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It does not yet have a dedicated thread; feel free to create one, giving it the same name as the title of this page" style="color:black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; 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="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="compact_objects">Compact objects</h4> <div class="hide"><div> <p><strong>objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">d \in C</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>d</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C(d,-)</annotation></semantics></math> commutes with certain <a class="existingWikiWord" href="/nlab/show/colimits">colimits</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/coproduct-preserving+representable">coproduct-preserving representable</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connected+object">connected object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+object">compact object</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+element">compact object in a (0,1)-category</a>, <a class="existingWikiWord" href="/nlab/show/compact+object+in+an+%28%E2%88%9E%2C1%29-category">compact object in an (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/finite+object">finite object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/small+object">small object</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tiny+object">tiny object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinitesimal+object">atomic object</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/amazing+right+adjoint">amazing right adjoint</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/accessible+category">accessible category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/presentable+category">presentable category</a></li> </ul> </li> </ul> <h2 id="models">Models</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+topos">compact topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+topological+space">compact topological space</a></p> </li> </ul> <h2 id="relative_version">Relative version</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/proper+geometric+morphism">proper geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/proper+map">proper map</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/compact+object+-+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='#properties'>Properties</a></li> <ul> <li><a href='#stability_and_closure_properties'>Stability and closure properties</a></li> <li><a href='#properness_and_beckchevalley_conditions'>Properness and Beck-Chevalley conditions</a></li> <li><a href='#CompactSites'>Compact sites</a></li> <li><a href='#StronglyCompactSites'>Strongly compact sites</a></li> </ul> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#compact_toposes'>Compact toposes</a></li> <li><a href='#StronglyCompactToposes'>Strongly compact toposes</a></li> <li><a href='#finite_objects'>Finite objects</a></li> <li><a href='#GeometricStacks'>Geometric stacks</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#References'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>The notion of <em>compact topos</em> is the generalization from <a class="existingWikiWord" href="/nlab/show/topology">topology</a> to <a class="existingWikiWord" href="/nlab/show/topos+theory">topos theory</a> of the notion of <em><a class="existingWikiWord" href="/nlab/show/compact+topological+space">compact topological space</a></em>.</p> <p>More generally, over a general <a class="existingWikiWord" href="/nlab/show/base+topos">base topos</a>, the notion of <em>proper geometric morphism</em> is the generalization to morphisms between toposes of the notion of <em><a class="existingWikiWord" href="/nlab/show/proper+map">proper map</a></em> between <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a>.</p> <h2 id="Definition">Definition</h2> <div class="num_defn" id="CompactTopos"> <h6 id="definition_2">Definition</h6> <p>A <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>ℰ</mi></mrow><annotation encoding="application/x-tex">\mathcal{E}</annotation></semantics></math> is called a <strong><a class="existingWikiWord" href="/nlab/show/compact+topos">compact topos</a></strong> if the <a class="existingWikiWord" href="/nlab/show/direct+image">direct image</a> of the <a class="existingWikiWord" href="/nlab/show/global+section+geometric+morphism">global section geometric morphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi><mo>:</mo><mi>ℰ</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">\Gamma : \mathcal{E} \to Set</annotation></semantics></math> preserves <a class="existingWikiWord" href="/nlab/show/directed+colimit">directed</a> <a class="existingWikiWord" href="/nlab/show/joins">joins</a> of <a class="existingWikiWord" href="/nlab/show/subterminal+objects">subterminal objects</a>.