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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> <blockquote> <p>This article is about ends (and coends) in <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a>. For ends in <a class="existingWikiWord" href="/nlab/show/topology">topology</a>, see at <em><a class="existingWikiWord" href="/nlab/show/end+compactification">end compactification</a></em>.</p> </blockquote> <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="enriched_category_theory">Enriched category theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a></strong></p> <h2 id="background">Background</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a>, <a class="existingWikiWord" href="/nlab/show/closed+monoidal+category">closed monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cosmos">cosmos</a>, <a class="existingWikiWord" href="/nlab/show/multicategory">multicategory</a>, <a class="existingWikiWord" href="/nlab/show/bicategory">bicategory</a>, <a class="existingWikiWord" href="/nlab/show/double+category">double category</a>, <a class="existingWikiWord" href="/nlab/show/virtual+double+category">virtual double category</a></p> </li> </ul> <h2 id="basic_concepts">Basic concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+category">enriched category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+functor">enriched functor</a>, <a class="existingWikiWord" href="/nlab/show/profunctor">profunctor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+natural+transformation">enriched natural transformation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+adjoint+functor">enriched adjoint functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+product+category">enriched product category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+functor+category">enriched functor category</a></p> </li> </ul> <h2 id="universal_constructions">Universal constructions</h2> <ul> <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> </ul> <h2 id="extra_stuff_structure_property">Extra stuff, structure, property</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/copowering">copowering</a> (<a class="existingWikiWord" href="/nlab/show/tensoring">tensoring</a>)</p> <p><a class="existingWikiWord" href="/nlab/show/powering">powering</a> (<a class="existingWikiWord" href="/nlab/show/cotensoring">cotensoring</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+enriched+category">monoidal enriched category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cartesian+closed+enriched+category">cartesian closed enriched category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+cartesian+closed+enriched+category">locally cartesian closed enriched category</a></p> </li> </ul> </li> </ul> <h3 id="homotopical_enrichment">Homotopical enrichment</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+homotopical+category">enriched homotopical category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+model+category">enriched model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+homotopical+presheaves">model structure on homotopical presheaves</a></p> </li> </ul> </div></div> <h4 id="limits_and_colimits">Limits and colimits</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/limit">limits and colimits</a></strong></p> <h2 id="1categorical">1-Categorical</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/limit">limit and colimit</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/limits+and+colimits+by+example">limits and colimits by example</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/commutativity+of+limits+and+colimits">commutativity of limits and colimits</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/small+limit">small limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/filtered+colimit">filtered colimit</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/directed+colimit">directed colimit</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/sequential+colimit">sequential colimit</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sifted+colimit">sifted colimit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connected+limit">connected limit</a>, <a class="existingWikiWord" href="/nlab/show/wide+pullback">wide pullback</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/preserved+limit">preserved limit</a>, <a class="existingWikiWord" href="/nlab/show/reflected+limit">reflected limit</a>, <a class="existingWikiWord" href="/nlab/show/created+limit">created limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/product">product</a>, <a class="existingWikiWord" href="/nlab/show/fiber+product">fiber product</a>, <a class="existingWikiWord" href="/nlab/show/base+change">base change</a>, <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a>, <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a>, <a class="existingWikiWord" href="/nlab/show/pushout">pushout</a>, <a class="existingWikiWord" href="/nlab/show/cobase+change">cobase change</a>, <a class="existingWikiWord" href="/nlab/show/equalizer">equalizer</a>, <a class="existingWikiWord" href="/nlab/show/coequalizer">coequalizer</a>, <a class="existingWikiWord" href="/nlab/show/join">join</a>, <a class="existingWikiWord" href="/nlab/show/meet">meet</a>, <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a>, <a class="existingWikiWord" href="/nlab/show/initial+object">initial object</a>, <a class="existingWikiWord" href="/nlab/show/direct+product">direct product</a>, <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/finite+limit">finite limit</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/exact+functor">exact functor</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kan+extension">Kan extension</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Yoneda+extension">Yoneda