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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='#definition'>Definition</a></li> <ul> <li><a href='#special_case_of_sheaf_topoi'>Special case of sheaf topoi</a></li> <ul> <li><a href='#testing_sheaf_morphisms_on_stalks'>Testing sheaf morphisms on stalks</a></li> <li><a href='#example'>Example</a></li> </ul> </ul> <li><a href='#examples'>Examples</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="definition">Definition</h2> <p>Recall that a <a class="existingWikiWord" href="/nlab/show/point+of+a+topos">point</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> of a <a class="existingWikiWord" href="/nlab/show/topos">topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/geometric+morphism">geometric morphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>:</mo><mi>Set</mi><mo>→</mo><mi>E</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> x : Set \to E \,. </annotation></semantics></math></div> <p>The <strong>stalk</strong> at <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> of an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi><mo>∈</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">e \in E</annotation></semantics></math> is the image 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">e</annotation></semantics></math> under the corresponding <a class="existingWikiWord" href="/nlab/show/inverse+image">inverse image</a> morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>x</mi> <mo>*</mo></msup><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>E</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex"> x^* \,\colon\, E \to Set </annotation></semantics></math></div> <p>i.e.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>stalk</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>≔</mo><mspace width="thinmathspace"></mspace><msup><mi>x</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> stalk_x(e) \,\coloneqq\, x^*(e) \,. </annotation></semantics></math></div> <h3 id="special_case_of_sheaf_topoi">Special case of sheaf topoi</h3> <p>If <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">E</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/category+of+sheaves">category of sheaves</a> on the <a class="existingWikiWord" href="/nlab/show/category+of+open+subsets">category of open subsets</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Op</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Op(X)</annotation></semantics></math> 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></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>E</mi><mspace width="thinmathspace"></mspace><mo>=</mo><mspace width="thinmathspace"></mspace><mi>Sh</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> E \,=\, Sh(X) \,, </annotation></semantics></math></div> <p>then the <a class="existingWikiWord" href="/nlab/show/point+of+a+topos">topos points</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">E</annotation></semantics></math> come precisely from the ordinary points</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo lspace="verythinmathspace">:</mo><mo>*</mo><mo>→</mo><mi>X</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>Top</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> (x \colon {*} \to X) \in Top \,, </annotation></semantics></math></div> <p>of the 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>, where the <a class="existingWikiWord" href="/nlab/show/direct+image">direct image</a> morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mo>*</mo></msub><mo lspace="verythinmathspace">:</mo><mi>Set</mi><mo>→</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> x_* \colon Set \to Sh(X) </annotation></semantics></math></div> <p>sends every set to the <a class="existingWikiWord" href="/nlab/show/skyscraper+sheaf">skyscraper sheaf</a> at this point 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>. By the general <a class="existingWikiWord" href="/nlab/show/Kan+extension">Kan extension</a> formula for the <a class="existingWikiWord" href="/nlab/show/inverse+image">inverse image</a> (see there) one finds in this case for any <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>F</mi><mo>∈</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F \in Sh(X)</annotation></semantics></math> the stalk</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mi>stalk</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><munder><mi>colim</mi><mrow><mo stretchy="false">(</mo><mo>*</mo><mo>→</mo><msup><mi>x</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>∈</mo><mo stretchy="false">(</mo><msub><mi>const</mi> <mo>*</mo></msub><mo>,</mo><msup><mi>x</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></mrow></munder><mi>F</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><munder><mi>colim</mi><mfrac linethickness="0"><mrow><mrow><mi>V</mi><mo>⊂</mo><mi>X</mi></mrow></mrow><mrow><mrow><mi>x</mi><mo>∈</mo><mi>V</mi></mrow></mrow></mfrac></munder><mi>F</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} stalk_x(F) & = \underset{ ({*} \to x^{-1}(V)) \in (const_{*}, x^{-1}) }{colim} F(V) \\ & = \underset{ { V \subset X } \atop { x \in V } }{ colim } F(V) \end{aligned} \,. </annotation></semantics></math></div> <p>So for sheaves on (open subsets of) topological spaces the stalk at a given point is the colimit over all values of the sheaf on open subsets containing this point.