</p> <p>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>ℱ</mi><mo>→</mo><mi>ℰ</mi></mrow><annotation encoding="application/x-tex">f : \mathcal{F} \to \mathcal{E}</annotation></semantics></math> is called <strong>proper</strong> if it exhibits <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> as a <a class="existingWikiWord" href="/nlab/show/compact+topos">compact topos</a> over <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>. (The <a class="existingWikiWord" href="/nlab/show/stack+semantics">stack semantics</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> can be used to formalize this.)</p> </div> <div class="num_defn" id="StronglyCompactTopos"> <h6 id="definition_3">Definition</h6> <p>A topos is called <strong>strongly compact</strong> if <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> commutes even with all <a class="existingWikiWord" href="/nlab/show/filtered+colimits">filtered colimits</a>.</p> <p>A geometric morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>ℱ</mi><mo>→</mo><mi>ℰ</mi></mrow><annotation encoding="application/x-tex">f : \mathcal{F} \to \mathcal{E}</annotation></semantics></math> is called <strong>tidy</strong> if it exhibits <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> as a strongly compact topos over <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>(<a href="#MoerdijkVermeulen">MV, p. 53</a>)</p> <p>This are the first stages of a notion that in <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos+theory">(∞,1)-topos theory</a> continue as follows</p> <div class="num_defn" id="StronglyCompactTopos"> <h6 id="definition_4">Definition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>κ</mi></mrow><annotation encoding="application/x-tex">\kappa</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/regular+cardinal">regular cardinal</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><mo>≤</mo><mi>n</mi><mo>≤</mo><mn>∞</mn></mrow><annotation encoding="application/x-tex"> -1 \leq n \leq \infty</annotation></semantics></math>. Then an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> is <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>κ</mi></mrow><annotation encoding="application/x-tex">\kappa</annotation></semantics></math>-compact of height <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></em> if the <a class="existingWikiWord" href="/nlab/show/global+section+geometric+morphism">global section geometric morphism</a> preserves <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>κ</mi></mrow><annotation encoding="application/x-tex">\kappa</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/filtered+%28%E2%88%9E%2C1%29-category">filtered</a> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-colimits">(∞,1)-colimits</a> of <a class="existingWikiWord" href="/nlab/show/n-truncated">n-truncated</a> objects.</p> <p>Accordingly a geometric morphism is <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>κ</mi></mrow><annotation encoding="application/x-tex">\kappa</annotation></semantics></math>-proper of height <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></em> if it exhibits a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>κ</mi></mrow><annotation encoding="application/x-tex">\kappa</annotation></semantics></math>-compact of height <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> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-topos over a <a class="existingWikiWord" href="/nlab/show/base+%28%E2%88%9E%2C1%29-topos">base (∞,1)-topos</a>.</p> </div> <p>In this terminology</p> <ul> <li> <p>a topos <em>compact of height (-1)</em> is the same as a <em>compact topos</em>;</p> </li> <li> <p>a topos <em>compact of height 0</em> is the same as a <em>strongly compact topos</em>;</p> </li> </ul> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>An <a class="existingWikiWord" href="/nlab/show/n-coherent+%28%E2%88%9E%2C1%29-topos">n-coherent (∞,1)-topos</a> is compact of height <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> in the sense of def. <a class="maruku-ref" href="#StronglyCompactTopos"></a>, this is (<a class="existingWikiWord" href="/nlab/show/Rational+and+p-adic+Homotopy+Theory">Lurie XIII, prop. 2.3.9</a>).</p> </div> <h2 id="properties">Properties</h2> <h3 id="stability_and_closure_properties">Stability and closure properties</h3> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <ol> <li> <p>Any <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence</a> is proper and the class of proper maps is closed under composition.</p> </li> <li> <p>If in the <a class="existingWikiWord" href="/nlab/show/diagram">diagram</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>A</mi></mtd> <mtd><mover><mo>→</mo><mi>p</mi></mover></mtd> <mtd><mi>B</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mi>f</mi></msup></mtd> <mtd><msup><mo>↙</mo> <mi>g</mi></msup></mtd></mtr> <mtr><mtd><mi>C</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ A&\xrightarrow{p}&B \\ \downarrow^f&\swarrow^g \\ C } </annotation></semantics></math></div> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/surjective+geometric+morphism">surjective geometric morphism</a> and <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 proper then so is <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>.