extension</a></li> </ul> </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 and coend</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fibered+limit">fibered limit</a></p> </li> </ul> <h2 id="2categorical">2-Categorical</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-limit">2-limit</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/inserter">inserter</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/isoinserter">isoinserter</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equifier">equifier</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/inverter">inverter</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/PIE-limit">PIE-limit</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-pullback">2-pullback</a>, <a class="existingWikiWord" href="/nlab/show/comma+object">comma object</a></p> </li> </ul> <h2 id="1categorical_2">(∞,1)-Categorical</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-limit">(∞,1)-limit</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-pullback">(∞,1)-pullback</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fiber+sequence">fiber sequence</a></li> </ul> </li> </ul> </li> </ul> <h3 id="modelcategorical">Model-categorical</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+Kan+extension">homotopy Kan extension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+limit">homotopy limit</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+product">homotopy product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+equalizer">homotopy equalizer</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+totalization">homotopy totalization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+end">homotopy end</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+colimit">homotopy colimit</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coproduct">homotopy coproduct</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coequalizer">homotopy coequalizer</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+cofiber">homotopy cofiber</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+pushout">homotopy pushout</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+realization">homotopy realization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coend">homotopy coend</a></p> </li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/infinity-limits+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <ul> <li><a href='#in_ordinary_category_theory'>In ordinary category theory</a></li> <ul> <li><a href='#definition_via_extranatural_transformations'>Definition via extranatural transformations</a></li> <li><a href='#explicit_definition'>Explicit definition</a></li> <li><a href='#ends_as_right_adjoint_functors'>Ends as right adjoint functors</a></li> </ul> <li><a href='#in_enriched_category_theory'>In enriched category theory</a></li> <ul> <li><a href='#end_of_valued_functors'>End of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-valued functors</a></li> <li><a href='#ends_of_valued_functors_for_'>Ends 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>-valued functors for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>∈</mo><mi>V</mi><mo lspace="0em" rspace="thinmathspace">Cat</mo></mrow><annotation encoding="application/x-tex">C \in V\Cat</annotation></semantics></math></a></li> <li><a href='#end_as_an_equalizer'>End as an equalizer</a></li> <ul> <li><a href='#ordinary_ends_as_equalizers'>Ordinary ends as equalizers</a></li> <li><a href='#enriched_ends_over_valued_functors_as_equalizers'>Enriched ends over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-valued functors as equalizers</a></li> </ul> <li><a href='#end_as_a_weighted_limit'>End as a weighted limit</a></li> <li><a href='#connecting_the_two_definitions'>Connecting the two definitions</a></li> </ul> </ul> <li><a href='#Properties'>Properties</a></li> <ul> <li><a href='#SetCoendsAsColimits'><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>-enriched coends as ordinary colimits</a></li> <li><a href='#commutativity_of_ends_and_coends'>Commutativity of ends and coends</a></li> </ul> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#natural_transformations'>Natural transformations</a></li> <li><a href='#enriched_functor_categories'>Enriched functor categories</a></li> <li><a href='#kan_extension'>Kan extension</a></li> <li><a href='#geometric_realization'>Geometric realization</a></li> <li><a href='#tensor_product_of_functors'>Tensor product of functors</a></li> </ul> <li><a href='#coend_calculus'>(Co)end calculus</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>An <em>end</em> is a special kind of <a class="existingWikiWord" href="/nlab/show/limit">limit</a> over a <a class="existingWikiWord" href="/nlab/show/functor">functor</a> of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>×</mo><mi>C</mi><mo>→</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">F : C^{op} \times C \to D</annotation></semantics></math> (sometimes called a <em><a class="existingWikiWord" href="/nlab/show/bifunctor">bifunctor</a></em>).</p> <p>If we think of such a functor in the sense of <a class="existingWikiWord" href="/nlab/show/profunctors">profunctors</a> as encoding a left and right <a class="existingWikiWord" href="/nlab/show/action">action</a> on the object</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>c</mi><mo>∈</mo><mi>C</mi></mrow></munder><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>,</mo></mrow><annotation encoding="application/x-tex"> \prod_{c \in C} F(c,c), </annotation></semantics></math></div> <p>then the <em>end</em> of the functor picks out the universal <a class="existingWikiWord" href="/nlab/show/subobject">subobject</a> on which the left and right actions coincide. Dually, the <em>coend</em> 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 the universal quotient of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mrow><mi>c</mi><mo>∈</mo><mi>C</mi></mrow></msub><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\coprod_{c \in C} F(c,c)</annotation></semantics></math> that forces the two actions 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> on that object to be equal.