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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> </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='#morita_equivalent_sites'>Morita equivalent sites</a></li> <li><a href='#subcanonical_sites'>Subcanonical sites</a></li> </ul> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#classes_of_sites'>Classes of sites</a></li> <li><a href='#specific_sites'>Specific sites</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>A <em>site</em> is a <a class="existingWikiWord" href="/nlab/show/locally+presentable+category">presentation</a> of a <a class="existingWikiWord" href="/nlab/show/sheaf+topos">sheaf topos</a> as a structure <a class="existingWikiWord" href="/nlab/show/free+cocompletion">freely generated under colimits</a> from a category, subject to the relation that certain <a class="existingWikiWord" href="/nlab/show/covering">covering</a> colimits are preserved.</p> <p>As such, sites generalise <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> and <a class="existingWikiWord" href="/nlab/show/locales">locales</a>, which present <a class="existingWikiWord" href="/nlab/show/localic+topos">localic</a> sheaf toposes. More precisely, sites generalise and <a class="existingWikiWord" href="/nlab/show/categorify">categorify</a> <a class="existingWikiWord" href="/nlab/show/posites">posites</a>, which present localic toposes but also present locales themselves in a decategorified manner.</p> <p>In technical terms, a <em>site</em> is a <a class="existingWikiWord" href="/nlab/show/small+category">small category</a> equipped with a <a class="existingWikiWord" href="/nlab/show/coverage">coverage</a> or <a class="existingWikiWord" href="/nlab/show/Grothendieck+topology">Grothendieck topology</a>. The <a class="existingWikiWord" href="/nlab/show/category+of+sheaves">category of sheaves</a> over a site is a <a class="existingWikiWord" href="/nlab/show/sheaf+topos">sheaf topos</a> and the site is a <em>site of definition</em> for this topos.</p> <h2 id="definition">Definition</h2> <div class="num_defn" id="Site"> <h6 id="definition_2">Definition</h6> <p>A <em>site</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mi>J</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(C,J)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/category">category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> equipped with a <a class="existingWikiWord" href="/nlab/show/coverage">coverage</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math>.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℰ</mi></mrow><annotation encoding="application/x-tex">\mathcal{E}</annotation></semantics></math> a topos equipped with an <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence of categories</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ℰ</mi><mo>≃</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mi>J</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathcal{E} \simeq Sh(C,J) </annotation></semantics></math></div> <p>to the <a class="existingWikiWord" href="/nlab/show/sheaf+topos">sheaf topos</a> over a site, one says that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mi>J</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(C,J)</annotation></semantics></math> is a <strong>site of definition</strong> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℰ</mi></mrow><annotation encoding="application/x-tex">\mathcal{E}</annotation></semantics></math>.</p> </div> <p>Some classes of sites have their special names</p> <div class="num_defn" id="TypesOfSites"> <h6 id="definition_3">Definition</h6> <p>A site is called</p> <ul> <li> <p>a <a class="existingWikiWord" href="/nlab/show/small+site">small site</a>, <a class="existingWikiWord" href="/nlab/show/large+site">large site</a>, <em>essentially small site</em> if the underlying category is a <a class="existingWikiWord" href="/nlab/show/small+category">small category</a>, <a class="existingWikiWord" href="/nlab/show/large+category">large category</a>, <a class="existingWikiWord" href="/nlab/show/essentially+small+category">essentially small category</a>, respectively;</p> </li> <li> <p>a <strong><a class="existingWikiWord" href="/nlab/show/cartesian+site">cartesian site</a></strong> if the underlying category is <a class="existingWikiWord" href="/nlab/show/finitely+complete+category">finitely complete</a> (which the <a class="existingWikiWord" href="/nlab/show/Elephant">Elephant</a> calls a <a class="existingWikiWord" href="/nlab/show/cartesian+category">cartesian category</a>);</p> </li> <li> <p>a <strong><a class="existingWikiWord" href="/nlab/show/standard+site">standard site</a></strong> if it is a cartesian site equipped with a <a class="existingWikiWord" href="/nlab/show/subcanonical+coverage">subcanonical coverage</a>.</p> </li> </ul> </div> <p>The term <a class="existingWikiWord" href="/nlab/show/standard+site">standard site</a> appears in (<a href="#Johnstone">Johnstone, example A2.1.11</a>).</p> <div class="num_remark" id="SmallAndLarge"> <h6 id="remark">Remark</h6> <p>Often a site is required to be a <a class="existingWikiWord" href="/nlab/show/small+category">small category</a>. But also <a class="existingWikiWord" href="/nlab/show/large+site">large site</a>s play a role.</p> </div> <div class="num_remark" id="CoveragesAndTopologies"> <h6 id="remark_2">Remark</h6> <p>Every <a class="existingWikiWord" href="/nlab/show/coverage">coverage</a> induces a <a class="existingWikiWord" href="/nlab/show/Grothendieck+topology">Grothendieck topology</a>. Often sites are defined to be categories equipped with a Grothendieck topology. Some constructions and properties are more elegantly handled with coverages, some with Grothendieck topologies.</p> <p>Notice that there are many equivalent ways to define a <a class="existingWikiWord" href="/nlab/show/Grothendieck+topology">Grothendieck topology</a>, for instance in terms of a system of <a class="existingWikiWord" href="/nlab/show/local+isomorphisms">local isomorphisms</a>, or in terms of a system of <a class="existingWikiWord" href="/nlab/show/dense+monomorphisms">dense monomorphisms</a> in the <a class="existingWikiWord" href="/nlab/show/category+of+presheaves">category of presheaves</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PSh</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSh(S)</annotation></semantics></math>.