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href="/nlab/show/category">category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> </li> </ul> </li> </ul> <h2 id="toposes">Toposes</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%280%2C1%29-topos">(0,1)-topos</a>, <a class="existingWikiWord" href="/nlab/show/Heyting+algebra">Heyting algebra</a>, <a class="existingWikiWord" href="/nlab/show/locale">locale</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pretopos">pretopos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topos">topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+topos">Grothendieck topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+presheaves">category of presheaves</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/representable+functor">representable presheaf</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+sheaves">category of sheaves</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/site">site</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sieve">sieve</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/coverage">coverage</a>, <a class="existingWikiWord" href="/nlab/show/Grothendieck+pretopology">pretopology</a>, <a class="existingWikiWord" href="/nlab/show/Grothendieck+topology">topology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sheafification">sheafification</a></p> </li> </ul> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quasitopos">quasitopos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/base+topos">base topos</a>, <a class="existingWikiWord" href="/nlab/show/indexed+topos">indexed topos</a></p> </li> </ul> <h2 id="toc_internal_logic">Internal Logic</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/categorical+semantics">categorical semantics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/internal+logic">internal logic</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/subobject+classifier">subobject classifier</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/natural+numbers+object">natural numbers object</a></p> </li> </ul> </li> </ul> <h2 id="topos_morphisms">Topos morphisms</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/logical+morphism">logical morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+morphism">geometric morphism</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/direct+image">direct image</a>/<a class="existingWikiWord" href="/nlab/show/inverse+image">inverse image</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/global+section">global sections</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+embedding">geometric embedding</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/surjective+geometric+morphism">surjective geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/essential+geometric+morphism">essential geometric morphism</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+connected+geometric+morphism">locally connected geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connected+geometric+morphism">connected geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/totally+connected+geometric+morphism">totally connected geometric morphism</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%C3%A9tale+geometric+morphism">étale geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/open+geometric+morphism">open geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/proper+geometric+morphism">proper geometric morphism</a>, <a class="existingWikiWord" href="/nlab/show/compact+topos">compact topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/separated+geometric+morphism">separated geometric morphism</a>, <a class="existingWikiWord" href="/nlab/show/Hausdorff+topos">Hausdorff topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+geometric+morphism">local geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bounded+geometric+morphism">bounded geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/base+change">base change</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/localic+geometric+morphism">localic geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hyperconnected+geometric+morphism">hyperconnected geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/atomic+geometric+morphism">atomic geometric morphism</a></p> </li> </ul> </li> </ul> <h2 id="extra_stuff_structure_properties">Extra stuff, structure, properties</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+locale">topological locale</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/localic+topos">localic topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/petit+topos">petit topos/gros topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+connected+topos">locally connected topos</a>, <a class="existingWikiWord" href="/nlab/show/connected+topos">connected topos</a>, <a class="existingWikiWord" href="/nlab/show/totally+connected+topos">totally connected topos</a>, <a class="existingWikiWord" href="/nlab/show/strongly+connected+topos">strongly connected topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+topos">local topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohesive+topos">cohesive topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/classifying+topos">classifying topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+topos">smooth topos</a></p> </li> </ul> <h2 id="cohomology_and_homotopy">Cohomology and homotopy</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+groups+in+an+%28infinity%2C1%29-topos">homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">model structure on simplicial presheaves</a></p> </li> </ul> <h2 id="in_higher_category_theory">In higher category theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+topos+theory">higher topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%280%2C1%29-topos">(0,1)-topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/%280%2C1%29-site">(0,1)-site</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-topos">2-topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-site">2-site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-sheaf">2-sheaf</a>, <a class="existingWikiWord" href="/nlab/show/stack">stack</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-site">(∞,1)-site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-sheaf">(∞,1)-sheaf</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stack">∞-stack</a>, <a class="existingWikiWord" href="/nlab/show/derived+stack">derived stack</a></p> </li> </ul> </li> </ul> <h2 id="theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Diaconescu%27s+theorem">Diaconescu's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Barr%27s+theorem">Barr's theorem</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/topos+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> </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='#example'>Example</a></li> <ul> <li><a href='#remark'>Remark</a></li> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#remark_2'>Remark</a></li> </ul> <li><a href='#open_localizations'>Open localizations</a></li> <li><a href='#ext_int'>Open subtoposes associated to a subtopos</a></li> <li><a href='#related_pages'>Related pages</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>The concept of an <strong>open subtopos</strong> generalizes the concept of an open subspace from topology to toposes.</p> <h2 id="definition">Definition</h2> <p>Let <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> be a <a class="existingWikiWord" href="/nlab/show/subterminal+object">subterminal object</a> of a <a class="existingWikiWord" href="/nlab/show/topos">topos</a>. Then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>o</mi> <mi>U</mi></msub><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo>≔</mo><mo stretchy="false">(</mo><mi>U</mi><mo>⇒</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">o_U(V)\coloneqq (U\Rightarrow V)</annotation></semantics></math> defines a <a class="existingWikiWord" href="/nlab/show/Lawvere-Tierney+topology">Lawvere-Tierney topology</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℰ</mi></mrow><annotation encoding="application/x-tex">\mathcal{E}</annotation></semantics></math>, whose corresponding <a class="existingWikiWord" href="/nlab/show/subtopos">subtopos</a> is called the <em>open subtopos</em> associated to <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>.</p> <p>The reflector into the topos of sheaves can be constructed explicitly as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>O</mi> <mi>U</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>X</mi> <mi>U</mi></msup></mrow><annotation encoding="application/x-tex">O_U(X) = X^U</annotation></semantics></math>.</p> <p>A <em>topology</em> that is of this form for some subterminal object <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 called <em>open</em>.</p> <h2 id="example">Example</h2> <p>In case <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℰ</mi><mo>=</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{E}=Sh(X)</annotation></semantics></math> is the topos of sheaves on a topological space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, a subterminal object is just an open subset <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and the open subtopos corresponding to it is equivalent to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sh</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh(U)</annotation></semantics></math>.