</p> <p>A classical example of an <em>end</em> is the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-object of <a class="existingWikiWord" href="/nlab/show/natural+transformations">natural transformations</a> between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/enriched+functors">enriched functors</a> in <a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a>. Perhaps the most common way in which ends and coends arise is through homs and tensor products of (generalized) modules, and their close cousins, <a class="existingWikiWord" href="/nlab/show/weighted+limits">weighted limits</a> and <a class="existingWikiWord" href="/nlab/show/weighted+colimits">weighted colimits</a>. These concepts are fundamental in enriched category theory.</p> <h2 id="definition">Definition</h2> <h3 id="in_ordinary_category_theory">In ordinary category theory</h3> <h4 id="definition_via_extranatural_transformations">Definition via extranatural transformations</h4> <p>In ordinary <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a>, given a <a class="existingWikiWord" href="/nlab/show/functor">functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>×</mo><mi>C</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">F: C^{op} \times C \to X</annotation></semantics></math>, an <strong>end</strong> 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> 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> is an object <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> 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> equipped with a <a class="existingWikiWord" href="/nlab/show/universal+construction">universal</a> <a class="existingWikiWord" href="/nlab/show/extranatural+transformation">extranatural transformation</a> from <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> to <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>. This means that, given any extranatural transformation from an 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> 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> to <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>, there exists a unique map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>→</mo><mi>e</mi></mrow><annotation encoding="application/x-tex">x \to e</annotation></semantics></math> which respects the extranatural transformations.</p> <p>In more detail: the end 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 traditionally denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∫</mo> <mrow><mi>c</mi><mo>:</mo><mi>C</mi></mrow></msub><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\int_{c: C} F(c, c)</annotation></semantics></math>, and the components of the universal extranatural transformation,</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mi>c</mi></msub><mo>:</mo><msub><mo>∫</mo> <mrow><mi>c</mi><mo>:</mo><mi>C</mi></mrow></msub><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>→</mo><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>,</mo></mrow><annotation encoding="application/x-tex">\pi_c: \int_{c: C} F(c, c) \to F(c, c),</annotation></semantics></math></div> <p>are called the <em>projection maps</em> of the end. Then, given any extranatural transformation with components</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>θ</mi> <mi>c</mi></msub><mo>:</mo><mi>x</mi><mo>→</mo><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>,</mo></mrow><annotation encoding="application/x-tex">\theta_c: x \to F(c, c),</annotation></semantics></math></div> <p>there exists a unique map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>x</mi><mo>→</mo><mi>e</mi></mrow><annotation encoding="application/x-tex">f: x \to e</annotation></semantics></math> such that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>θ</mi> <mi>c</mi></msub><mo>=</mo><msub><mi>π</mi> <mi>c</mi></msub><mi>f</mi></mrow><annotation encoding="application/x-tex">\theta_c = \pi_c f</annotation></semantics></math></div> <p>for every object <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 <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> <p>The notion of <strong>coend</strong> is dual to the notion of end. The coend 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 written <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><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>,</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\int^{c: C} F(c, c)</annotation></semantics></math>, and comes equipped with a universal extranatural transformation with components</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ι</mi> <mi>c</mi></msub><mo lspace="verythinmathspace">:</mo><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>c</mi><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>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex">\iota_c \colon F(c,c) \to \int^{c: C} F(c,c).</annotation></semantics></math></div> <h4 id="explicit_definition">Explicit definition</h4> <p>We unwrap the definition of an extranatural transformation to obtain a more explicit description of an end.