</p> </div> <div class="num_defn"> <h6 id="definition_4">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mi>J</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(C,J)</annotation></semantics></math> a site, we write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Sh</mi> <mi>J</mi></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh_J(C)</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/category+of+sheaves">category of sheaves</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> with respect to the <a class="existingWikiWord" href="/nlab/show/coverage">coverage</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math>.</p> </div> <h2 id="properties">Properties</h2> <h3 id="morita_equivalent_sites">Morita equivalent sites</h3> <p>Many inequivalent sites may have equivalent <a class="existingWikiWord" href="/nlab/show/sheaf+topos">sheaf topos</a>es.</p> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>Every <a class="existingWikiWord" href="/nlab/show/sheaf+topos">sheaf topos</a> has a <a href="#TypesOfSites">standard site</a> of definition.</p> </div> <p>This appears as (<a href="#Johnstone">Johnstone, theorem C2.2.8 (iii)</a>).</p> <div class="num_remark"> <h6 id="remark_3">Remark</h6> <p>By <a href="classifying+topos#SheafToposesAreClassifyingForTheirTheoryOfLocalAlgegras">this corollary</a> at <em><a class="existingWikiWord" href="/nlab/show/classifying+topos">classifying topos</a></em> this means that every sheaf topos is the <a class="existingWikiWord" href="/nlab/show/classifying+topos">classifying topos</a> for some <a class="existingWikiWord" href="/nlab/show/theory">theory</a> of <a class="existingWikiWord" href="/nlab/show/local+algebras">local algebras</a>.</p> </div> <h3 id="subcanonical_sites">Subcanonical sites</h3> <div class="num_prop" id="CharacterizationOfSubcanonicalSites"> <h6 id="proposition_2">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℰ</mi></mrow><annotation encoding="application/x-tex">\mathcal{E}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/sheaf+topos">sheaf topos</a>, the <a class="existingWikiWord" href="/nlab/show/essentially+small+category">essentially small</a> sites of definition <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mi>J</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C}, J)</annotation></semantics></math> of <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> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/subcanonical+coverage">subcanonical coverage</a> are precisely the <a class="existingWikiWord" href="/nlab/show/full+subcategories">full subcategories</a> on <a class="existingWikiWord" href="/nlab/show/generating+families">generating families</a> of objects equipped with the coverages induced from the <a class="existingWikiWord" href="/nlab/show/canonical+coverage">canonical coverage</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>.</p> </div> <p>This appears as (<a href="#Johnstone">Johnstone, prop. C2.2.16</a>).</p> <h2 id="examples">Examples</h2> <h3 id="classes_of_sites">Classes of sites</h3> <ul> <li> <p>Every <a class="existingWikiWord" href="/nlab/show/frame">frame</a> is canonically a site, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> is covered by <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 stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{U_i\}</annotation></semantics></math> precisely if it is their <a class="existingWikiWord" href="/nlab/show/join">join</a>.</p> <p>A subclass of examples is the <a class="existingWikiWord" href="/nlab/show/category+of+open+subsets">category of open subsets</a> of a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>.</p> <p>This are examples of <a class="existingWikiWord" href="/nlab/show/posite">posite</a>s/<a class="existingWikiWord" href="/nlab/show/%280%2C1%29-site">(0,1)-site</a>.</p> </li> <li> <p>Various categories come with canonical structures of sites on them:</p> <ul> <li> <p>For every category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> there is its <a class="existingWikiWord" href="/nlab/show/canonical+coverage">canonical coverage</a>.</p> </li> <li> <p>On every <a class="existingWikiWord" href="/nlab/show/regular+category">regular category</a> there is its <a class="existingWikiWord" href="/nlab/show/regular+coverage">regular coverage</a>.</p> </li> <li> <p>On every <a class="existingWikiWord" href="/nlab/show/coherent+category">coherent category</a> there is its <a class="existingWikiWord" href="/nlab/show/coherent+coverage">coherent coverage</a>.</p> </li> <li> <p>Generalizing the previous two examples, on an <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-ary+regular+category">∞-ary regular category</a> there is 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>-canonical coverage.</p> </li> </ul> <p>If the category in question is the <a class="existingWikiWord" href="/nlab/show/syntactic+category">syntactic category</a> of a <a class="existingWikiWord" href="/nlab/show/theory">theory</a>, the corresponding site is the <a class="existingWikiWord" href="/nlab/show/syntactic+site">syntactic site</a>.</p> </li> <li> <p>For every site there is the corresponding <a class="existingWikiWord" href="/nlab/show/double+negation+topology">double negation topology</a> that forces the sheaf topos to a <a class="existingWikiWord" href="/nlab/show/Boolean+topos">Boolean topos</a>.</p> </li> </ul> <p>Other classes of sites are listed in the following.