</p> </li> <li> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math> is proper 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> is a <a class="existingWikiWord" href="/nlab/show/geometric+embedding">geometric embedding</a> then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> is proper.</p> </li> <li> <p>Any <a class="existingWikiWord" href="/nlab/show/hyperconnected+geometric+morphism">hyperconnected geometric morphism</a> is proper.</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>F</mi><mo>→</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">f:F\to G</annotation></semantics></math> is proper iff its <a class="existingWikiWord" href="/nlab/show/localic+reflection">localic reflection</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Sh</mi> <mi>G</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">Sh_G(X)\to G</annotation></semantics></math> is, i.e. iff <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/compact+locale">compact</a> <a class="existingWikiWord" href="/nlab/show/internal+locale">internal locale</a> in <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>.</p> </li> <li> <p>If in a pullback square the bottom morphism is open and surjective and the left morphism is proper then so is the right.</p> </li> </ol> </div> <p>(<a href="#MoerdijkVermeulen">VM, I.1, I.2</a>)</p> <div class="num_prop"> <h6 id="proposition_2">Proposition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> of a proper geometric morphism is again proper.</p> <p>The pullback of a tidy geometric morphism is again tidy.</p> </div> <p>(<a href="#MoerdijkVermeulen">VM, theorem 5.8</a>)</p> <h3 id="properness_and_beckchevalley_conditions">Properness and Beck-Chevalley conditions</h3> <p>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></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> of toposes is said to satisfy the <em>stable</em> (weak) Beck-Chevalley condition if any pullback 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> satisfies the (weak) <a class="existingWikiWord" href="/nlab/show/Beck-Chevalley+condition">Beck-Chevalley condition</a> ((weak)BCC).</p> <div class="num_prop"> <h6 id="proposition_3">Proposition</h6> <p>A map satisfies the stable weak BCC iff it is proper.</p> </div> <p>(<a href="#MoerdijkVermeulen">MV, Corollary I.5.9</a>)</p> <h3 id="CompactSites">Compact sites</h3> <p>We discuss classes of <a class="existingWikiWord" href="/nlab/show/sites">sites</a> such that their <a class="existingWikiWord" href="/nlab/show/sheaf+topos">sheaf topos</a> is a compact topos, def. <a class="maruku-ref" href="#CompactTopos"></a> (<a href="#MoerdijkVermeulen">VM, I.5</a>).</p> <p>(…)</p> <h3 id="StronglyCompactSites">Strongly compact sites</h3> <p>We discuss classes of <a class="existingWikiWord" href="/nlab/show/sites">sites</a> such that their <a class="existingWikiWord" href="/nlab/show/sheaf+topos">sheaf topos</a> is a strongly compact topos, def. <a class="maruku-ref" href="#CompactTopos"></a> (<a href="#MoerdijkVermeulen">VM, III.4</a>).</p> <p>(…)</p> <h2 id="examples">Examples</h2> <h3 id="compact_toposes">Compact toposes</h3> <div class="num_prop"> <h6 id="proposition_4">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/topos">topos</a> and <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>H</mi></mstyle></mrow><annotation encoding="application/x-tex">X \in \mathbf{H}</annotation></semantics></math> an <a class="existingWikiWord" href="/nlab/show/object">object</a>. If</p> <ul> <li> <p><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/compact+topological+space">compact topological space</a>-object” in that:</p> <p>for every set of morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><mi>X</mi><msub><mo stretchy="false">}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\{U_i \to X\}_{i \in I}</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\coprod_{i \in I} U_i \to X</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/effective+epimorphism">effective epimorphism</a>, there is a <a class="existingWikiWord" href="/nlab/show/finite+set">finite subset</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi><mo>⊂</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">J \subset I</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mrow><mi>i</mi><mo>∈</mo><mi>J</mi></mrow></msub><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\coprod_{i \in J} U_i \to X</annotation></semantics></math> is still an <a class="existingWikiWord" href="/nlab/show/effective+epimorphism">effective epimorphism</a>;</p> </li> </ul> <p>then</p> <ul> <li>The <a class="existingWikiWord" href="/nlab/show/slice+topos">slice topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbf{H}_{/X}</annotation></semantics></math> is a compact topos, def. <a class="maruku-ref" href="#CompactTopos"></a>.</li> </ul> </div> <div class="num_remark"> <h6 id="remark_2">Remark</h6> <p>Beware 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 a “compact topological space-object” is different from it being a <a class="existingWikiWord" href="/nlab/show/compact+object">compact object</a> (the difference being that between compactness of height (-1) and height 0). For the latter case see prop. <a class="maruku-ref" href="#SliceOverCompactIsStronglyCompact"></a> below.