</p> <div class="num_defn" id="wedge"> <h6 id="definition_2">Definition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>×</mo><mi>C</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">F: C^{op} \times C \to X</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/functor">functor</a>. A <strong>wedge</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi><mo>:</mo><mi>w</mi><mo>→</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">e: w \to F</annotation></semantics></math> <strong>over F</strong> is an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>w</mi></mrow><annotation encoding="application/x-tex">w</annotation></semantics></math> and maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mi>c</mi></msub><mo>:</mo><mi>w</mi><mo>→</mo><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">e_c: w\to F(c, c)</annotation></semantics></math> for each object <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, given any morphism <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>→</mo><mi>c</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">f: c \to c'</annotation></semantics></math>, the following diagram commutes:</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><mover><mo>→</mo><mrow><msub><mi>e</mi> <mrow><mi>c</mi><mo>′</mo></mrow></msub></mrow></mover></mtd> <mtd><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo>′</mo><mo>,</mo><mi>c</mi><mo>′</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mo></mo><mpadded width="0" lspace="-100%width"><mrow><msub><mi>e</mi> <mi>c</mi></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>c</mi><mo>′</mo><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo></mtd> <mtd><munder><mo>→</mo><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></munder></mtd> <mtd><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>c</mi><mo>′</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ w & \overset{e_{c'}}{\to} & F(c', c')\\ ^\mathllap{e_c}\downarrow & & \downarrow^\mathrlap{F(f, c')}\\ F(c, c) & \underset{F(c, f)}{\to} & F(c, c') } </annotation></semantics></math></div></div> <p>Given a wedge <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi><mo>:</mo><mi>w</mi><mo>→</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">e: w \to F</annotation></semantics></math> and a map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>v</mi><mo>→</mo><mi>w</mi></mrow><annotation encoding="application/x-tex">f: v \to w</annotation></semantics></math>, we obtain a wedge <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi><mi>f</mi><mo>:</mo><mi>v</mi><mo>→</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">e f: v \to F</annotation></semantics></math> by composition. We define the end as follows:</p> <div class="num_defn"> <h6 id="definition_3">Definition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>×</mo><mi>C</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">F: C^{op} \times C \to X</annotation></semantics></math> be a functor. An <strong>end</strong> 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 a universal wedge over <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>, i.e., a wedge <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi><mo>:</mo><mi>w</mi><mo>→</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">e: w \to F</annotation></semantics></math> such that each other wedge <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi><mo>′</mo><mo>:</mo><mi>w</mi><mo>′</mo><mo>→</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">e': w' \to F</annotation></semantics></math> factors through <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> via a unique map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>w</mi><mo>′</mo><mo>→</mo><mi>w</mi></mrow><annotation encoding="application/x-tex">w' \to w</annotation></semantics></math>.</p> </div> <p>Dually, a cowedge is given by maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>→</mo><mi>w</mi></mrow><annotation encoding="application/x-tex">F(c, c) \to w</annotation></semantics></math> satisfying similar commutativity conditions, and a coend is a universal cowedge.</p> <h4 id="ends_as_right_adjoint_functors">Ends as right adjoint functors</h4> <p>In complete analogy to how <a class="existingWikiWord" href="/nlab/show/limits">limits</a> are right adjoint functors to the <a class="existingWikiWord" href="/nlab/show/diagonal+functor">diagonal functor</a> for the diagram, ends are right adjoint functors to the hom functor.</p> <p>In more detail, suppose <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 <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> are categories.</p> <p>If every diagram <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mi>op</mi></msup><mo>×</mo><mi>C</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">C^{op}\times C\to X</annotation></semantics></math> admits an end, then we have a functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>end</mi><mo lspace="verythinmathspace">:</mo><mi>Fun</mi><mo stretchy="false">(</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>×</mo><mi>C</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">end\colon Fun(C^{op}\times C,X)\to X</annotation></semantics></math></div> <p>whose left adjoint is the hom functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>hom</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>Fun</mi><mo stretchy="false">(</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>×</mo><mi>C</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">hom\colon X\to Fun(C^{op}\times C,X)</annotation></semantics></math></div> <p>that sends an object <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> to the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>hom</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>×</mo><mi>C</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">hom(x)\colon C^{op}\times C\to X</annotation></semantics></math> that sends <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(c,d)</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mrow><mi>hom</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo></mrow></msub><mi>x</mi><mo>=</mo><mi>hom</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">\coprod_{hom(c,d)}x=hom(c,d)\otimes x</annotation></semantics></math>. (For coends, one uses <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>x</mi> <mrow><mi>hom</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex">x^{hom(c,d)}</annotation></semantics></math> instead.)