</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/big+site">big site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dense+sub-site">dense sub-site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/large+site">large site</a>, <a class="existingWikiWord" href="/nlab/show/essentially+small+site">essentially small site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/concrete+site">concrete site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+connected+site">locally connected site</a> / <a class="existingWikiWord" href="/nlab/show/locally+%E2%88%9E-connected+site">locally ∞-connected site</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/connected+site">connected site</a> / <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-connected+site">∞-connected site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/strongly+connected+site">strongly connected site</a> / <a class="existingWikiWord" href="/nlab/show/strongly+%E2%88%9E-connected+site">strongly ∞-connected site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/totally+connected+site">totally connected site</a> / <a class="existingWikiWord" href="/nlab/show/totally+%E2%88%9E-connected+site">totally ∞-connected site</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+site">local site</a> / <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-local+site">∞-local site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohesive+site">cohesive site</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-cohesive+site">∞-cohesive site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/atomic+site">atomic site</a></p> </li> </ul> <h3 id="specific_sites">Specific sites</h3> <ul> <li> <p>Sites for <a class="existingWikiWord" href="/nlab/show/big+topos">big topos</a>es defining notions of <a class="existingWikiWord" href="/nlab/show/higher+geometry">geometry</a> are:</p> <ul> <li> <p>The sites that define the <a class="existingWikiWord" href="/nlab/show/higher+geometry">geometry</a> called <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a> are <a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mrow></mrow> <mi>smooth</mi></msub></mrow><annotation encoding="application/x-tex">{}_{smooth}</annotation></semantics></math>, <a class="existingWikiWord" href="/nlab/show/SmoothMfd">SmoothMfd</a>, etc, equipped with the <a class="existingWikiWord" href="/nlab/show/open+cover">open cover</a> <a class="existingWikiWord" href="/nlab/show/coverage">coverage</a>. Or more generally <a class="existingWikiWord" href="/nlab/show/smooth+loci">smooth loci</a> etc.</p> </li> <li> <p>The sites that induce <span class="newWikiWord">topological geometry<a href="/nlab/new/topological+geometry">?</a></span> are small versions of <a class="existingWikiWord" href="/nlab/show/Top">Top</a> equipped with the <a class="existingWikiWord" href="/nlab/show/open+cover">open cover</a> <a class="existingWikiWord" href="/nlab/show/coverage">coverage</a>.</p> </li> <li> <p>The sites that induce the <a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a> modeled on <a class="existingWikiWord" href="/nlab/show/Euclidean+topology">Euclidean topology</a> are the large site of <a class="existingWikiWord" href="/nlab/show/paracompact+manifold">paracompact manifold</a>s and its <a class="existingWikiWord" href="/nlab/show/dense+sub-site">dense sub-site</a> <a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mrow></mrow> <mi>top</mi></msub></mrow><annotation encoding="application/x-tex">{}_{top}</annotation></semantics></math>.</p> </li> </ul> </li> <li> <p>The sites that define the <a class="existingWikiWord" href="/nlab/show/higher+geometry">geometry</a> called <a class="existingWikiWord" href="/nlab/show/algebraic+geometry">algebraic geometry</a> are site structures on categories of formal duals of <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a>s or commutative <a class="existingWikiWord" href="/nlab/show/associative+algebra">associative algebra</a>s</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fpqc-site">fpqc-site</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/fppf-site">fppf-site</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/syntomic+site">syntomic site</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/%C3%A9tale+site">étale site</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Nisnevich+site">Nisnevich site</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Zariski+site">Zariski site</a></p> <p><a class="existingWikiWord" href="/nlab/show/crystalline+site">crystalline site</a></p> </li> </ul> </li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/posite">posite</a></p> </li> <li> <p><strong>site</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/internal+site">internal site</a></li> <li><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-ary+site">∞-ary site</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-site">2-site</a>, <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-site">(2,1)-site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-site">(∞,1)-site</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/model+site">model site</a>, <a class="existingWikiWord" href="/nlab/show/simplicial+site">simplicial site</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/morphism+of+sites">morphism of sites</a>, <a class="existingWikiWord" href="/nlab/show/covering+lifting+property">covering lifting property</a></p> </li> </ul> <h2 id="references">References</h2> <p>In</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Peter+Johnstone">Peter Johnstone</a>, <em><a class="existingWikiWord" href="/nlab/show/Elephant">Sketches of an Elephant</a></em></li> </ul> <p>sites are discussed in section C2.1.</p> </body></html> </div> <div class="revisedby"> <p> Last revised on August 4, 2016 at 07:27:42. See the <a href="/nlab/history/site" 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/site" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/941/#Item_11">Discuss</a><span class="backintime"><a href="/nlab/revision/site/57" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/site" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/site" accesskey="S" class="navlink" id="history" rel="nofollow">History (57 revisions)</a> <a href="/nlab/show/site/cite" style="color: black">Cite</a> <a href="/nlab/print/site" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/site" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>