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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="category_theory">Category Theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></strong></p> <h2 id="sidebar_concepts">Concepts</h2> <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> <li> <p><a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cat">Cat</a></p> </li> </ul> <h2 id="sidebar_universal_constructions">Universal constructions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+construction">universal construction</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/representable+functor">representable functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+functor">adjoint functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/limit">limit</a>/<a class="existingWikiWord" href="/nlab/show/colimit">colimit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/weighted+limit">weighted limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/end">end</a>/<a class="existingWikiWord" href="/nlab/show/coend">coend</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kan+extension">Kan extension</a></p> </li> </ul> </li> </ul> <h2 id="sidebar_theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+construction">Grothendieck construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+functor+theorem">adjoint functor theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monadicity+theorem">monadicity theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+lifting+theorem">adjoint lifting theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gabriel-Ulmer+duality">Gabriel-Ulmer duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freyd-Mitchell+embedding+theorem">Freyd-Mitchell embedding theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relation+between+type+theory+and+category+theory">relation between type theory and category theory</a></p> </li> </ul> <h2 id="sidebar_extensions">Extensions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf+and+topos+theory">sheaf and topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a></p> </li> </ul> <h2 id="sidebar_applications">Applications</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/applications+of+%28higher%29+category+theory">applications of (higher) category theory</a></li> </ul> <div> <p> <a href="/nlab/edit/category+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="topos_theory">Topos Theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/topos+theory">topos theory</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Toposes">Toposes</a></li> </ul> <h2 id="background">Background</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category">category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> </li> </ul> </li> </ul> <h2 id="toposes">Toposes</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%280%2C1%29-topos">(0,1)-topos</a>, <a class="existingWikiWord" href="/nlab/show/Heyting+algebra">Heyting algebra</a>, <a class="existingWikiWord" href="/nlab/show/locale">locale</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pretopos">pretopos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topos">topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+topos">Grothendieck topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+presheaves">category of presheaves</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/representable+functor">representable presheaf</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+sheaves">category of sheaves</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/site">site</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sieve">sieve</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/coverage">coverage</a>, <a class="existingWikiWord" href="/nlab/show/Grothendieck+pretopology">pretopology</a>, <a class="existingWikiWord" href="/nlab/show/Grothendieck+topology">topology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sheafification">sheafification</a></p> </li> </ul> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quasitopos">quasitopos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/base+topos">base topos</a>, <a class="existingWikiWord" href="/nlab/show/indexed+topos">indexed topos</a></p> </li> </ul> <h2 id="internal_logic">Internal Logic</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/categorical+semantics">categorical semantics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/internal+logic">internal logic</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/subobject+classifier">subobject classifier</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/natural+numbers+object">natural numbers object</a></p> </li> </ul> </li> </ul> <h2 id="topos_morphisms">Topos morphisms</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/logical+morphism">logical morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+morphism">geometric morphism</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/direct+image">direct image</a>/<a class="existingWikiWord" href="/nlab/show/inverse+image">inverse image</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/global+section">global sections</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+embedding">geometric embedding</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/surjective+geometric+morphism">surjective geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/essential+geometric+morphism">essential geometric morphism</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+connected+geometric+morphism">locally connected geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connected+geometric+morphism">connected geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/totally+connected+geometric+morphism">totally connected geometric morphism</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%C3%A9tale+geometric+morphism">étale geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/open+geometric+morphism">open geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/proper+geometric+morphism">proper geometric morphism</a>, <a class="existingWikiWord" href="/nlab/show/compact+topos">compact topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/separated+geometric+morphism">separated geometric morphism</a>, <a class="existingWikiWord" href="/nlab/show/Hausdorff+topos">Hausdorff topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+geometric+morphism">local geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bounded+geometric+morphism">bounded geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/base+change">base change</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/localic+geometric+morphism">localic geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hyperconnected+geometric+morphism">hyperconnected geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/atomic+geometric+morphism">atomic geometric morphism</a></p> </li> </ul> </li> </ul> <h2 id="extra_stuff_structure_properties">Extra stuff, structure, properties</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+locale">topological locale</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/localic+topos">localic topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/petit+topos">petit topos/gros topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+connected+topos">locally connected topos</a>, <a class="existingWikiWord" href="/nlab/show/connected+topos">connected topos</a>, <a class="existingWikiWord" href="/nlab/show/totally+connected+topos">totally connected topos</a>, <a class="existingWikiWord" href="/nlab/show/strongly+connected+topos">strongly connected topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+topos">local topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohesive+topos">cohesive topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/classifying+topos">classifying topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+topos">smooth topos</a></p> </li> </ul> <h2 id="cohomology_and_homotopy">Cohomology and homotopy</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+groups+in+an+%28infinity%2C1%29-topos">homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">model structure on simplicial presheaves</a></p> </li> </ul> <h2 id="in_higher_category_theory">In higher category theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+topos+theory">higher topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%280%2C1%29-topos">(0,1)-topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/%280%2C1%29-site">(0,1)-site</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-topos">2-topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-site">2-site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-sheaf">2-sheaf</a>, <a class="existingWikiWord" href="/nlab/show/stack">stack</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-site">(∞,1)-site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-sheaf">(∞,1)-sheaf</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stack">∞-stack</a>, <a class="existingWikiWord" href="/nlab/show/derived+stack">derived stack</a></p> </li> </ul> </li> </ul> <h2 id="theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Diaconescu%27s+theorem">Diaconescu's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Barr%27s+theorem">Barr's theorem</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/topos+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> </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='#remarks'>Remarks</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#limits_and_colimits'>Limits and colimits</a></li> <li><a href='#PresheavesAreColimitsOfRepresentables'>Presheaves are colimits of representables</a></li> </ul> <li><a href='#special_cases'>Special cases</a></li> <li><a href='#examples'>Examples</a></li> <li><a href='#related_concepts'>Related concepts</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>Given a small category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> of “primitive objects”, we can think of a functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mspace width="mediummathspace"></mspace><msup><mi>C</mi> <mi>op</mi></msup><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">F:\: C^{op} \to Set</annotation></semantics></math> as being a more complex object built out of primitive objects:</p> <ul> <li> <p>Given a primitive object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> 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">C</annotation></semantics></math>, we interpret <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(X)</annotation></semantics></math> as a set representing the ways <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> occurs