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>The <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbf{H}_{/X}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/identity">identity</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>id</mi> <mi>X</mi></msub><mo>:</mo><mi>X</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">id_X : X \to X</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>. A <a class="existingWikiWord" href="/nlab/show/subterminal+object">subterminal object</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbf{H}_{/X}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/monomorphism">monomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>↪</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">U \hookrightarrow X</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/global+section+geometric+morphism">global section geometric morphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Γ</mi> <mi>X</mi></msub><mo>:</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow></msub><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">\Gamma_X : \mathbf{H}_{/X} \to Set</annotation></semantics></math> sends an object <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>X</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[E \to X]</annotation></semantics></math> to its set of <a class="existingWikiWord" href="/nlab/show/sections">sections</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Γ</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mi>E</mi><mo>→</mo><mi>X</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>=</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>E</mi><mo stretchy="false">)</mo><msub><mo>×</mo> <mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">{</mo><msub><mi>id</mi> <mi>X</mi></msub><mo stretchy="false">}</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Gamma_X([E \to X]) = \mathbf{H}(X, E) \times_{\mathbf{H}(X,X)} \{id_X\} \,. </annotation></semantics></math></div> <p>Therefore it sends all subterminal object in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbf{H}_{/X}</annotation></semantics></math> to the <a class="existingWikiWord" href="/nlab/show/empty+set">empty set</a> except the terminal 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> itself, which is sent to the singleton set.</p> <p>So let <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> now be a compact-topological-space-object and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mo>•</mo></msub><mo>:</mo><mi>I</mi><mo>→</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow></msub></mrow><annotation encoding="application/x-tex">U_\bullet : I \to \mathbf{H}_{/X}</annotation></semantics></math> is directed system of subterminals.</p> <p>If their union <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∨</mo> <mi>i</mi></msub><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\vee_i U_i</annotation></semantics></math> does not cover <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>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Γ</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><msub><mo>∨</mo> <mi>i</mi></msub><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mi>∅</mi></mrow><annotation encoding="application/x-tex">\Gamma_X(\vee_i U_i) = \emptyset</annotation></semantics></math>. But then also none of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">U_i</annotation></semantics></math> can be <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> itself, and hence also <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Γ</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mi>∅</mi></mrow><annotation encoding="application/x-tex">\Gamma_X(U_i) = \emptyset</annotation></semantics></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">i \in I</annotation></semantics></math> and so <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∨</mo> <mi>i</mi></msub><msub><mi>Γ</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mi>∅</mi></mrow><annotation encoding="application/x-tex">\vee_i \Gamma_X(U_i) = \emptyset</annotation></semantics></math>. On the other hand, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∨</mo> <mi>i</mi></msub><msub><mi>U</mi> <mi>i</mi></msub><mo>=</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\vee_i U_i = X</annotation></semantics></math> then the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><mi>X</mi><msub><mo stretchy="false">}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\{U_i \to X\}_{i \in I}</annotation></semantics></math> form a cover, hence then by assumption there is a finite subset <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><mi>X</mi><msub><mo stretchy="false">}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>J</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\{U_i \to X\}_{i \in J}</annotation></semantics></math> which still covers. By the assumption that the system <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">U_\bullet</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/directed+set">directed