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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> <ul> <li><a href='#of_bundles'>Of bundles</a></li> <li><a href='#of_sheaves_on_topological_spaces'>Of sheaves on topological spaces</a></li> <li><a href='#GeneralTopos'>Of objects in a general Grothendieck topos</a></li> <li><a href='#of_objects_in_an_topos'>Of objects in an <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</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>The <em>global sections</em> of a <a class="existingWikiWord" href="/nlab/show/bundle">bundle</a> are simply its <a class="existingWikiWord" href="/nlab/show/sections">sections</a>. When bundles are replaced by their <a class="existingWikiWord" href="/nlab/show/sheaves">sheaves</a> of <a class="existingWikiWord" href="/nlab/show/local+sections">local sections</a>, then forming global sections corresponds to the <a class="existingWikiWord" href="/nlab/show/direct+image">direct image</a> operation on sheaves with respect to the morphism to the <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal</a> <a class="existingWikiWord" href="/nlab/show/site">site</a>. This definition generalizes to objects in a general <a class="existingWikiWord" href="/nlab/show/topos">topos</a> and <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a>.</p> <h2 id="definition">Definition</h2> <p>We start describing the more explicit notions of global sections of bundles and then work our way towards the more abstract notions in terms of <a class="existingWikiWord" href="/nlab/show/topos">topos</a> theory.</p> <h3 id="of_bundles">Of bundles</h3> <p>A <strong>global <a class="existingWikiWord" href="/nlab/show/section">section</a></strong> of a <a class="existingWikiWord" href="/nlab/show/bundle">bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mover><mo>→</mo><mi>p</mi></mover><mi>B</mi></mrow><annotation encoding="application/x-tex">E \overset{p}\to B</annotation></semantics></math> is simply a <a class="existingWikiWord" href="/nlab/show/section">section</a> of <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>, that is a map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo lspace="verythinmathspace">:</mo><mi>B</mi><mo>→</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">s\colon B \to E</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>∘</mo><mi>s</mi><mo>=</mo><msub><mo lspace="0em" rspace="thinmathspace">id</mo> <mi>B</mi></msub></mrow><annotation encoding="application/x-tex">p \circ s = \id_B</annotation></semantics></math>.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>E</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>s</mi></mpadded></msup><mo>↗</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>p</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>B</mi></mtd> <mtd><mover><mo>→</mo><mi>id</mi></mover></mtd> <mtd><mi>B</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && E \\ & {}^{\mathllap{s}}\nearrow & \downarrow^{\mathrlap{p}} \\ B &\stackrel{id}{\to}& B } \,. </annotation></semantics></math></div> <p>The adjective ‘global’ here is used to distinguish from a <strong><a class="existingWikiWord" href="/nlab/show/local+section">local section</a></strong>: a <em>generalised</em> section over some subspace <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>:</mo><mi>U</mi><mo>↪</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">i : U \hookrightarrow B</annotation></semantics></math> which is a section of the map 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> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>i</mi> <mo>*</mo></msup><mi>E</mi><mo>:</mo><mo>=</mo><mi>E</mi><msub><mo stretchy="false">|</mo> <mi>U</mi></msub><mo>→</mo><mi>U</mi></mrow><annotation encoding="application/x-tex"> i^* E := E|_U \to U </annotation></semantics></math></div> <p>from the <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>i</mi> <mo>*</mo></msup><mi>E</mi><mo>:</mo><mo>=</mo><mi>E</mi><msub><mo stretchy="false">|</mo> <mi>U</mi></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>E</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>U</mi></mtd> <mtd><mover><mo>→</mo><mi>i</mi></mover></mtd> <mtd><mi>B</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ i^* E := E|_U &\to& E \\ \downarrow && \downarrow \\ U &\stackrel{i}{\to}& B } \,. </annotation></semantics></math></div> <p>Compare the notion of <a class="existingWikiWord" href="/nlab/show/global+point">global point</a>, which is really the special case when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a> (where the generalised section corresponds to a <a class="existingWikiWord" href="/nlab/show/generalised+element">generalised element</a>). On the other hand, a global section of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mover><mo>→</mo><mi>p</mi></mover><mi>B</mi></mrow><annotation encoding="application/x-tex">E \overset{p}\to B</annotation></semantics></math> 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> <em>is</em> simply a global point in the <a class="existingWikiWord" href="/nlab/show/slice+category">slice category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo stretchy="false">/</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}/B</annotation></semantics></math>.</p> <p>One often writes</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>U</mi></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><msub><mi>Hom</mi> <mrow><mi>𝒞</mi><mo stretchy="false">/</mo><mi>U</mi></mrow></msub><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mi>E</mi><msub><mo stretchy="false">|</mo> <mi>U</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \Gamma_U(E) := Hom_{\mathcal{C}/U}(U , E|_U) </annotation></semantics></math></div> <p>for the <strong>set of global sections</strong> over <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> (or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Gamma(U,E)</annotation></semantics></math> or similar).