inside <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></p> </li> <li> <p>Given a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mspace width="mediummathspace"></mspace><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f:\: X \to Y</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">C</annotation></semantics></math>, we interpret <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>:</mo><mspace width="mediummathspace"></mspace><mi>F</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>→</mo><mi>F</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(f):\: F(Y) \to F(X)</annotation></semantics></math> as the function mapping each occurrence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> in <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> to the corresponding suboccurrence <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> (included in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math> through <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>) of <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 <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></p> </li> </ul> <p>Such functors are called presheaves.</p> <h2 id="definition">Definition</h2> <p>A <strong>presheaf</strong> on a <a class="existingWikiWord" href="/nlab/show/small+category">small category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mspace width="mediummathspace"></mspace><msup><mi>C</mi> <mi>op</mi></msup><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex"> F:\: C^{op} \to Set </annotation></semantics></math></div> <p>from the <a class="existingWikiWord" href="/nlab/show/opposite+category">opposite category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">C^{op}</annotation></semantics></math> 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">C</annotation></semantics></math> to the category <a class="existingWikiWord" href="/nlab/show/Set">Set</a> of <a class="existingWikiWord" href="/nlab/show/set">set</a>s. Equivalently this may be thought of as a <a class="existingWikiWord" href="/nlab/show/contravariant+functor">contravariant functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mspace width="mediummathspace"></mspace><mi>C</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">F:\: C \to Set</annotation></semantics></math>.</p> <p>More generally, given any category <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>, an <strong><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>-valued presheaf</strong> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is a functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mspace width="mediummathspace"></mspace><msup><mi>C</mi> <mi>op</mi></msup><mo>→</mo><mi>S</mi><mo>.</mo></mrow><annotation encoding="application/x-tex"> F:\: C^{op} \to S. </annotation></semantics></math></div> <p>While, hence, presheaves are just <a class="existingWikiWord" href="/nlab/show/functors">functors</a> (on <a class="existingWikiWord" href="/nlab/show/small+categories">small categories</a>), one says “presheaf” to indicate a specific perspective or interest, namely interest in the <em><a class="existingWikiWord" href="/nlab/show/sheafification">sheafification</a></em> of the functor/presheaf, or at least interest in the <a class="existingWikiWord" href="/nlab/show/functor+category">functor category</a> as a <a class="existingWikiWord" href="/nlab/show/topos">topos</a> (the <a class="existingWikiWord" href="/nlab/show/presheaf+topos">presheaf topos</a>). Hence “presheaf” is a <a class="existingWikiWord" href="/nlab/show/concept+with+an+attitude">concept with an attitude</a>.</p> <p>Historically, the initial applications of presheaves and sheaves involved cases like <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">S = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/CRing">CRing</a> (the category of <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative rings</a>), <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">S = </annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Ab">Ab</a> (<a class="existingWikiWord" href="/nlab/show/abelian+groups">abelian groups</a>), <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">S = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Mod">RMod</a> (<a class="existingWikiWord" href="/nlab/show/modules">modules</a>), etc. Later, especially with the development of <a class="existingWikiWord" href="/nlab/show/topos+theory">topos theory</a>, the primary importance of the <a class="existingWikiWord" href="/nlab/show/sheaf+topos">category of set-valued (pre)sheaves</a> as a <a class="existingWikiWord" href="/nlab/show/topos">topos</a> was recognized; these other cases could be considered algebraic objects which live in the topos. This article and the one on <a class="existingWikiWord" href="/nlab/show/sheaf+topos">sheaf topos</a> recognize these later developments by making the set-valued case the default (in other words, presheaf or sheaf without further qualification is understood to refer to the set-valued case).