</p> <p>This immediately implies a Fubini theorem for ends and coends.</p> <h3 id="in_enriched_category_theory">In enriched category theory</h3> <p>There is a definition of <em>end</em> in <a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a> as follows.</p> <h4 id="end_of_valued_functors">End of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-valued functors</h4> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a>, and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> be a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/enriched+category">enriched category</a>. Assuming <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> is also <a class="existingWikiWord" href="/nlab/show/closed+monoidal+category">closed monoidal</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> may be considered as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-enriched. In that case, suppose <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>⊗</mo><mi>C</mi><mo>→</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">F: C^{op} \otimes C \to V</annotation></semantics></math> is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/enriched+functor">enriched functor</a>.</p> <p>Then there is a covariant <a class="existingWikiWord" href="/nlab/show/action">action</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> on <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>, with components</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>λ</mi> <mrow><mi>c</mi><mo>,</mo><mi>d</mi><mo>,</mo><mi>e</mi></mrow></msub><mo>:</mo><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>C</mi><mo stretchy="false">(</mo><mi>d</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo><mo>→</mo><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo><mo>,</mo></mrow><annotation encoding="application/x-tex">\lambda_{c, d, e}: F(c, d) \otimes C(d, e) \to F(c, e),</annotation></semantics></math></div> <p>(where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>d</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C(d, e)</annotation></semantics></math> is customary notation for the <a class="existingWikiWord" href="/nlab/show/hom-object">hom-object</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><mi>d</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\hom_C(d, e)</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> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>), and a contravariant action 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> on <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>, with components</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ρ</mi> <mrow><mi>c</mi><mo>,</mo><mi>d</mi><mo>,</mo><mi>e</mi></mrow></msub><mo>:</mo><mi>F</mi><mo stretchy="false">(</mo><mi>d</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>C</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo>→</mo><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex">\rho_{c, d, e}: F(d, e) \otimes C(c, d) \to F(c, e).</annotation></semantics></math></div> <p>In detail, the covariant action is the <a class="existingWikiWord" href="/nlab/show/adjunct">adjunct</a> of the morphism</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>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><mi>C</mi><mo stretchy="false">(</mo><mi>d</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">[</mo><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo>,</mo><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>∈</mo><msub><mi>Hom</mi> <mi>V</mi></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">(</mo><mi>d</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">[</mo><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo>,</mo><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (F(c,-) \colon C(d,e) \to [F(c,d), F(c,e)]) \in Hom_V(C(d,e),[F(c,d), F(c,e)]) </annotation></semantics></math></div> <p>under the <a class="existingWikiWord" href="/nlab/show/closed+monoidal+category">Hom-adjunction</a></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>V</mi></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">(</mo><mi>d</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">[</mo><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo>,</mo><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mo>≃</mo></mover><msub><mi>Hom</mi> <mi>V</mi></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">(</mo><mi>d</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo>,</mo><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Hom_V(C(d,e),[F(c,d), F(c,e)]) \stackrel{\simeq}{\longrightarrow} Hom_V(C(d,e)\otimes F(c,d),F(c,e)) </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>. Similarly for the contravariant action.</p> <div class="un_remark"> <h6 id="remark">Remark</h6> <p>Even if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> is not closed monoidal, we can still define a <strong>notion</strong> of covariant <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>-action, sometimes called a “left” <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 class="existingWikiWord" href="/nlab/show/module">module</a>, as consisting of a function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo lspace="verythinmathspace">:</mo><mi>Ob</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Ob</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F \colon Ob(C) \to Ob(V)</annotation></semantics></math> together with an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ob</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo>×</mo><mi>Ob</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ob(V) \times Ob(V)</annotation></semantics></math>-indexed collection of