</p> <h3 id="of_sheaves_on_topological_spaces">Of sheaves on topological spaces</h3> <p>Every <a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>Op</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A \in Sh(X) = Sh(Op(X))</annotation></semantics></math> on (the <a class="existingWikiWord" href="/nlab/show/site">site</a> that is given by 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> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is the sheaf of <em><a class="existingWikiWord" href="/nlab/show/local+sections">local sections</a></em> of its <a class="existingWikiWord" href="/nlab/show/etale+space">etale space</a> <a class="existingWikiWord" href="/nlab/show/bundle">bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">E \to X</annotation></semantics></math> in that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>:</mo><mi>U</mi><mo>↦</mo><msub><mi>Γ</mi> <mi>U</mi></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> A : U \mapsto \Gamma_U(E) </annotation></semantics></math></div> <p>for every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>∈</mo><mi>Op</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U \in Op(X)</annotation></semantics></math>. For this reasons one often speaks of the value of a <a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a> on some object as a set of sections, even if the corresponding bundle is never mentioned and doesn’t really matter.</p> <p>The set of global sections on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is</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><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><mi>A</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>Hom</mi> <mrow><mi>Sh</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \Gamma_X(A) = A(X) = Hom_{Sh(X)}(X, A) \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \in Sh(X)</annotation></semantics></math> denotes the <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a> of the <a class="existingWikiWord" href="/nlab/show/category+of+sheaves">category of sheaves</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sh</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh(X)</annotation></semantics></math>. Often this is written just using different notation</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><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>Hom</mi> <mrow><mi>Sh</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mo>*</mo><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \Gamma_X(A) = Hom_{Sh(X)}(*,A) </annotation></semantics></math></div> <p>One notices 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><mi>Sh</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">\Gamma_X(-) : Sh(X) \to Set</annotation></semantics></math> defined this way is the <a class="existingWikiWord" href="/nlab/show/direct+image">direct image</a> functor on <a class="existingWikiWord" href="/nlab/show/Grothendieck+toposes">Grothendieck toposes</a> that is induced from the canonical morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">X \to *</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>s (now “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo></mrow><annotation encoding="application/x-tex">*</annotation></semantics></math>” really denotes the <a class="existingWikiWord" href="/nlab/show/point">point</a> topological space!) and hence from the corresponding morphism of <a class="existingWikiWord" href="/nlab/show/site">site</a>s.</p> <p>Again, this expression for global sections induces a relative version, e.g. for sheaves on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/relative+scheme">schemes</a>, the direct image functor goes into the base scheme <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>).</p> <h3 id="GeneralTopos">Of objects in a general Grothendieck topos</h3> <p>The definition of global sections of sheaves on topological spaces in terms of the <a class="existingWikiWord" href="/nlab/show/direct+image">direct image</a> of the canonical morphism to the terminal <a class="existingWikiWord" href="/nlab/show/site">site</a> generalizes to <a class="existingWikiWord" href="/nlab/show/Grothendieck+topos">sheaf toposes</a> over arbitrary <a class="existingWikiWord" href="/nlab/show/sites">sites</a>.</p> <p>For every <a class="existingWikiWord" href="/nlab/show/Grothendieck+topos">Grothendieck topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒯</mi></mrow><annotation encoding="application/x-tex">\mathcal{T}</annotation></semantics></math>, there is a <a class="existingWikiWord" href="/nlab/show/geometric+morphism">geometric morphism</a> (the <em><a class="existingWikiWord" href="/nlab/show/terminal+geometric+morphism">terminal geometric morphism</a></em>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Γ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>𝒯</mi><mover><mo>→</mo><mo>←</mo></mover><mi>Set</mi><mo>:</mo><mi>LConst</mi></mrow><annotation encoding="application/x-tex"> \Gamma \;\colon\; \mathcal{T} \stackrel{\leftarrow}{\to} Set : LConst </annotation></semantics></math></div> <p>called the <strong>global sections</strong> functor. It is given by the <a class="existingWikiWord" href="/nlab/show/hom-set">hom-set</a> out of the <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Γ</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>=</mo><msub><mi>Hom</mi> <mi>𝒯</mi></msub><mo stretchy="false">(</mo><mo>*</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \Gamma(-) = Hom_{\mathcal{T}}({*}, -) </annotation></semantics></math></div> <p>and hence assigns to each object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mi>𝒯</mi></mrow><annotation encoding="application/x-tex">A\in \mathcal{T}</annotation></semantics></math> its set of <a class="existingWikiWord" href="/nlab/show/global+element">global element</a>s <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>Hom</mi> <mi>𝒯</mi></msub><mo stretchy="false">(</mo><mo>*</mo><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Gamma(A) = Hom_{\mathcal{T}}(*,A)</annotation></semantics></math>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>LConst</mi><mo>:</mo><mi>Set</mi><mo>→</mo><mi>𝒯</mi></mrow><annotation encoding="application/x-tex">LConst : Set \to \mathcal{T}</annotation></semantics></math> of the global section functor is the canonical <a class="existingWikiWord" href="/nlab/show/Set">Set</a>-<a class="existingWikiWord" href="/nlab/show/copower">tensoring</a> functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>⊗</mo><mo>:</mo><mi>Set</mi><mo>×</mo><mi>𝒯</mi><mo>→</mo><mi>𝒯</mi></mrow><annotation