</p> <p>The <strong><a class="existingWikiWord" href="/nlab/show/category+of+presheaves">category of presheaves</a></strong> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>, usually denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Set</mi> <mrow><msup><mi>C</mi> <mi>op</mi></msup></mrow></msup></mrow><annotation encoding="application/x-tex">Set^{C^{op}}</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[C^{op},Set]</annotation></semantics></math>, but often abbreviated as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>C</mi><mo>^</mo></mover></mrow><annotation encoding="application/x-tex">\widehat{C}</annotation></semantics></math>, has:</p> <ul> <li> <p>functors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mspace width="mediummathspace"></mspace><msup><mi>C</mi> <mi>op</mi></msup><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">F:\: C^{op} \to Set</annotation></semantics></math> as objects;</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformations</a> between such functors as morphisms.</p> </li> </ul> <p>As such, it is an example of a <a class="existingWikiWord" href="/nlab/show/functor+category">functor category</a>.</p> <h2 id="remarks">Remarks</h2> <ul> <li> <p>Speaking of functors as presheaves indicates operations that one wants to do apply to these functors, or certain properties that one wants to check.</p> <ul> <li> <p>when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>=</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">S = Set</annotation></semantics></math>, and especially one is interested in the <a class="existingWikiWord" href="/nlab/show/Yoneda+embedding">Yoneda embedding</a> of a category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> into its presheaf category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[C^{op}, Set]</annotation></semantics></math> for purposes of studying, for instance, <a class="existingWikiWord" href="/nlab/show/limit">limit</a>s, <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a>s, <a class="existingWikiWord" href="/nlab/show/ind-object">ind-object</a>s, and <a class="existingWikiWord" href="/nlab/show/pro-object">pro-object</a>s 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">C</annotation></semantics></math>;</p> </li> <li> <p>or when there is the structure of a <a class="existingWikiWord" href="/nlab/show/site">site</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>, such that it makes sense to ask if a given presheaf is actually a <a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a>.</p> </li> </ul> </li> <li> <p>One generally useful way to think of presheaves is in the sense of <a class="existingWikiWord" href="/nlab/show/space+and+quantity">space and quantity</a>.</p> </li> <li> <p>In the case where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>=</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">S = Set</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/small+category">small</a>, an important general principle is that the presheaf category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[C^{op},Set]</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/free+cocompletion">free cocompletion</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">C</annotation></semantics></math>; see <a class="existingWikiWord" href="/nlab/show/Yoneda+extension">Yoneda extension</a>. Intuitively, it is formed by taking <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> and ‘freely throwing in small colimits’. The category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is contained in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[C^{op},Set]</annotation></semantics></math> via the <a class="existingWikiWord" href="/nlab/show/Yoneda+embedding">Yoneda embedding</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>:</mo><mspace width="mediummathspace"></mspace><mi>C</mi><mo>→</mo><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> Y:\: C \to [C^{op},Set]</annotation></semantics></math></div> <p>The Yoneda embedding sends each object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">c \in C</annotation></semantics></math> to the presheaf</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>=</mo><mi>hom</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> F(-) = hom(-, c) </annotation></semantics></math></div> <p>Presheaves of this form, or isomorphic to those of this form, are called <a class="existingWikiWord" href="/nlab/show/representable+functors">representable</a>; among their properties, representable presheaves always turn colimits into limits, in the sense that a representable functor from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">C^{op}</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Set</mi></mrow><annotation encoding="application/x-tex">Set</annotation></semantics></math> turns colimits 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">C</annotation></semantics></math> (i.e., limits in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">C^{op}</annotation></semantics></math>) into limits in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Set</mi></mrow><annotation encoding="application/x-tex">Set</annotation></semantics></math> (i.e., colimits in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Set</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">Set^{op}</annotation></semantics></math>). In general, such continuity is a necessary but not sufficient criterion for representability; however, nicely enough, it <em>is</em> sufficient when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> itself is a presheaf category. To see this, suppose <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> is such a presheaf on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>=</mo><mo stretchy="false">[</mo><msup><mi>D</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">C = [D^{op}, Set]</annotation></semantics></math>, and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>=</mo><mi>K</mi><mi>Y</mi></mrow><annotation encoding="application/x-tex">G = K Y</annotation></semantics></math>, a presheaf on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>. By the <a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a>, we have a natural isomorphism between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>D</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>,</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">[D^{op}, Set](Y(-), G)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mi>Y</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K Y(-)</annotation></semantics></math>. But by the free cocompletion property of the Yoneda embedding, a colimit-preserving functor on presheaves is entirely determined by its precomposition with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math>; accordingly, our isomorphism must extend to an identification of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">[C^{op}, Set](-, G)</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(-)</annotation></semantics></math>, thus establishing the representability of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>.</p> </li> </ul> <h2 id="properties">Properties</h2> <h3 id="limits_and_colimits">Limits and colimits</h3> <p>Any <a class="existingWikiWord" href="/nlab/show/category+of+presheaves">category of presheaves</a> is <a class="existingWikiWord" href="/nlab/show/complete+category">complete</a> and <a class="existingWikiWord" href="/nlab/show/cocomplete+category">cocomplete</a>, with both <a class="existingWikiWord" href="/nlab/show/limit">limits</a> and <a class="existingWikiWord" href="/nlab/show/colimit">colimits</a> being computed <em>pointwise</em>. That is, to compute the limit or colimit of a diagram <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mspace width="mediummathspace"></mspace><mi>D</mi><mo>→</mo><msup><mi>Set</mi> <mrow><msup><mi>C</mi> <mi>op</mi></msup></mrow></msup></mrow><annotation encoding="application/x-tex">F:\: D \to Set^{C^op}</annotation></semantics></math>, we think of it as a functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mspace width="mediummathspace"></mspace><mi>D</mi><mo>×</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">F:\: D \times C^{op} \to Set</annotation></semantics></math> and take the limit or colimit in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> variable.</p> <h3 id="PresheavesAreColimitsOfRepresentables">Presheaves are colimits of representables</h3> <p> <div class='num_prop' id='EveryPresheafIsColimitOfRepresentables'> <h6>Proposition</h6> <p>Every presheaf is a <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a> of <a class="existingWikiWord" href="/nlab/show/representable+functor">representable presheaves</a>.</p> </div> </p> <p>An elegant way to express this colimit for a presheaf <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mspace width="mediummathspace"></mspace><msup><mi>C</mi> <mi>op</mi></msup><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">F:\: C^{op} \to Set</annotation></semantics></math> is in terms of the <a class="existingWikiWord" href="/nlab/show/coend">coend</a> identity</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>=</mo><msup><mo>∫</mo> <mrow><mi>c</mi><mo>∈</mo><mi>C</mi></mrow></msup><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>×</mo><msub><mi>hom</mi> <mi>C</mi></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> F(-) = \int^{c \in C} F(c) \times hom_C(-,c) \,, </annotation></semantics></math></div> <p>which follows by <a class="existingWikiWord" href="/nlab/show/Yoneda+reduction">Yoneda reduction</a>. See also at <em><a class="existingWikiWord" href="/nlab/show/co-Yoneda+lemma">co-Yoneda lemma</a></em>.</p> <p>More concretely: let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>:</mo><mspace width="mediummathspace"></mspace><mi>C</mi><mo>→</mo><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">Y:\: C \to [C^{op}, Set]</annotation></semantics></math> denote the <a class="existingWikiWord" href="/nlab/show/Yoneda+embedding">Yoneda embedding</a> and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mi>F</mi></msub><mo>≔</mo><mi>Y</mi><mo stretchy="false">/</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">C_F \coloneqq Y/F</annotation></semantics></math> be the corresponding <a class="existingWikiWord" href="/nlab/show/comma+category">comma category</a>, the <a class="existingWikiWord" href="/nlab/show/category+of+elements">category of elements</a> 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>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mi>F</mi></msub><mo>≔</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mi>Y</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mrow><mi>Y</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd></mtd> <mtd><mi>Y</mi><mo