morphisms</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><mi>c</mi><mo stretchy="false">)</mo><mo>×</mo><mi>C</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo>→</mo><mi>F</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(c) \times C(c, d) \to F(d)</annotation></semantics></math></div> <p>satisfying some evident unit and associativity axioms, and regard this notion as a stand-in for the notion of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>→</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">C \to V</annotation></semantics></math>. Similarly, we have an evident notion of contravariant <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>-action as a stand-in for a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mi>op</mi></msup><mo>→</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">C^{op} \to V</annotation></semantics></math>. Notice that we don’t even require symmetry to make sense of this. Finally, we can combine these notions into one 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>-bimodule, where we have a function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo lspace="verythinmathspace">:</mo><mi>Ob</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>×</mo><mi>Ob</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Ob</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F \colon Ob(C) \times Ob(C) \to Ob(V)</annotation></semantics></math> together with a collection of morphisms</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>F</mi><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>C</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo>→</mo><mi>F</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C(a, b) \otimes F(b, c) \otimes C(c, d) \to F(a, d)</annotation></semantics></math></div> <p>with evident axioms for a bimodule structure, as a stand-in for a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-functor of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mi>op</mi></msup><mo>⊗</mo><mi>C</mi><mo>→</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">C^{op} \otimes C \to V</annotation></semantics></math>.</p> </div> <p>A <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>- <strong>extranatural transformation</strong></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>θ</mi><mo>:</mo><mi>v</mi><mover><mo>→</mo><mo>•</mo></mover><mi>F</mi></mrow><annotation encoding="application/x-tex">\theta: v \stackrel{\bullet}{\to} F</annotation></semantics></math></div> <p>from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi></mrow><annotation encoding="application/x-tex">v</annotation></semantics></math> to <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> consists of a family of arrows in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</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>θ</mi> <mi>c</mi></msub><mo>:</mo><mi>v</mi><mo>→</mo><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>,</mo></mrow><annotation encoding="application/x-tex">\theta_c: v \to F(c, c),</annotation></semantics></math></div> <p>indexed over objects <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 <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 for every pair of objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(c, d)</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>, the composites of (1) and (2) agree:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>v</mi><mo>⊗</mo><mi>C</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msub><mi>θ</mi> <mi>c</mi></msub><mo>⊗</mo><mn>1</mn></mrow></mover><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>C</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msub><mi>λ</mi> <mrow><mi>c</mi><mo>,</mo><mi>c</mi><mo>,</mo><mi>d</mi></mrow></msub></mrow></mover><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mspace width="2em"></mspace><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">v \otimes C(c, d) \stackrel{\theta_c \otimes 1}{\to} F(c, c) \otimes C(c, d) \stackrel{\lambda_{c, c, d}}{\to} F(c, d) \qquad (1)</annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>v</mi><mo>⊗</mo><mi>C</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msub><mi>θ</mi> <mi>d</mi></msub><mo>⊗</mo><mn>1</mn></mrow></mover><mi>F</mi><mo stretchy="false">(</mo><mi>d</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>C</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msub><mi>ρ</mi> <mrow><mi>c</mi><mo>,</mo><mi>d</mi><mo>,</mo><mi>d</mi></mrow></msub></mrow></mover><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mspace width="2em"></mspace><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">v \otimes C(c, d) \stackrel{\theta_d \otimes 1}{\to} F(d, d) \otimes C(c, d) \stackrel{\rho_{c, d, d}}{\to} F(c, d) \qquad (2)</annotation></semantics></math></div> <p>A <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-enriched <strong>end</strong> 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 an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∫</mo> <mrow><mi>c</mi><mo>:</mo><mi>C</mi></mrow></msub><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\int_{c: C} F(c, c)</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> equipped with a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-extranatural transformation</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>π</mi><mo>:</mo><msub><mo>∫</mo> <mrow><mi>c</mi><mo>:</mo><mi>C</mi></mrow></msub><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mo>•</mo></mover><mi>F</mi></mrow><annotation encoding="application/x-tex">\pi: \int_{c: C} F(c, c) \stackrel{\bullet}{\to} F</annotation></semantics></math></div> <p>such any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-extranatural transformation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi></mrow><annotation encoding="application/x-tex">v</annotation></semantics></math> to <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 obtained by pulling back the components of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>v</mi><mo>→</mo><msub><mo>∫</mo> <mrow><mi>c</mi><mo>:</mo><mi>C</mi></mrow></msub><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f: v \to \int_{c: C} F(c, c)</annotation></semantics></math>, for some unique map <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>. That is,</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>θ</mi> <mi>c</mi></msub><mo>=</mo><msub><mi>π</mi> <mi>c</mi></msub><mi>f</mi></mrow><annotation encoding="application/x-tex">\theta_c = \pi_c f</annotation></semantics></math></div> <p>for all objects <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 <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> <h4 id="ends_of_valued_functors_for_">Ends 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>-valued functors for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>∈</mo><mi>V</mi><mo lspace="0em" rspace="thinmathspace">Cat</mo></mrow><annotation encoding="application/x-tex">C \in V\Cat</annotation></semantics></math></h4> <p>If <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 any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-enriched category and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>⊗</mo><mi>C</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">F: C^{op} \otimes C \to X</annotation></semantics></math> is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-enriched functor, then the <strong>end</strong> 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> 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> is, if it exists, an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∫</mo> <mrow><mi>c</mi><mo>:</mo><mi>C</mi></mrow></msub><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\int_{c: C} F(c, c)</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> that <a class="existingWikiWord" href="/nlab/show/representable+functor">represents</a> the functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mo>∫</mo> <mrow><mi>c</mi><mo>:</mo><mi>C</mi></mrow></msub><mi>X</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \int_{c: C} X(-,F(c,c))\,. </annotation></semantics></math></div> <p>That means that the end <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∫</mo> <mrow><mi>c</mi><mo>:</mo><mi>C</mi></mrow></msub><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\int_{c: C} F(c,c)</annotation></semantics></math> comes equipped with an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ob</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ob(C)</annotation></semantics></math>-indexed family of arrows</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mi>c</mi></msub><mo>:</mo><mi>I</mi><mo>→</mo><mi>X</mi><mo stretchy="false">(</mo><msub><mo>∫</mo> <mrow><mi>c</mi><mo>:</mo><mi>C</mi></mrow></msub><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>,</mo><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \pi_c: I \to X(\int_{c: C} F(c, c), F(c, c)) </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>, such that, for every 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> 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>, the family of maps</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 stretchy="false">(</mo><mi>x</mi><mo>,</mo><msub><mi>π</mi> <mi>c</mi></msub><mo stretchy="false">)</mo><mo>:</mo><mi>X</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><msub><mo>∫</mo> <mrow><mi>c</mi><mo>:</mo><mi>C</mi></mrow></msub><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><mi>X</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> X(x, \pi_c): X(x, \int_{c: C} F(c, c)) \to X(x, F(c, c)) </annotation></semantics></math></div> <p>are the projection maps realizing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><msub><mo>∫</mo> <mrow><mi>c</mi><mo>:</mo><mi>C</mi></mrow></msub><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X(x, \int_{c: C} F(c, c))</annotation></semantics></math> as the corresponding end <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∫</mo> <mrow><mi>c</mi><mo>:</mo><mi>C</mi></mrow></msub><mi>X</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\int_{c: C} X(x, F(c, c))</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>.</p> <h4 id="end_as_an_equalizer">End as an equalizer</h4> <h5 id="ordinary_ends_as_equalizers">Ordinary ends as equalizers</h5> <p>Now we motivate and define the <em>end</em> in <a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a> in terms of <a class="existingWikiWord" href="/nlab/show/equalizers">equalizers</a>.</p> <p>Recall from the discussion at the end of <a class="existingWikiWord" href="/nlab/show/limit">limit</a> that the <a class="existingWikiWord" href="/nlab/show/limit">limit</a> over an (ordinary, i.e., not enriched) <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><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>is given by the <a class="existingWikiWord" href="/nlab/show/equalizer">equalizer</a> of</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>c</mi><mo>∈</mo><mi>Obj</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow></munder><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>f</mi><mo>∈</mo><mi>Mor</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></munder><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>∘</mo><msub><mi>p</mi> <mrow><mi>t</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo></mrow></mover><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>f</mi><mo>∈</mo><mi>Mor</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow></munder><mi>F</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \prod_{c \in Obj(C)} F(c) \stackrel{\prod_{f \in Mor(c)} (F(f) \circ p_{t(f)}) }{\to} \prod_{f \in