encoding="application/x-tex"> \otimes : Set \times \mathcal{T} \to \mathcal{T} </annotation></semantics></math></div> <p>applied to the <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>const</mi><mo>=</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>⊗</mo><mo>*</mo><mo>:</mo><mi>Set</mi><mo>→</mo><mi>𝒯</mi></mrow><annotation encoding="application/x-tex"> const = (-)\otimes {*} : Set \to \mathcal{T} </annotation></semantics></math></div> <p>which sends a set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> to the <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><mi>S</mi><mo stretchy="false">|</mo></mrow><annotation encoding="application/x-tex">|S|</annotation></semantics></math> copies of the terminal object</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>⊗</mo><mo>*</mo><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow></munder><mo>*</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> S \otimes {*} = \coprod_{s \in S} {*} \,. </annotation></semantics></math></div> <p>This is called the <strong>constant object</strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒯</mi></mrow><annotation encoding="application/x-tex">\mathcal{T}</annotation></semantics></math> on the set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>. Notably when <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/Grothendieck+topos">sheaf topos</a> this is the <strong><a class="existingWikiWord" href="/nlab/show/constant+sheaf">constant sheaf</a></strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>LConst</mi> <mi>S</mi></msub></mrow><annotation encoding="application/x-tex">LConst_S</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒯</mi><mover><mover><mo>→</mo><mi>Γ</mi></mover><mover><mo>←</mo><mi>LConst</mi></mover></mover><mi>Set</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathcal{T} \stackrel{\stackrel{LConst}{\leftarrow}}{\overset{\Gamma}{\to}} Set \,. </annotation></semantics></math></div> <p>If the 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/locally+connected+topos">locally connected topos</a> then the left adjoint functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>LConst</mi></mrow><annotation encoding="application/x-tex">LConst</annotation></semantics></math> is also a right adjoint, its left adjoint being the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>0</mn></msub><mo>:</mo><mi>𝒯</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">\Pi_0 : \mathcal{T} \to Set</annotation></semantics></math> that sends an object to its set of connected components.</p> <h3 id="of_objects_in_an_topos">Of objects in an <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</h3> <p>The previous abstract definition generalizes straightforwardly to every context of <a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a> where the required notions of <a class="existingWikiWord" href="/nlab/show/adjoint+functor">adjoint functor</a> etc. are provided.</p> <p>Notably in <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> theory the global section functor on an <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stack">∞-stack</a> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/hom-functor">hom-functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Γ</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><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><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>=</mo><msub><mi>Sh</mi> <mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>Sh</mi> <mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo><mo>=</mo><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex"> \Gamma(-) \;\coloneqq\; \mathbf{H}(*,-) \colon \; \mathbf{H} = Sh_{(\infty,1)}(C) \to Sh_{(\infty,1)}(*) = \infty Grpd </annotation></semantics></math></div> <p>of morphisms out of the terminal object.</p> <p>This is indeed again the <a class="existingWikiWord" href="/nlab/show/terminal+geometric+morphism">terminal geometric morphism</a>.</p> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> be an <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stack">∞-stack</a> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a>. Then the <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Geom</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>,</mo><mn>∞</mn><mi>Grpd</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Geom(\mathbf{H}, \infty Grpd)</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/geometric+morphism">geometric</a> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a>s is <a class="existingWikiWord" href="/nlab/show/contractible">contractible</a>.</p> <p>So <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">\infty Grpd</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a> in the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> of <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a>es and geometric morphisms.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>This is <a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">HTT 6.3.4.1</a></p> </div> <p>If the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> is a <a class="existingWikiWord" href="/nlab/show/locally+contractible+%28%E2%88%9E%2C1%29-topos">locally contractible (∞,1)-topos</a> then this is an <a class="existingWikiWord" href="/nlab/show/essential+geometric+morphism">essential geometric morphism</a>.</p> <p>The composite <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi><mo>∘</mo><mi>LConst</mi></mrow><annotation encoding="application/x-tex">\Gamma \circ LConst</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/shape+of+an+%28%E2%88%9E%2C1%29-topos">shape of</a> <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> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/local+section">local section</a>, <a class="existingWikiWord" href="/nlab/show/flat+section">flat section</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/ex-space">ex-space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/global+element">global element</a></p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on February 3, 2023 at 13:45:42. 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