</p> <p>As one would expect from the topological situation, for any topos <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℰ</mi></mrow><annotation encoding="application/x-tex">\mathcal{E}</annotation></semantics></math>, the empty subtopos (given by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>o</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo>≔</mo><mo stretchy="false">(</mo><mo>⊥</mo><mo>⇒</mo><mi>V</mi><mo stretchy="false">)</mo><mo>=</mo><mo>⊤</mo></mrow><annotation encoding="application/x-tex">o(V) \coloneqq (\bot \Rightarrow V) = \top</annotation></semantics></math>) and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℰ</mi></mrow><annotation encoding="application/x-tex">\mathcal{E}</annotation></semantics></math> itself (given by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>o</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo>≔</mo><mo stretchy="false">(</mo><mo>⊤</mo><mo>⇒</mo><mi>V</mi><mo stretchy="false">)</mo><mo>=</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">o(V) \coloneqq (\top \Rightarrow V) = V</annotation></semantics></math>) are open subtoposes 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> <h3 id="remark">Remark</h3> <p>The <a class="existingWikiWord" href="/nlab/show/subterminal+object">subterminal object</a> <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> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℰ</mi></mrow><annotation encoding="application/x-tex">\mathcal{E}</annotation></semantics></math> is associated with a <a class="existingWikiWord" href="/nlab/show/closed+subtopos">closed subtopos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Sh</mi> <mrow><mi>c</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>ℰ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh_{c(U)}(\mathcal{E})</annotation></semantics></math> as well e.g. in the case of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℰ</mi><mo>=</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{E}=Sh(X)</annotation></semantics></math> on a space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> this yields <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sh</mi><mo stretchy="false">(</mo><mi>X</mi><mo>∖</mo><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh(X\setminus U)</annotation></semantics></math>.</p> <p>Moreover, given a <a class="existingWikiWord" href="/nlab/show/Lawvere-Tierney+topology">Lawvere-Tierney topology</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> on a topos <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℰ</mi></mrow><annotation encoding="application/x-tex">\mathcal{E}</annotation></semantics></math> with corresponding subtopos <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>ℰ</mi><mo stretchy="false">)</mo><mo>↪</mo><mi>ℰ</mi></mrow><annotation encoding="application/x-tex">Sh_j(\mathcal{E})\hookrightarrow\mathcal{E}</annotation></semantics></math>, we a get a canonical <a class="existingWikiWord" href="/nlab/show/subterminal+object">subterminal object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ext</mi><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">ext(j)</annotation></semantics></math> associated to <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> by taking the <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>-closure of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi><mo>↣</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">O\rightarrowtail 1</annotation></semantics></math>. The corresponding closed and open subtoposes associated to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ext</mi><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">ext(j)</annotation></semantics></math> provide a ‘<em>closure</em>’ <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Sh</mi> <mrow><mi>c</mi><mo stretchy="false">(</mo><mi>ext</mi><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>ℰ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh_{c(ext(j))}(\mathcal{E})</annotation></semantics></math>, respectively, an ‘<em>exterior</em>’ <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Sh</mi> <mrow><mi>o</mi><mo stretchy="false">(</mo><mi>ext</mi><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>ℰ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh_{o(ext(j))}(\mathcal{E})</annotation></semantics></math> for <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>ℰ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh_j(\mathcal{E})</annotation></semantics></math> (cf. <a href="#SGA4">SGA4</a>, p.461). More on this <a href="#ext_int">below</a>.</p> <h2 id="properties">Properties</h2> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Sh</mi> <mi>j</mi></msub><mo stretchy="false">(</mo><mi>ℰ</mi><mo stretchy="false">)</mo><mo>↪</mo><mi>ℰ</mi></mrow><annotation encoding="application/x-tex">Sh_j(\mathcal{E})\hookrightarrow\mathcal{E}</annotation></semantics></math> be a subtopos with corresponding topology <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>. The following are equivalent:</p> <ul> <li><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 open.</li> <li>The <a class="existingWikiWord" href="/nlab/show/associated+sheaf+functor">associated sheaf functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo>:</mo><mi>ℰ</mi><mo>→</mo><msub><mi>Sh</mi> <mi>j</mi></msub><mo stretchy="false">(</mo><mi>ℰ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L:\mathcal{E}\to Sh_j(\mathcal{E})</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/logical+functor">logical</a>.