set</a> it also contains the union <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>=</mo><msub><mo>∨</mo> <mrow><mi>i</mi><mo>∈</mo><mi>J</mi></mrow></msub><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">X = \vee_{i \in J} U_i</annotation></semantics></math>. Therefore <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∨</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><msub><mi>Γ</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>Γ</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">\vee_{i \in I} \Gamma_X(U_i) = \Gamma_X(X) = *</annotation></semantics></math> is the singleton, as is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Γ</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><msub><mo>∨</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>Γ</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Gamma_X(\vee_{i \in I} U_i) = \Gamma_X(X)</annotation></semantics></math>. So <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Γ</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\Gamma_X</annotation></semantics></math> preserves directed unions of subterminals and hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbf{H}_{/X}</annotation></semantics></math> is a compact topos.</p> </div> <h3 id="StronglyCompactToposes">Strongly compact toposes</h3> <p>The following propositions say in summary that</p> <ol> <li> <p>the <em><a class="existingWikiWord" href="/nlab/show/petit+topos">petit topos</a></em> over a <a class="existingWikiWord" href="/nlab/show/compact+topological+space">compact topological space</a> that is also <a class="existingWikiWord" href="/nlab/show/Hausdorff+topological+space">Hausdorff</a> is strongly compact.</p> </li> <li> <p>the <em><a class="existingWikiWord" href="/nlab/show/gros+topos">gros topos</a></em> over a <a class="existingWikiWord" href="/nlab/show/compact+object">compact object</a> is strongly compact.</p> </li> </ol> <p>See also (<a href="#MoerdijkVermeulen">VM, III.1</a>).</p> <div class="num_prop"> <h6 id="proposition_5">Proposition</h6> <p>Examples of strongly compact toposes <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>, def. <a class="maruku-ref" href="#StronglyCompactTopos"></a>, include the following.</p> <ol> <li> <p>Every <a class="existingWikiWord" href="/nlab/show/coherent+topos">coherent topos</a> is strongly compact.</p> </li> <li> <p>The <a class="existingWikiWord" href="/nlab/show/sheaf+topos">sheaf topos</a> over a <a class="existingWikiWord" href="/nlab/show/compact+topological+space">compact</a> <a class="existingWikiWord" href="/nlab/show/Hausdorff+space">Hausdorff</a> <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> is strongly compact.</p> </li> </ol> </div> <p>(<a href="#MoerdijkVermeulen">MV, Examples III.1.1</a>)</p> <div class="num_prop" id="SliceOverCompactIsStronglyCompact"> <h6 id="proposition_6">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> be a topos over <a class="existingWikiWord" href="/nlab/show/Set">Set</a> and <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>H</mi></mstyle></mrow><annotation encoding="application/x-tex">X \in \mathbf{H}</annotation></semantics></math> an object. Then the following are equivalent</p> <ol> <li> <p><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/compact+object">compact object</a> (in the sense that the <a class="existingWikiWord" href="/nlab/show/hom+functor">hom functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{H}(X,-)</annotation></semantics></math> preserves <a class="existingWikiWord" href="/nlab/show/filtered+colimits">filtered colimits</a>)</p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/slice+topos">slice topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbf{H}_{/X}</annotation></semantics></math> is strongly compact, def. <a class="maruku-ref" href="#StronglyCompactTopos"></a>.</p> </li> </ol> </div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>The <a class="existingWikiWord" href="/nlab/show/direct+image">direct image</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Γ</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\Gamma_X</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/global+section+geometric+morphism">global section geometric morphism</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><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>×</mo><mi>X</mi><mo>⊣</mo><msub><mi>Γ</mi> <mi>X</mi></msub><mo stretchy="false">)</mo><mo>:</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow></msub><mover><munder><mo>→</mo><mrow><msub><mstyle mathvariant="bold"><mi>Γ</mi></mstyle> <mi>X</mi></msub></mrow></munder><mover><mo>←</mo><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>×</mo><mi>X</mi></mrow></mover></mover><mstyle mathvariant="bold"><mi>H</mi></mstyle><mover><munder><mo>→</mo><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mo>*</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></munder><mover><mo>←</mo><mi>Δ</mi></mover></mover><mi>Set</mi></mrow><annotation encoding="application/x-tex"> ((-) \times X \dashv \Gamma_X) : \mathbf{H}_{/X} \stackrel{\overset{(-) \times