</p> <p>By the general definition of colimits in <a class="existingWikiWord" href="/nlab/show/Set">Set</a> described at <a class="existingWikiWord" href="/nlab/show/limits+and+colimits+by+example">limits and colimits by example</a>, the elements in this colimit can in turn be described as <a class="existingWikiWord" href="/nlab/show/equivalence+classes">equivalence classes</a> represented pairs <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(z, V)</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">x \in V</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>z</mi><mo>∈</mo><mi>F</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">z \in F(V)</annotation></semantics></math>, where the equivalence relation says that two such pairs <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>z</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>V</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(z_1, V_1)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>z</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>V</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(z_2, V_2)</annotation></semantics></math> coincide if there is a third pair <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(z,U)</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>⊂</mo><msub><mi>V</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">U \subset V_1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>⊂</mo><msub><mi>V</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">U \subset V_2</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>z</mi><mo>=</mo><msub><mi>z</mi> <mn>1</mn></msub><msub><mo stretchy="false">|</mo> <mi>U</mi></msub><mo>=</mo><msub><mi>z</mi> <mn>2</mn></msub><msub><mo stretchy="false">|</mo> <mi>U</mi></msub></mrow><annotation encoding="application/x-tex">z = z_1|_U = z_2|_U</annotation></semantics></math>.</p> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>=</mo><mi>C</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F = C(-)</annotation></semantics></math> a sheaf of functions 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>, such an equivalence class, hence such an element in a stalk of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> is called a function <strong>germ</strong>.</p> <h4 id="testing_sheaf_morphisms_on_stalks">Testing sheaf morphisms on stalks</h4> <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">E</annotation></semantics></math> a topos with <a class="existingWikiWord" href="/nlab/show/point+of+a+topos">enough points</a>, the behaviour of morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">f : A \to B</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">E</annotation></semantics></math> can be tested on stalks:</p> <div class="num_theorem"> <h6 id="theorem">Theorem</h6> <p>A morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">f : A \to B</annotation></semantics></math> of sheaves 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 a</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monomorphism">monomorphism</a></p> </li> <li> <p>resp. <a class="existingWikiWord" href="/nlab/show/epimorphism">epimorphism</a></p> </li> <li> <p>resp. <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a></p> </li> </ul> <p>if and only if every induced map of stalk sets <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>stalk</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>:</mo><msub><mi>stalk</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>stalk</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">stalk_x(f) : stalk_x(A) \to stalk_x(B)</annotation></semantics></math> is, for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x \in X</annotation></semantics></math></p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>The statement for isomorphisms follows from the identification of sheaves with <a class="existingWikiWord" href="/nlab/show/etale+space">etale space</a>s (e.g. section II, 6, corollary 3 in MacLane-Moerdijk, <a class="existingWikiWord" href="/nlab/show/Sheaves+in+Geometry+and+Logic">Sheaves in Geometry and Logic</a>). The statement for epimorphisms/monomorphisms is proposition 6 there.</p> </div> <h4 id="example">Example</h4> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> be a smooth manifold and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega^n(X)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Z</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Z^{n+1}(X)</annotation></semantics></math> be the sheaves of differential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-forms and that of <em>closed</em> differential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1)</annotation></semantics></math>-forms 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>, respectively, for some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math>. Let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>:</mo><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>Z</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex"> d : \Omega^n(X) \to Z^{n+1} </annotation></semantics></math></div> <p>be the morphism of sheaves that is given on each open subset by the deRham differential.</p> <p>Then:</p> <ul> <li> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">U \subset X</annotation></semantics></math> the map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>U</mi></msub><mo>:</mo><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>Z</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">d_U : \Omega^n(U) \to Z^{n+1}(U)</annotation></semantics></math> need not be <a class="existingWikiWord" href="/nlab/show/epimorphism">epi</a>, since not every closed form is exact;</p> </li> <li> <p>but by the <a class="existingWikiWord" href="/nlab/show/Poincare+lemma">Poincare lemma</a> every closed form is <em>locally</em> exact, so that for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x \in X</annotation></semantics></math> the map of stalks <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>x</mi></msub><mo>:</mo><msub><mi>stalk</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><msub><mi>stalk</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><msup><mi>Z</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">d_x : stalk_x(\Omega^n(X)) \to stalk_x(Z^{n+1}(X))</annotation></semantics></math> is an epimorphism.</p> </li> </ul> <p>Accordingly, the morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>:</mo><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>Z</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">d : \Omega^n(X) \to Z^{n+1}(X)</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/epimorphism">epimorphism</a> of sheaves.</p> <p>This kind of example plays a crucial role in the computation of <a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a>, see the examples listed there.