</li> <li><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>ℰ</mi><mo stretchy="false">)</mo><mo>↪</mo><mi>ℰ</mi></mrow><annotation encoding="application/x-tex">Sh_j(\mathcal{E})\hookrightarrow\mathcal{E}</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/open+geometric+morphism">open geometric morphism</a>.</li> </ul> </div> <p>See Johnstone (<a href="#Johnstone80">1980</a>, pp.219-220; <a href="#Johnstone02">2002</a>, pp.609-610).</p> <div class="num_prop"> <h6 id="proposition_2">Proposition</h6> <p>Let <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> a <a class="existingWikiWord" href="/nlab/show/subterminal+object">subterminal object</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Sh</mi> <mrow><mi>c</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>ℰ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh_{c(U)}(\mathcal{E})</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Sh</mi> <mrow><mi>o</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>ℰ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh_{o(U)}(\mathcal{E})</annotation></semantics></math> the corresponding closed, resp. open subtoposes. Then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Sh</mi> <mrow><mi>c</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>ℰ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh_{c(U)}(\mathcal{E})</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Sh</mi> <mrow><mi>o</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>ℰ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh_{o(U)}(\mathcal{E})</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/complement">complements</a> for each other in the <a class="existingWikiWord" href="/nlab/show/lattice+of+subtoposes">lattice of subtoposes</a>.</p> </div> <p>See Johnstone <a href="#Johnstone">(2002, pp.212,215)</a>.</p> <h3 id="remark_2">Remark</h3> <p>Whereas, general <a class="existingWikiWord" href="/nlab/show/open+geometric+morphisms">open morphisms</a> are only bound to preserve <a class="existingWikiWord" href="/nlab/show/first+order+logic">first order logic</a>, open <em>inclusions</em> preserve also <a class="existingWikiWord" href="/nlab/show/higher+order+logic">higher order logic</a> since their <a class="existingWikiWord" href="/nlab/show/inverse+image">inverse images</a> are logical.</p> <p>That the inverse image is logical is a special case of the general fact that the pullback functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℰ</mi><mo>→</mo><mi>ℰ</mi><mo stretchy="false">/</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathcal{E}\to\mathcal{E}/X</annotation></semantics></math> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">X\to 1</annotation></semantics></math> is logical for arbitrary objects <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>. In particular, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Sh</mi> <mrow><mi>o</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>ℰ</mi><mo stretchy="false">)</mo><mo>≅</mo><mi>ℰ</mi><mo stretchy="false">/</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">Sh_{o(U)}(\mathcal{E})\cong\mathcal{E}/U</annotation></semantics></math>.</p> <h2 id="open_localizations">Open localizations</h2> <p>Subtoposes of a topos <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℰ</mi></mrow><annotation encoding="application/x-tex">\mathcal{E}</annotation></semantics></math> correspond to <a class="existingWikiWord" href="/nlab/show/localization">localizations</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> i.e. replete, reflective subcategories whose reflector preserves finite limits. Just as this notion makes sense more generally for categories with finite limits, the notion of open localization makes sense more generally for <a class="existingWikiWord" href="/nlab/show/locally+presentable+category">locally presentable categories</a>:</p> <p>Given a locally <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math>-presentable 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">\mathcal{C}</annotation></semantics></math> with subcategory of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math>-presentable objects <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">\mathcal{P}</annotation></semantics></math>, the subobject <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Ω</mi> <mi>𝒞</mi></msub></mrow><annotation encoding="application/x-tex">\Omega_\mathcal{C}</annotation></semantics></math> of the subobject classifier of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Set</mi> <mrow><msup><mi>𝒫</mi> <mi>op</mi></msup></mrow></msup></mrow><annotation encoding="application/x-tex">Set^{\mathcal{P}^{op}}</annotation></semantics></math> given by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Ω</mi> <mi>𝒞</mi></msub><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo></mrow><annotation encoding="application/x-tex">\Omega_\mathcal{C}(P):=</annotation></semantics></math> set of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math>-exact subpresheaves of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒫</mi><mo stretchy="false">(</mo><mtext>_</mtext><mo>,</mo><mi>P</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{P}(\text{_},P)</annotation></semantics></math>, classifies subobjects of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>. Furthermore, localizations of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> correspond to topologies <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi><mo>:</mo><msub><mi>Ω</mi> <mi>𝒞</mi></msub><mo>→</mo><msub><mi>Ω</mi> <mi>𝒞</mi></msub></mrow><annotation encoding="application/x-tex">j:\Omega_\mathcal{C}\to\Omega_\mathcal{C}</annotation></semantics></math>.