X}{\leftarrow}}{\underset{\mathbf{\Gamma}_X}{\to}} \mathbf{H} \stackrel{\overset{\Delta}{\leftarrow}}{\underset{\mathbf{H}(*,-)}{\to}} Set </annotation></semantics></math></div> <p>is given by the <a class="existingWikiWord" href="/nlab/show/hom+functor">hom functor</a> out of the <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a>. The terminal object in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbf{H}_{/X}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/identity">identity</a> <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>id</mi> <mi>X</mi></msub><mo>:</mo><mi>X</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">id_X : X \to X</annotation></semantics></math>. So the terminal geometric morphism takes any <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>X</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[E \to X]</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbf{H}_{/X}</annotation></semantics></math> to the set of <a class="existingWikiWord" href="/nlab/show/sections">sections</a>, given by the <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> of the <a class="existingWikiWord" href="/nlab/show/hom+set">hom set</a> along the inclusion of the <a class="existingWikiWord" href="/nlab/show/identity">identity</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Γ</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mi>E</mi><mo>→</mo><mi>X</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>=</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>E</mi><mo stretchy="false">)</mo><msub><mo>×</mo> <mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">{</mo><mi>id</mi><mo stretchy="false">}</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Gamma_X([E \to X]) = \mathbf{H}(X,E) \times_{\mathbf{H}(X,X)} \{id\} \,. </annotation></semantics></math></div> <p>By the discussion at <em><a href="overcategory#LimitsAndColimits">overcategory – limits and colimits</a></em> we have that <a class="existingWikiWord" href="/nlab/show/colimits">colimits</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbf{H}_{/X}</annotation></semantics></math> are computed in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>. So if <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>X</mi><mo stretchy="false">]</mo><mo>≃</mo><munder><mi>lim</mi><mrow><msub><mo>⟶</mo> <mi>i</mi></msub></mrow></munder><mrow><mo stretchy="false">[</mo><msub><mi>E</mi> <mi>i</mi></msub><mo>→</mo><mi>X</mi><mo stretchy="false">]</mo></mrow></mrow><annotation encoding="application/x-tex">[E \to X] \simeq \underset{\longrightarrow_i}{\lim}{[E_i \to X]}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/filtered+colimit">filtered colimit</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbf{H}_{/X}</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>≃</mo><munder><mi>lim</mi><mrow><msub><mo>⟶</mo> <mi>i</mi></msub></mrow></munder><mrow><msub><mi>E</mi> <mi>i</mi></msub></mrow></mrow><annotation encoding="application/x-tex">E \simeq \underset{\longrightarrow_i}{\lim}{E_i }</annotation></semantics></math> is a filtered colimit in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>.</p> <p>If now <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>H</mi></mstyle></mrow><annotation encoding="application/x-tex">X \in \mathbf{H}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/compact+object">compact object</a>, then this commutes over this colimit and hence</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>Γ</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mi>E</mi><mo>→</mo><mi>X</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><munder><mi>lim</mi><mrow><msub><mo>⟶</mo> <mi>i</mi></msub></mrow></munder><msub><mi>E</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><msub><mo>×</mo> <mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">{</mo><mi>id</mi><mo stretchy="false">}</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mo stretchy="false">(</mo><munder><mi>lim</mi><mrow><msub><mo>⟶</mo> <mi>i</mi></msub></mrow></munder><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>E</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mo>×</mo> <mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">{</mo><mi>id</mi><mo stretchy="false">}</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><munder><mi>lim</mi><mrow><msub><mo>⟶</mo> <mi>i</mi></msub></mrow></munder><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>E</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><msub><mo>×</mo> <mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">{</mo><mi>id</mi><mo stretchy="false">}</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><munder><mi>lim</mi><mrow><msub><mo>⟶</mo> <mi>i</mi></msub></mrow></munder><msub><mi>Γ</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mo stretchy="false">[</mo><msub><mi>E</mi> <mi>i</mi></msub><mo>→</mo><mi>X</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} \Gamma_X([E \to X]) &= \mathbf{H}(X,\underset{\longrightarrow_i}{\lim} E_i) \times_{\mathbf{H}(X,X)} \{id\} \\ & \simeq (\underset{\longrightarrow_i}{\lim}\mathbf{H}(X, E_i)) \times_{\mathbf{H}(X,X)} \{id\} \\ &\simeq \underset{\longrightarrow_i}{\lim} (\mathbf{H}(X, E_i) \times_{\mathbf{H}(X,X)} \{id\}) \\ & \simeq \underset{\longrightarrow_i}{\lim} \Gamma_X([E_i \to X]) \end{aligned} \,, </annotation></semantics></math></div> <p>where in the second but last step we used that in the <a class="existingWikiWord" href="/nlab/show/topos">topos</a> <a class="existingWikiWord" href="/nlab/show/Set">Set</a> <a class="existingWikiWord" href="/nlab/show/universal+colimits">colimits are preserved by pullback</a>.