</p> <h2 id="examples">Examples</h2> <p>For a <a class="existingWikiWord" href="/nlab/show/locally+ringed+topos">locally ringed topos</a> with <a class="existingWikiWord" href="/nlab/show/structure+sheaf">structure sheaf</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒪</mi></mrow><annotation encoding="application/x-tex">\mathcal{O}</annotation></semantics></math>, the stalk of the <a class="existingWikiWord" href="/nlab/show/multiplicative+group">multiplicative group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔾</mi> <mi>m</mi></msub></mrow><annotation encoding="application/x-tex">\mathbb{G}_m</annotation></semantics></math> at a point <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 multiplicative group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>𝒪</mi> <mi>x</mi> <mo>×</mo></msubsup></mrow><annotation encoding="application/x-tex">\mathcal{O}_x^\times</annotation></semantics></math> in the stalk <a class="existingWikiWord" href="/nlab/show/local+ring">local ring</a> of the structure sheaf. (e.g. <a href="#Milne">Milne, example 6.13</a>)</p> <h2 id="related_concepts">Related concepts</h2> <div> <p><strong>Examples of sequences of local structures</strong></p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/geometry">geometry</a></th><th>point</th><th>first order <a class="existingWikiWord" href="/nlab/show/infinitesimal+object">infinitesimal</a></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊂</mo></mrow><annotation encoding="application/x-tex">\subset</annotation></semantics></math></th><th><a class="existingWikiWord" href="/nlab/show/formal+geometry">formal</a> = arbitrary order infinitesimal</th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊂</mo></mrow><annotation encoding="application/x-tex">\subset</annotation></semantics></math></th><th>local = <a class="existingWikiWord" href="/nlab/show/stalk">stalk</a>wise</th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊂</mo></mrow><annotation encoding="application/x-tex">\subset</annotation></semantics></math></th><th>finite</th></tr></thead><tbody><tr><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>←</mo></mrow><annotation encoding="application/x-tex">\leftarrow</annotation></semantics></math> <strong><a class="existingWikiWord" href="/nlab/show/differentiation">differentiation</a></strong></td><td style="text-align: left;"></td><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/integration">integration</a></strong> <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></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/derivative">derivative</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Taylor+series">Taylor series</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/germ">germ</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/curve">curve</a> (path)</td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/tangent+vector">tangent vector</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/jet">jet</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/germ">germ</a> of <a class="existingWikiWord" href="/nlab/show/curve">curve</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/curve">curve</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/smooth+space">smooth space</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/infinitesimal+neighbourhood">infinitesimal neighbourhood</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/formal+neighbourhood">formal neighbourhood</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/germ+of+a+space">germ of a space</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/open+neighbourhood">open neighbourhood</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/function+algebra">function algebra</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/square-0+ring+extension">square-0 ring extension</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/nilpotent+ring+extension">nilpotent ring extension</a>/<a class="existingWikiWord" href="/nlab/show/formal+completion">formal completion</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/ring+extension">ring extension</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/arithmetic+geometry">arithmetic geometry</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔽</mi> <mi>p</mi></msub></mrow><annotation encoding="application/x-tex">\mathbb{F}_p</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/finite+field">finite field</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mi>p</mi></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_p</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/p-adic+integers">p-adic integers</a></td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mrow><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_{(p)}</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/localization+of+a+ring">localization at (p)</a></td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/integers">integers</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Lie+theory">Lie theory</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/formal+group">formal group</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/local+Lie+group">local Lie group</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Poisson+manifold">Poisson manifold</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/formal+deformation+quantization">formal deformation quantization</a></td><td style="text-align: left;"></td><td style="text-align: left;">local strict deformation quantization</td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/strict+deformation+quantization">strict deformation quantization</a></td></tr> </tbody></table> </div> <h2 id="references">References</h2> <ul id="Milne"> <li><a class="existingWikiWord" href="/nlab/show/James+Milne">James Milne</a>, section 6 of <em><a class="existingWikiWord" href="/nlab/show/Lectures+on+%C3%89tale+Cohomology">Lectures on Étale Cohomology</a></em></li> </ul> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ofer+Gabber">Ofer Gabber</a>, Shane Kelly, <em>Points in algebraic geometry</em>, J. Pure App. Alg. <strong>219</strong>:10 (2015) 4667-4680 <a href="https://doi.org/10.1016/j.jpaa.2015.03.001">doi</a></li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on October 15, 2023 at 13:43:15. 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