stretchy="false">(</mo><mi>V</mi><mo>′</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mi>f</mi></msub><mo>↘</mo></mtd> <mtd></mtd> <mtd><msub><mo>↙</mo> <mrow><mi>f</mi><mo>′</mo></mrow></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>F</mi></mtd></mtr></mtable></mrow><mo>}</mo></mrow></mrow><annotation encoding="application/x-tex"> C_F \coloneqq \left\lbrace \array{ Y(V) &&\stackrel{Y(g)}{\to}&& Y(V') \\ & {}_f\searrow && \swarrow_{f'} \\ && F } \right\rbrace </annotation></semantics></math></div> <p>and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><mspace width="mediummathspace"></mspace><msub><mi>C</mi> <mi>F</mi></msub><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">p:\: C_F \to C</annotation></semantics></math> the canonical forgetful functor. Then the colimit over representables expression <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</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>≃</mo><msub><mi>colim</mi> <mrow><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo>→</mo><mi>F</mi><mo stretchy="false">)</mo><mo>∈</mo><msub><mi>C</mi> <mi>F</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>Y</mi><mo>∘</mo><mi>p</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> F \simeq colim_{(Y(V) \to F) \in C_F} (Y\circ p) \,. </annotation></semantics></math></div> <p>This is often written with some convenient abuse of notation as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>≃</mo><msub><mi>colim</mi> <mrow><mi>V</mi><mo>→</mo><mi>F</mi></mrow></msub><mi>V</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> F \simeq colim_{V \to F} V \,. </annotation></semantics></math></div> <p>Notice that these formulas can also be understood as those for the left <a class="existingWikiWord" href="/nlab/show/Kan+extension">Kan extension</a> (see there) 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> along the identity functor.</p> <div class="proof"> <h6 id="proof">Proof</h6> <p>Notice that for every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>∈</mo><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">B \in [C^{op}, Set]</annotation></semantics></math> and using the property of the <a class="existingWikiWord" href="/nlab/show/hom-functor">hom-functor</a> we have</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>Hom</mi> <mrow><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>colim</mi> <mrow><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo>→</mo><mi>F</mi><mo stretchy="false">)</mo><mo>∈</mo><msub><mi>C</mi> <mi>F</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>Y</mi><mo>∘</mo><mi>p</mi><mo stretchy="false">)</mo><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>≃</mo><msub><mi>lim</mi> <mrow><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo>→</mo><mi>F</mi><mo stretchy="false">)</mo><mo>∈</mo><msub><mi>C</mi> <mi>F</mi></msub></mrow></msub><msub><mi>Hom</mi> <mrow><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow></msub><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><msub><mi>lim</mi> <mrow><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo>→</mo><mi>F</mi><mo stretchy="false">)</mo><mo>∈</mo><msub><mi>C</mi> <mi>F</mi></msub></mrow></msub><mi>B</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} Hom_{[C^{op}, Set]}(colim_{(Y(V) \to F) \in C_F} (Y\circ p),B) &\simeq lim_{(Y(V) \to F) \in C_F} Hom_{[C^{op}, Set]}(Y(V),B) \\ & \simeq lim_{(Y(V) \to F) \in C_F} B(V) \end{aligned} </annotation></semantics></math></div> <p>by the <a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a>.</p> <p>By the definition of <a class="existingWikiWord" href="/nlab/show/limit">limit</a> we have that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>⋯</mi><mo>=</mo><msub><mi>Hom</mi> <mrow><mo stretchy="false">[</mo><msubsup><mi>C</mi> <mi>F</mi> <mi>op</mi></msubsup><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow></msub><mo stretchy="false">(</mo><mi>pt</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo><mo>,</mo></mrow><annotation encoding="application/x-tex">\cdots=Hom_{[C_F^{op}, Set]}(pt,B),</annotation></semantics></math></div> <p>so for each natural transformation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>∈</mo><msub><mi>Hom</mi> <mrow><mo stretchy="false">[</mo><msubsup><mi>C</mi> <mi>F</mi> <mi>op</mi></msubsup><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow></msub><mo stretchy="false">(</mo><mi>pt</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\alpha \in Hom_{[C_F^{op}, Set]}(pt,B)</annotation></semantics></math> and each object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>:</mo><mspace width="mediummathspace"></mspace><mi>Y</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo>→</mo><mi>F</mi><mo>∈</mo><msub><mi>C</mi> <mi>F</mi></msub></mrow><annotation encoding="application/x-tex">h:\: Y(V)\to F\in C_F</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>α</mi> <mi>h</mi></msub></mrow><annotation encoding="application/x-tex">\alpha_h</annotation></semantics></math> is a map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mo>*</mo><mo stretchy="false">}</mo><mo>→</mo><mi>B</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\{*\}\to B(V)</annotation></semantics></math>, that is, it is an element of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B(V)</annotation></semantics></math>. However, by Yoneda, we know that each object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>:</mo><mspace