</p> <p>At this level of generality, an open subtopos of a Grothendieck topos corresponds to the notion of an <strong>open localization</strong> of a <a class="existingWikiWord" href="/nlab/show/locally+presentable+category">locally presentable category</a> that is studied in <a href="#BK91">Borceux-Korotenski (1991)</a>. The main result in their paper is the following</p> <div class="num_prop"> <h6 id="proposition_3">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>l</mi><mo>⊣</mo><mi>i</mi><mo>:</mo><mi>𝒟</mi><mo>↪</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">l\dashv i: \mathcal{D}\hookrightarrow\mathcal{C}</annotation></semantics></math> be a localization of a <a class="existingWikiWord" href="/nlab/show/locally+presentable+category">locally presentable 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">\mathcal{C}</annotation></semantics></math>. The localization is called <em>open</em> if the following equivalent conditions are satisfied:</p> <ul> <li> <p>The corresponding <a class="existingWikiWord" href="/nlab/show/closure+operator">closure operator</a> admits a universal dense interior operator.</p> </li> <li> <p>The associated <a class="existingWikiWord" href="/nlab/show/Lawvere-Tierney+topology">topology</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>j</mi> <mi>𝒟</mi></msub><mo>:</mo><msub><mi>Ω</mi> <mi>𝒞</mi></msub><mo>→</mo><msub><mi>Ω</mi> <mi>𝒞</mi></msub></mrow><annotation encoding="application/x-tex">j_\mathcal{D}:\Omega_\mathcal{C}\to\Omega_\mathcal{C}</annotation></semantics></math> has a left adjoint.</p> </li> <li> <p>The localization is <a class="existingWikiWord" href="/nlab/show/essential+geometric+morphism">essential</a> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>⊣</mo><mi>l</mi></mrow><annotation encoding="application/x-tex">k\dashv l</annotation></semantics></math> and, additionally, the first of the following diagrams<sup id="fnref:1"><a href="#fn:1" rel="footnote">1</a></sup> being a pullback implies the second being a pullback, too:</p> </li> </ul> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>D</mi></mtd> <mtd><mover><mo>→</mo><mi>d</mi></mover></mtd> <mtd><mi>D</mi><mo>′</mo></mtd></mtr> <mtr><mtd><mi>f</mi><mo stretchy="false">↓</mo></mtd> <mtd><mi>p</mi><mo>.</mo><mi>b</mi><mo>.</mo></mtd> <mtd><mo stretchy="false">↓</mo><mi>g</mi></mtd></mtr> <mtr><mtd><mi>lC</mi></mtd> <mtd><munder><mo>→</mo><mi>lc</mi></munder></mtd> <mtd><mi>lC</mi><mo>′</mo></mtd></mtr></mtable></mrow><mspace width="2em"></mspace><mo>⇒</mo><mspace width="2em"></mspace><mrow><mtable><mtr><mtd><mi>kD</mi></mtd> <mtd><mover><mo>→</mo><mi>kd</mi></mover></mtd> <mtd><mi>kD</mi><mo>′</mo></mtd></mtr> <mtr><mtd><mover><mi>f</mi><mo stretchy="false">¯</mo></mover><mo stretchy="false">↓</mo></mtd> <mtd><mi>p</mi><mo>.</mo><mi>b</mi><mo>.</mo></mtd> <mtd><mo stretchy="false">↓</mo><mover><mi>g</mi><mo stretchy="false">¯</mo></mover></mtd></mtr> <mtr><mtd><mi>C</mi></mtd> <mtd><munder><mo>→</mo><mi>c</mi></munder></mtd> <mtd><mi>C</mi><mo>′</mo></mtd></mtr></mtable></mrow><mspace width="1em"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex">\array{ D&\overset{d}{\to}&D'\\ f\downarrow&p.b.&\downarrow g\\ lC&\underset{lc}{\to}&lC' } \qquad\Rightarrow \qquad \array{ kD&\overset{kd}{\to}&kD'\\ \bar{f}\downarrow&p.b.&\downarrow \bar{g}\\ C&\underset{c}{\to}&C' }\quad . </annotation></semantics></math></div></div> <p>In case <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> is a topos, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>:</mo><mi>𝒟</mi><mo>↪</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">i:\mathcal{D}\hookrightarrow\mathcal{C}</annotation></semantics></math> is an open subtopos.</p> <p>Open localizations are special cases of <strong>essential localizations</strong>, and are in general better behaved than the latter. For example, the meet of two essential localizations in the lattice of essential localizations does not coincide with their meet in the lattice of localizations. Compare this with the following</p> <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><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/locally+presentable+category">locally presentable category</a>. The meet of two open localizations in the lattice of localizations is an open localization.</p> </div> <p>cf. <a href="#BK91">Borceux-Korotenski (1991, p.235)</a>.