi>f</mi> <mo>*</mo></msub><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>𝒵</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && \mathcal{X} \\ && \downarrow^{\mathrlap{(g^* \dashv g_*)}} \\ \mathcal{Y} &\stackrel{(f^* \dashv f_*)}{\to}& \mathcal{Z} } </annotation></semantics></math></div> <p>be a <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a> of <a class="existingWikiWord" href="/nlab/show/toposes">toposes</a>. Then its <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> in the <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-category">(2,1)-category</a> version of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Topos</mi></mrow><annotation encoding="application/x-tex">Topos</annotation></semantics></math> is computed, roughly, by the <a class="existingWikiWord" href="/nlab/show/pushout">pushout</a> of their <a class="existingWikiWord" href="/nlab/show/sites">sites</a> of definition.</p> <p>More in detail: there exist <a class="existingWikiWord" href="/nlab/show/sites">sites</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>𝒟</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde \mathcal{D}</annotation></semantics></math>, <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>, and <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> with <a class="existingWikiWord" href="/nlab/show/finite+limit">finite limit</a>s and <a class="existingWikiWord" href="/nlab/show/morphisms+of+sites">morphisms of sites</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></mtd> <mtd></mtd> <mtd><mi>𝒟</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↑</mo> <mpadded width="0"><mi>g</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mover><mi>𝒟</mi><mo stretchy="false">˜</mo></mover></mtd> <mtd><mover><mo>←</mo><mi>f</mi></mover></mtd> <mtd><mi>𝒞</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && \mathcal{D} \\ && \uparrow^{\mathrlap{g}} \\ \tilde \mathcal{D} &\stackrel{f}{\leftarrow}& \mathcal{C} } </annotation></semantics></math></div> <p>such that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>𝒳</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo stretchy="false">(</mo><msup><mi>g</mi> <mo>*</mo></msup><mo>⊣</mo><msub><mi>g</mi> <mo>*</mo></msub><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>𝒴</mi></mtd> <mtd><mover><mo>→</mo><mrow><mo stretchy="false">(</mo><msup><mi>f</mi> <mo>*</mo></msup><mo>⊣</mo><msub><mi>f</mi> <mo>*</mo></msub><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>𝒵</mi></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mo>≃</mo><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mrow><mo>(</mo><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>Sh</mi><mo stretchy="false">(</mo><mi>𝒟</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo stretchy="false">(</mo><msub><mi>Lan</mi> <mi>g</mi></msub><mo>⊣</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>∘</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Sh</mi><mo stretchy="false">(</mo><mover><mi>𝒟</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><mo stretchy="false">(</mo><msub><mi>Lan</mi> <mi>f</mi></msub><mo>⊣</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>∘</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>Sh</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \left( \array{ && \mathcal{X} \\ && \downarrow^{\mathrlap{(g^* \dashv g_*)}} \\ \mathcal{Y} &\stackrel{(f^* \dashv f_*)}{\to}& \mathcal{Z} } \right) \,\,\, \simeq \,\,\, \left( \array{ && Sh(\mathcal{D}) \\ && \downarrow^{\mathrlap{(Lan_g \dashv (-)\circ g)}} \\ Sh(\tilde \mathcal{D}) &\stackrel{(Lan_f \dashv (-)\circ f)}{\to}& Sh(\mathcal{C}) } \right) \,. </annotation></semantics></math></div> <p>Let then</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><mover><mi>𝒟</mi><mo stretchy="false">˜</mo></mover><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mi>𝒞</mi></munder><mi>𝒟</mi></mtd> <mtd><mover><mo>←</mo><mrow><mi>f</mi><mo>′</mo></mrow></mover></mtd> <mtd><mi>𝒟</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mi>g</mi><mo>′</mo></mrow></mpadded></msup><mo stretchy="false">↑</mo></mtd> <mtd><msub><mo>⇙</mo> <mo>≃</mo></msub></mtd> <mtd><msup><mo stretchy="false">↑</mo> <mpadded width="0"><mi>g</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mover><mi>𝒟</mi><mo stretchy="false">˜</mo></mover></mtd> <mtd><mover><mo>←</mo><mi>f</mi></mover></mtd> <mtd><mi>𝒞</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mo>∈</mo><msup><mi>Cat</mi> <mi>lex</mi></msup></mrow><annotation encoding="application/x-tex"> \array{ \tilde \mathcal{D} \coprod_{\mathcal{C}} \mathcal{D} &\stackrel{f'}{\leftarrow}& \mathcal{D} \\ {}^{\mathllap{g'}}\uparrow &\swArrow_{\simeq}& \uparrow^{\mathrlap{g}} \\ \tilde \mathcal{D} &\stackrel{f}{\leftarrow}& \mathcal{C} } \,\,\,\,\, \in Cat^{lex} </annotation></semantics></math></div> <p>be the <a class="existingWikiWord" href="/nlab/show/pushout">pushout</a> of the underlying <a class="existingWikiWord" href="/nlab/show/categories">categories</a> in the <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mrow></mrow> <mi>lex</mi></msup><mo>⊂</mo><mi>Cat</mi></mrow><annotation encoding="application/x-tex">{}^{lex} \subset Cat</annotation></semantics></math> of categories with finite limits.