width="mediummathspace"></mspace><mi>Y</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo>→</mo><mi>F</mi><mo>∈</mo><msub><mi>C</mi> <mi>F</mi></msub></mrow><annotation encoding="application/x-tex">h:\: Y(V)\to F\in C_F</annotation></semantics></math> specifies a unique element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>∈</mo><mi>F</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">h\in F(V)</annotation></semantics></math>. Then rephrasing this, <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> specifies a <a class="existingWikiWord" href="/nlab/show/function">function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo>→</mo><mi>B</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(V)\to B(V)</annotation></semantics></math>. The naturality of this assignment is guaranteed by the naturality of the map <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>. Then <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> induces a natural transformation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>k</mi> <mi>α</mi></msup><mo>:</mo><mspace width="mediummathspace"></mspace><mi>F</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">k^\alpha:\: F\to B</annotation></semantics></math>. It’s easy to check that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> defines an isomorphism:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mrow><mo stretchy="false">[</mo><msubsup><mi>C</mi> <mi>F</mi> <mi>op</mi></msubsup><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow></msub><mo stretchy="false">(</mo><mi>pt</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>Hom</mi> <mrow><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow></msub><mo stretchy="false">(</mo><mi>F</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Hom_{[C_F^{op}, Set]}(pt,B) \simeq Hom_{[C^{op}, Set]}(F,B) \,. </annotation></semantics></math></div> <p>Since this holds for all <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>, the claim follows, again using the <a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a>.</p> </div> <h2 id="special_cases">Special cases</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/representable+functor">representable presheaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/concrete+presheaf">concrete presheaf</a></p> </li> </ul> <h2 id="examples">Examples</h2> <p>Examples for presheaves are abundant. Here is a non-representative selection of some examples.</p> <ul> <li> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/locally+small">locally small</a> category, every object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">c \in C</annotation></semantics></math> gives rise to the <a class="existingWikiWord" href="/nlab/show/representable+functor">representable presheaf</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mi>C</mi></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>:</mo><mspace width="mediummathspace"></mspace><msup><mi>C</mi> <mi>op</mi></msup><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">Hom_C(-, c):\: C^{op} \to Set</annotation></semantics></math>.</p> </li> <li> <p>More generally, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>:</mo><mspace width="mediummathspace"></mspace><mi>C</mi><mo>↪</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">i:\: C \hookrightarrow D</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/subcategory">subcategory</a> of a locally small category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>, every object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>∈</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">d \in D</annotation></semantics></math> gives rise to the presheaf</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mi>D</mi></msub><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo>:</mo><mspace width="mediummathspace"></mspace><msup><mi>C</mi> <mi>op</mi></msup><mo>→</mo><mi>Set</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Hom_D(i(-), d):\: C^{op} \to Set \,. </annotation></semantics></math></div> <p>Let’s spell this out in more detail: given a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>:</mo><mspace width="mediummathspace"></mspace><mi>V</mi><mo>→</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">\phi:\: V \to U</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">C</annotation></semantics></math>, we can take any morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mspace width="mediummathspace"></mspace><mi>i</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">f:\: i(U) \to X</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mi>D</mi></msub><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Hom_{D}(U,X)</annotation></semantics></math> and turn it into a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mover><mo>→</mo><mi>ϕ</mi></mover><mi>U</mi><mover><mo>→</mo><mi>f</mi></mover><mi>X</mi></mrow><annotation encoding="application/x-tex">V \stackrel{\phi}{\to} U \stackrel{f}{\to} X</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mi>D</mi></msub><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Hom_{D}(i(V),X)</annotation></semantics></math>. This determines a map of set</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup><mo>:</mo><mspace width="mediummathspace"></mspace><msub><mi>Hom</mi> <mi>D</mi></msub><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>Hom</mi> <mi>D</mi></msub><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> f^*:\: Hom_{D}(i(U),X) \to Hom_{D}(i(V),X) \,. </annotation></semantics></math></div> <p>So we have a functorial assignment of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>W</mi></mtd> <mtd></mtd> <mtd><mo>↦</mo></mtd> <mtd></mtd> <mtd><msub><mi>Hom</mi> <mi>Diff</mi></msub><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">(</mo><mi>W</mi><mo stretchy="false">)</mo><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mi>g</mi></msup></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↑</mo> <mrow><msup><mi>g</mi> <mo>*</mo></msup></mrow></msup></mtd></mtr> <mtr><mtd><mi>V</mi></mtd> <mtd></mtd> <mtd><mo>↦</mo></mtd> <mtd></mtd> <mtd><msub><mi>Hom</mi> <mi>Diff</mi></msub><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mi>f</mi></msup></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↑</mo> <mrow><msup><mi>f</mi> <mo>*</mo></msup></mrow></msup></mtd></mtr> <mtr><mtd><mi>U</mi></mtd> <mtd></mtd> <mtd><mo>↦</mo></mtd> <mtd></mtd> <mtd><msub><mi>Hom</mi> <mi>Diff</mi></msub><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ W && \mapsto && Hom_{Diff}(i(W),X) \\ \downarrow^g &&&& \uparrow^{g^*} \\ V && \mapsto && Hom_{Diff}(i(V),X) \\ \downarrow^f &&&& \uparrow^{f^*} \\ U && \mapsto && Hom_{Diff}(i(U),X) } \,. </annotation></semantics></math></div> <p>Of course <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math> here could be any functor whatsoever. Asking if such a presheaf is <a class="existingWikiWord" href="/nlab/show/representable+functor">representable</a> is asking for a right <a class="existingWikiWord" href="/nlab/show/adjoint+functor">adjoint functor</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math>.</p> </li> <li> <p>A <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a> is a presheaf on the <a class="existingWikiWord" href="/nlab/show/simplex+category">simplex category</a></p> <p>A <a class="existingWikiWord" href="/nlab/show/globular+set">globular set</a> is a presheaf on the <a class="existingWikiWord" href="/nlab/show/globe+category">globe category</a>.</p> <p>A <a class="existingWikiWord" href="/nlab/show/cubical+set">cubical set</a> is a presheaf on the <a class="existingWikiWord" href="/nlab/show/cube+category">cube category</a>.</p> </li> <li> <p>A <a class="existingWikiWord" href="/nlab/show/diffeological+space">diffeological space</a> is a <a class="existingWikiWord" href="/nlab/show/concrete+presheaf">concrete presheaf</a> on <a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a>.</p> </li> <li> <p>An important class of presheaves is those on a <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> or <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</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> </li> <li> <p>Traditional standard examples include: the presheaf of <a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a>s 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>, that assigns to each <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 set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mi>ℝ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\infty(U,\mathbb{R})</annotation></semantics></math> of smooth functions and to each inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>⊂</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">V \subset U</annotation></semantics></math> the corresponding restriction operation of functions. This is further a sheaf.</p> </li> <li> <p>Traditional standard example which is a presheaf but not a sheaf: the presheaf of exact 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>, that assigns to <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 set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ω</mi> <mi>exact</mi> <mo>•</mo></msubsup><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega^\bullet_{exact}(U)</annotation></semantics></math> of exact forms on <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> and to each inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>⊂</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">V \subset U</annotation></semantics></math> the corresponding restriction operation of functions. Here, and like above, the site is made up by open sets in <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> with inclusions as morphisms.</p> </li> </ul> <p>… etc. pp.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%280%2C1%29-presheaf">(0,1)-presheaf</a></p> </li> <li> <p><strong>presheaf</strong></p> <p><a class="existingWikiWord" href="/nlab/show/constant+presheaf">constant presheaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%282%2C1%29-presheaf">(2,1)-presheaf</a></p> <p><a class="existingWikiWord" href="/nlab/show/presheaf+of+groupoids">presheaf of groupoids</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-presheaf">(∞,1)-presheaf</a></p> <p><a class="existingWikiWord" href="/nlab/show/simplicial+presheaf">simplicial presheaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-presheaf">(∞,n)-presheaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a>, <a class="existingWikiWord" href="/nlab/show/Yoneda+extension">Yoneda extension</a></p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on October 9, 2021 at 06:43:46. 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