</p> <div class="num_prop" id="open_supremum"> <h6 id="proposition_5">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/locally+presentable+category">locally presentable category</a> where unions are <a class="existingWikiWord" href="/nlab/show/universal+colimit">universal</a>. The supremum of a family of open localizations in the lattice of all localizations is again an open localization.</p> </div> <p>cf. <a href="#BK91">Borceux-Korotenski (1991, p.236)</a>.</p> <p>This applies e.g. to Grothendieck toposes since they are locally presentable and colimits are universal.</p> <div class="num_prop" id="open_supremum"> <h6 id="proposition_6">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/locally+presentable+category">locally presentable category</a> where unions are <a class="existingWikiWord" href="/nlab/show/universal+colimit">universal</a>. The open localizations in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> constitute a <a class="existingWikiWord" href="/nlab/show/locale">locale</a>.</p> </div> <p>cf. <a href="#BK91">Borceux-Korotenski (1991, p.237)</a>.</p> <p>The following proposition closes the circle and recovers the primordial example of sheaf subtoposes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sh</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh(U)</annotation></semantics></math> on open subsets <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> as a special case:</p> <div class="num_prop"> <h6 id="proposition_7">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/locally+presentable+category">locally presentable category</a> in which colimits are <a class="existingWikiWord" href="/nlab/show/universal+colimit">universal</a>. Then the <a class="existingWikiWord" href="/nlab/show/locale">locale</a> of open localizations of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> is isomorphic to the locale of <a class="existingWikiWord" href="/nlab/show/subterminal+object">subterminal objects</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>.</p> </div> <p>cf. <a href="#BK91">Borceux-Korotenski (1991, p.238)</a>.</p> <h2 id="ext_int">Open subtoposes associated to a subtopos</h2> <p>Let <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>ℰ</mi><mo stretchy="false">)</mo><mo>↪</mo><mi>ℰ</mi></mrow><annotation encoding="application/x-tex">Sh_j(\mathcal{E})\hookrightarrow\mathcal{E}</annotation></semantics></math> be a subtopos of a (Grothendieck) topos with corresponding <a class="existingWikiWord" href="/nlab/show/Lawvere-Tierney+topology">topology</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>. From <a href="#open_supremum">prop. <a class="maruku-ref" href="#open_supremum"></a></a> it follows that the supremum of the family of all open subtoposes contained in <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>ℰ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh_j(\mathcal{E})</annotation></semantics></math> is open again and, since it coincides with the supremum in the lattice of all localizations, is contained in <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>ℰ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh_j(\mathcal{E})</annotation></semantics></math>. Clearly, it is the biggest open subtopos contained in <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>ℰ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh_j(\mathcal{E})</annotation></semantics></math> and therefore called the <strong>interior</strong> of <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>ℰ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh_j(\mathcal{E})</annotation></semantics></math>, denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Sh</mi> <mrow><mi>o</mi><mo stretchy="false">(</mo><mi>int</mi><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>ℰ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh_{o(int(j))}(\mathcal{E})</annotation></semantics></math> and the corresponding <a class="existingWikiWord" href="/nlab/show/subterminal+object">subterminal object</a> by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>int</mi><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">int(j)</annotation></semantics></math>.</p> <p>Whereas the other open subtopos <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Sh</mi> <mrow><mi>o</mi><mo stretchy="false">(</mo><mi>ext</mi><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>ℰ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh_{o(ext(j))}(\mathcal{E})</annotation></semantics></math> connected with <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>ℰ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh_j(\mathcal{E})</annotation></semantics></math> corresponding to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ext</mi><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">ext(j)</annotation></semantics></math> is the biggest open subtopos disjoint from <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>ℰ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh_j(\mathcal{E})</annotation></semantics></math> i.e. its