</p> <p>This shows that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Γ</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>:</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow></msub><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">\Gamma_X(-) : \mathbf{H}_{/X} \to Set</annotation></semantics></math> commutes over filtered colimits 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/compact+object">compact object</a>.</p> <p>Conversely, assume that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Γ</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Gamma_X(-)</annotation></semantics></math> commutes over all filtered colimits. For every (<a class="existingWikiWord" href="/nlab/show/filtered+category">filtered</a>) <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mo>•</mo></msub><mo>:</mo><mi>I</mi><mo>→</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">F_\bullet : I \to \mathbf{H}</annotation></semantics></math> there is the corresponding filtered diagram <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><msub><mi>F</mi> <mo>•</mo></msub><mo>:</mo><mi>I</mi><mo>→</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow></msub></mrow><annotation encoding="application/x-tex">X \times F_\bullet : I \to \mathbf{H}_{/X}</annotation></semantics></math>, where <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><msub><mi>F</mi> <mi>i</mi></msub><mo>→</mo><mi>X</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[X \times F_i \to X]</annotation></semantics></math> is the projection. As before, the product with <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> preserves forming colimits</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mi>lim</mi><mrow><msub><mo>⟶</mo> <mi>i</mi></msub></mrow></munder><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mi>X</mi><mo>×</mo><msub><mi>F</mi> <mi>i</mi></msub><mo>→</mo><mi>X</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>≃</mo><mo stretchy="false">[</mo><mi>X</mi><mo>×</mo><mo stretchy="false">(</mo><munder><mi>lim</mi><mrow><msub><mo>⟶</mo> <mi>i</mi></msub></mrow></munder><msub><mi>F</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>→</mo><mi>X</mi><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \underset{\longrightarrow_i}{\lim} ([X \times F_i \to X]) \simeq [X \times (\underset{\longrightarrow_i}{\lim} F_i) \to X] \,. </annotation></semantics></math></div> <p>Moreover, sections of a trivial <a class="existingWikiWord" href="/nlab/show/bundle">bundle</a> are maps into the fiber</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Γ</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mi>X</mi><mo>×</mo><msub><mi>F</mi> <mi>i</mi></msub><mo>→</mo><mi>X</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>≃</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>F</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Gamma_X([X \times F_i \to X]) \simeq \mathbf{H}(X,F_i) \,. </annotation></semantics></math></div> <p>So it follows 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 compact object:</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><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><munder><mi>lim</mi><mrow><msub><mo>⟶</mo> <mi>i</mi></msub></mrow></munder><msub><mi>F</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mtd> <mtd><mo>≃</mo><msub><mi>Γ</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mi>X</mi><mo>×</mo><mo stretchy="false">(</mo><munder><mi>lim</mi><mrow><msub><mo>⟶</mo> <mi>i</mi></msub></mrow></munder><msub><mi>F</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>→</mo><mi>X</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><msub><mi>Γ</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><munder><mi>lim</mi><mrow><msub><mo>⟶</mo> <mi>i</mi></msub></mrow></munder><mo stretchy="false">[</mo><mi>X</mi><mo>×</mo><msub><mi>F</mi> <mi>i</mi></msub><mo>→</mo><mi>X</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><munder><mi>lim</mi><mrow><msub><mo>⟶</mo> <mi>i</mi></msub></mrow></munder><msub><mi>Γ</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mi>X</mi><mo>×</mo><msub><mi>F</mi> <mi>i</mi></msub><mo>→</mo><mi>X</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><munder><mi>lim</mi><mrow><msub><mo>⟶</mo> <mi>i</mi></msub></mrow></munder><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>F</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} \mathbf{H}(X, \underset{\longrightarrow_i}{\lim} F_i) & \simeq \Gamma_X( [X \times (\underset{\longrightarrow_i}{\lim} F_i) \to X]) \\ & \simeq \Gamma_X(\underset{\longrightarrow_i}{\lim} [X \times F_i \to X]) \\ & \simeq \underset{\longrightarrow_i}{\lim} \Gamma_X( [X \times F_i \to X]) \\ & \simeq \underset{\longrightarrow_i}{\lim} \mathbf{H}(X,F_i) \end{aligned} \,. </annotation></semantics></math></div></div> <h3 id="finite_objects">Finite objects</h3> <div class="num_prop"> <h6 id="proposition_7">Proposition</h6> <p>An object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>𝒯</mi></mrow><annotation encoding="application/x-tex">X \in \mathcal{T}</annotation></semantics></math> in a topos <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> is a <a class="existingWikiWord" href="/nlab/show/Kuratowski+finite+object">Kuratowski finite object</a> precisely if the <a class="existingWikiWord" href="/nlab/show/%C3%A9tale+geometric+morphism">étale geometric morphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>𝒯</mi> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow></msub><mo>→</mo><mi>𝒯</mi></mrow><annotation encoding="application/x-tex"> \mathcal{T}_{/X} \to \mathcal{T} </annotation></semantics></math></div> <p>out of the <a class="existingWikiWord" href="/nlab/show/slice+topos">slice topos</a> is a proper geometric morphism. And precisely 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 even <em>decidable</em> is this a tidy geometric morphism.</p> </div> <p>(<a href="#MoerdijkVermeulen">Moerdijk-Vermeulen, examples III 1.4</a>)</p> <h3 id="GeometricStacks">Geometric stacks</h3> <p>A typical condition on a <a class="existingWikiWord" href="/nlab/show/geometric+stack">geometric stack</a> to qualify as an <a class="existingWikiWord" href="/nlab/show/orbifold">orbifold</a>/<a class="existingWikiWord" href="/nlab/show/Deligne-Mumford+stack">Deligne-Mumford stack</a> is that its <a class="existingWikiWord" href="/nlab/show/diagonal">diagonal</a> be proper. This is equivalent to the corresponding map of toposes being a proper geometric morphism (e.g. <a href="#Carchedi12">Carchedi 12, section 2</a>, <a href="#LurieSpectral">Lurie Spectral, section 3</a>).</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/separated+geometric+morphism">separated geometric morphism</a>, <a class="existingWikiWord" href="/nlab/show/Hausdorff+topos">Hausdorff topos</a></li> </ul> <h2 id="References">References</h2> <p>The theory of proper geometric morphisms is largly due to</p> <ul> <li> <p id="MoerdijkVermeulen"><a class="existingWikiWord" href="/nlab/show/Ieke+Moerdijk">Ieke Moerdijk</a>, Jacob Vermeulen, <em>Relative compactness conditions for toposes</em> (<a href="https://dspace.library.uu.nl/handle/1874/2374">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ieke+Moerdijk">Ieke Moerdijk</a>, Jacob Vermeulen, <em>Proper maps of toposes</em> , Memoirs of the American Mathematical Society, no. 705 (2000)</p> </li> </ul> <p>based on the <a class="existingWikiWord" href="/nlab/show/locale">localic</a> case discussed in</p> <ul> <li>Jacob Vermeulen, <em>Proper maps of locales</em>, J. Pure Applied Alg. 92 (1994)</li> </ul> <p>A textbook account is in section C3.2 of</p> <ul> <li><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></li> </ul> <p>Discussion with relation to properness of <a class="existingWikiWord" href="/nlab/show/geometric+stacks">geometric stacks</a> includes</p> <ul> <li id="Carchedi12"><a class="existingWikiWord" href="/nlab/show/David+Carchedi">David Carchedi</a>, section 2 of <em>Étale Stacks as Prolongations</em> (<a href="http://arxiv.org/abs/1212.2282">arXiv:1212.2282</a>)</li> </ul> <p>Discussion of higher compactness conditions in <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos+theory">(∞,1)-topos theory</a> is in section 3 of</p> <ul> <li id="LurieSpectral"><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, <em>Spectral Schemes</em> (<a href="http://www.math.harvard.edu/~lurie/papers/DAG-VII.pdf">pdf</a>)</li> </ul> <p>and in section 2.3 of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, <em><a class="existingWikiWord" href="/nlab/show/Rational+and+p-adic+Homotopy+Theory">Rational and p-adic Homotopy Theory</a></em></li> </ul> <p>and for the special case of <a class="existingWikiWord" href="/nlab/show/spectral+Deligne-Mumford+stacks">spectral Deligne-Mumford stacks</a> in section 1.4 of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, <em><a class="existingWikiWord" href="/nlab/show/Quasi-Coherent+Sheaves+and+Tannaka+Duality+Theorems">Quasi-Coherent Sheaves and Tannaka Duality Theorems</a></em></li> </ul> <p>More on proper geometric morphisms between <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-toposes"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mo stretchy="false">(</mo> <mn>∞</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <annotation encoding="application/x-tex">(\infty,1)</annotation> </semantics> </math>-toposes</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Louis+Martini">Louis Martini</a>, <a class="existingWikiWord" href="/nlab/show/Sebastian+Wolf">Sebastian Wolf</a>, <em>Proper morphisms of ∞-topoi</em> [<a href="https://arxiv.org/abs/2311.08051">arXiv:2311.08051</a>].</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on November 15, 2023 at 09:45:52. 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