</p> <p>Let moreover</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Sh</mi><mo stretchy="false">(</mo><mover><mi>𝒟</mi><mo stretchy="false">˜</mo></mover><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mi>𝒞</mi></munder><mi>𝒟</mi><mo stretchy="false">)</mo><mo>↪</mo><mi>PSh</mi><mo stretchy="false">(</mo><mover><mi>𝒟</mi><mo stretchy="false">˜</mo></mover><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mi>𝒞</mi></munder><mi>𝒟</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Sh(\tilde \mathcal{D} \coprod_{\mathcal{C}} \mathcal{D}) \hookrightarrow PSh(\tilde \mathcal{D} \coprod_{\mathcal{C}} \mathcal{D}) </annotation></semantics></math></div> <p>be the <a class="existingWikiWord" href="/nlab/show/reflective+subcategory">reflective subcategory</a> obtained by <a class="existingWikiWord" href="/nlab/show/localization">localization</a> at the class of morphisms generated by the inverse image <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Lan</mi> <mrow><mi>f</mi><mo>′</mo></mrow></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Lan_{f'}(-)</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/covering">covering</a>s of <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> and the inverse image <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Lan</mi> <mrow><mi>g</mi><mo>′</mo></mrow></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Lan_{g'}(-)</annotation></semantics></math> of the coverings of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>𝒟</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde \mathcal{D}</annotation></semantics></math>.</p> <p>Then</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>Sh</mi><mo stretchy="false">(</mo><mover><mi>𝒟</mi><mo stretchy="false">˜</mo></mover><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mi>𝒞</mi></munder><mi>𝒟</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>𝒳</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo stretchy="false">(</mo><msup><mi>g</mi> <mo>*</mo></msup><mo>⊣</mo><msub><mi>g</mi> <mo>*</mo></msub><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>𝒴</mi></mtd> <mtd><mover><mo>→</mo><mrow><mo stretchy="false">(</mo><msup><mi>f</mi> <mo>*</mo></msup><mo>⊣</mo><msub><mi>f</mi> <mo>*</mo></msub><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>𝒵</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ Sh(\tilde \mathcal{D} \coprod_{\mathcal{C}} \mathcal{D}) &\to& \mathcal{X} \\ \downarrow && \downarrow^{\mathrlap{(g^* \dashv g_*)}} \\ \mathcal{Y} &\stackrel{(f^* \dashv f_*)}{\to}& \mathcal{Z} } </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> square.</p> </div> <p>This appears for instance as (<a href="#Lurie">Lurie, prop. 6.3.4.6</a>).</p> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>For <a class="existingWikiWord" href="/nlab/show/localic+toposes">localic toposes</a> this reduces to the statement of <a class="existingWikiWord" href="/nlab/show/localic+reflection">localic reflection</a>: the pullback of toposes is given by the of the underlying <a class="existingWikiWord" href="/nlab/show/locales">locales</a> which in turn is the <a class="existingWikiWord" href="/nlab/show/pushout">pushout</a> of the corresponding <a class="existingWikiWord" href="/nlab/show/frames">frames</a>.</p> </div> <h3 id="free_loop_spaces">Free loop spaces</h3> <p>The <a class="existingWikiWord" href="/nlab/show/free+loop+space+object">free loop space object</a> of a topos in <a class="existingWikiWord" href="/nlab/show/Topos">Topos</a> is called the <a class="existingWikiWord" href="/nlab/show/isotropy+group+of+a+topos">isotropy group of a topos</a>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><strong>Topos</strong></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29Topos">(∞,1)Topos</a></p> </li> </ul> <h2 id="references">References</h2> <p>The characterization of colimits in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Topos</mi></mrow><annotation encoding="application/x-tex">Topos</annotation></semantics></math> is in</p> <ul id="Moerdijk"> <li><a class="existingWikiWord" href="/nlab/show/Ieke+Moerdijk">Ieke Moerdijk</a>, <em>The classifying topos of a continuous groupoid. I</em> Transaction of the American mathematical society Volume 310, Number 2, (1988) (<a href="http://www.ams.org/journals/tran/1988-310-02/S0002-9947-1988-0973173-9/S0002-9947-1988-0973173-9.pdf">pdf</a>)</li> </ul> <p>The fact that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Topos</mi></mrow><annotation encoding="application/x-tex">Topos</annotation></semantics></math> is extensive is in</p> <ul id="BungeLack"> <li><a class="existingWikiWord" href="/nlab/show/Marta+Bunge">Marta Bunge</a>, <a class="existingWikiWord" href="/nlab/show/Steve+Lack">Steve Lack</a>, <em>van Kampen theorem for toposes</em> (<a href="http://www.maths.usyd.edu.au/u/stevel/papers/vkt.ps.gz">ps</a>)</li> </ul> <p>Limits and colimits of toposes are discussed in 6.3.2-6.3.4 of</p> <ul id="Lurie"> <li><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, <em><a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">Higher Topos Theory</a></em> .</li> </ul> <p>There this is discussed for for <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a>es, but the statements are verbatim true also for ordinary toposes (in the <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-category">(2,1)-category</a> version of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Topos</mi></mrow><annotation encoding="application/x-tex">Topos</annotation></semantics></math>).</p> <p>The adjunction between toposes and locally presentable categories is discussed in</p> <ul id="BungeCarboni"> <li><a class="existingWikiWord" href="/nlab/show/Marta+Bunge">Marta Bunge</a>, <a class="existingWikiWord" href="/nlab/show/Aurelio+Carboni">Aurelio Carboni</a>, <em>The symmetric topos</em>, Journal of Pure and Applied Algebra 105:233-249, (1995)</li> </ul> <div class="property">category: <a class="category_link" href="/nlab/all_pages/category">category</a></div></body></html> </div> <div class="revisedby"> <p> Last revised on June 16, 2017 at 08:41:13. See the <a href="/nlab/history/Topos" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/Topos" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/2884/#Item_5">Discuss</a><span class="backintime"><a href="/nlab/revision/Topos/15" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/Topos" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/Topos" accesskey="S" class="navlink" id="history" rel="nofollow">History (15 revisions)</a> <a href="/nlab/show/Topos/cite" style="color: black">Cite</a> <a href="/nlab/print/Topos" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/Topos" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>