exterior. Then the sum <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Sh</mi> <mrow><mi>o</mi><mo stretchy="false">(</mo><mi>ext</mi><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>ℰ</mi><mo stretchy="false">)</mo><mo>∨</mo><msub><mi>Sh</mi> <mrow><mi>o</mi><mo stretchy="false">(</mo><mi>int</mi><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>ℰ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh_{o(ext(j))}(\mathcal{E}) \vee Sh_{o(int(j))}(\mathcal{E})</annotation></semantics></math> is open again and corresponds to the subterminal object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ext</mi><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo><mo>∨</mo><mi>int</mi><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">ext(j)\vee int(j)</annotation></semantics></math>. Its closed complement <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Sh</mi> <mrow><mi>c</mi><mo stretchy="false">(</mo><mi>ext</mi><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo><mo>∨</mo><mi>int</mi><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>ℰ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh_{c(ext(j)\vee int(j))}(\mathcal{E})</annotation></semantics></math> is called the boundary of <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>ℰ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh_j(\mathcal{E})</annotation></semantics></math> in (<a href="#SGA4">SGA 4</a>, p. 461).</p> <p>For some further details see at <a class="existingWikiWord" href="/nlab/show/dense+subtopos">dense subtopos</a>.</p> <h2 id="related_pages">Related pages</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/open+geometric+morphism">open geometric morphism</a></li> <li><a class="existingWikiWord" href="/nlab/show/locally+closed+subtopos">locally closed subtopos</a></li> <li><a class="existingWikiWord" href="/nlab/show/closed+subtopos">closed subtopos</a></li> <li><a class="existingWikiWord" href="/nlab/show/dense+subtopos">dense subtopos</a></li> <li><a class="existingWikiWord" href="/nlab/show/localization">localization</a></li> <li><a class="existingWikiWord" href="/nlab/show/level">level</a></li> <li><a class="existingWikiWord" href="/nlab/show/Artin+gluing">Artin gluing</a></li> <li><a class="existingWikiWord" href="/nlab/show/co-Heyting+boundary">co-Heyting boundary</a></li> </ul> <h2 id="references">References</h2> <ul> <li id="SGA4"> <p><a class="existingWikiWord" href="/nlab/show/M.+Artin">M. Artin</a>, <a class="existingWikiWord" href="/nlab/show/A.+Grothendieck">A. Grothendieck</a>, <a class="existingWikiWord" href="/nlab/show/J.+L.+Verdier">J. L. Verdier</a>, <em>Théorie des Topos et Cohomologie Etale des Schémas (<a class="existingWikiWord" href="/nlab/show/SGA4">SGA4</a>)</em>, LNM <strong>269</strong> Springer Heidelberg 1972. (Exposé IV 9.2, 9.3.4-9.4., pp.451ff)</p> </li> <li id="BK91"> <p><a class="existingWikiWord" href="/nlab/show/Francis+Borceux">F. Borceux</a>, M. Korostenski, <em>Open localizations</em> , JPAA <strong>74</strong> (1991) 229-238 <a href="https://doi.org/10.1016/0022-4049(91)90113-G">doi</a></p> </li> <li> <p>C. Getz, M. Korostenski, <em>Open localizations and factorization systems</em> , Quest. Math. <strong>17</strong> no.2 (1994) 225-230 <a href="https:/doi.org/10.1080/16073606.1994.9631761">doi</a></p> </li> <li id="Johnstone77"> <p><a class="existingWikiWord" href="/nlab/show/Peter+Johnstone">Peter Johnstone</a>, <em>Topos Theory</em> , Academic Press New York 1977. (Dover reprint 2014, 93-95)</p> </li> <li id="Johnstone80"> <p><a class="existingWikiWord" href="/nlab/show/Peter+Johnstone">Peter Johnstone</a>, <em>Open maps of toposes</em> , Manuscripta Math. <strong>31</strong> (1980) 217-247. (<a href="http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002222744">gdz</a>)</p> </li> <li id="Johnstone"> <p><a class="existingWikiWord" href="/nlab/show/Peter+Johnstone">Peter Johnstone</a>, <em><a class="existingWikiWord" href="/nlab/show/Sketches+of+an+Elephant">Sketches of an Elephant</a> vols. I,II</em>, Oxford UP 2002. (A4.5., pp.204-220; C3.1.5-7, pp.609f)</p> </li> </ul> <div class="footnotes"><hr /><ol><li id="fn:1"> <p>Here <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> corresponds to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>f</mi><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">\bar{f}</annotation></semantics></math>, resp. <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> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>g</mi><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">\bar{g}</annotation></semantics></math>, under the adjunction <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>⊣</mo><mi>l</mi></mrow><annotation encoding="application/x-tex">k\dashv l</annotation></semantics></math>. <a href="#fnref:1" rev="footnote">↩</a></p> </li></ol></div></body></html> </div> <div class="revisedby"> <p> Last revised on August 15, 2021 at 13:56:58. 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