Mor(C)} F(s(f)) </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>c</mi><mo>∈</mo><mi>Obj</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow></munder><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>f</mi><mo>∈</mo><mi>Mor</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></munder><mo stretchy="false">(</mo><msub><mi>p</mi> <mrow><mi>s</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo></mrow></mover><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>f</mi><mo>∈</mo><mi>Mor</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow></munder><mi>F</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \prod_{c \in Obj(C)} F(c) \stackrel{\prod_{f \in Mor(c)} (p_{s(f)}) }{\to} \prod_{f \in Mor(C)} F(s(f)) \,. </annotation></semantics></math></div> <p>If we want to generalize an expression like this to <a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a>, the explicit indexing over the set of morphisms has to be replaced by something that makes sense in an <a class="existingWikiWord" href="/nlab/show/enriched+category">enriched category</a>.</p> <p>To that end, observe that we have a canonical isomorphism (of sets, still)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mrow><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>1</mn></msub><mover><mo>→</mo><mi>f</mi></mover><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><mo>∈</mo><mi>Mor</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow></munder><mi>F</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>≃</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><msub><mi>c</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>c</mi> <mn>2</mn></msub><mo>∈</mo><mi>Obj</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow></munder><mi>F</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>1</mn></msub><msup><mo stretchy="false">)</mo> <mrow><mi>C</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \prod_{{(c_1 \stackrel{f}{\to} c_2)} \in Mor(C)} F(c_1) \simeq \prod_{c_1,c_2 \in Obj(C)} F(c_1)^{C(c_1,c_2)} \,. </annotation></semantics></math></div> <p>If we write for the <a class="existingWikiWord" href="/nlab/show/hom-set">hom-set</a> instead</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>C</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>,</mo><mi>F</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>:</mo><mo>=</mo><mi>F</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>1</mn></msub><msup><mo stretchy="false">)</mo> <mrow><mi>C</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex"> [C(c_1,c_2), F(c_1)] := F(c_1)^{C(c_1,c_2)} </annotation></semantics></math></div> <p>with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[-,-]</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a> in <a class="existingWikiWord" href="/nlab/show/Set">Set</a>, then the expression starts to make sense in any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/enriched+category">enriched category</a>.</p> <p>Still equivalently, but suggestively rewriting the above, we now obtain the limit over <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> as the <a class="existingWikiWord" href="/nlab/show/equalizer">equalizer</a> of</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>c</mi><mo>∈</mo><mi>Obj</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow></munder><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mover><mover><mo>→</mo><mi>λ</mi></mover><mover><mo>→</mo><mi>ρ</mi></mover></mover><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><msub><mi>c</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>c</mi> <mn>2</mn></msub><mo>∈</mo><mi>Obj</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow></munder><mo stretchy="false">[</mo><mi>C</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>,</mo><mi>F</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \prod_{c \in Obj(C)} F(c) \stackrel{\stackrel{\rho}{\to}}{\stackrel{\lambda}{\to}} \prod_{c_1,c_2 \in Obj(C)} [C(c_1,c_2),F(c_1)] \,, </annotation></semantics></math></div> <p>where in components</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ρ</mi> <mrow><msub><mi>c</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>c</mi> <mn>2</mn></msub></mrow></msub><mo>:</mo><mi>F</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">[</mo><mi>C</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>,</mo><mi>F</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> \rho_{c_1, c_2} : F(c_1) \to [C(c_1,c_2), F(c_1)] </annotation></semantics></math></div> <p>is the <a class="existingWikiWord" href="/nlab/show/adjunct">adjunct</a> of</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>→</mo><mo>*</mo><mo>→</mo><mo stretchy="false">[</mo><mi>F</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>,</mo><mi>F</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> C(c_1, c_2) \to * \to [F(c_1), F(c_1)] </annotation></semantics></math></div> <p>(with the last map the <a class="existingWikiWord" href="/nlab/show/adjunct">adjunct</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Id</mi> <mrow><mi>F</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">Id_{F(c_1)}</annotation></semantics></math>), and where</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>λ</mi> <mrow><msub><mi>c</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>c</mi> <mn>2</mn></msub></mrow></msub><mo>:</mo><mi>F</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">[</mo><mi>C</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>,</mo><mi>F</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> \lambda_{c_1, c_2} : F(c_2) \to [C(c_1,c_2), F(c_1)] </annotation></semantics></math></div> <p>is the <a class="existingWikiWord" href="/nlab/show/adjunct">adjunct</a> of</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mrow><msub><mi>c</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>c</mi> <mn>2</mn></msub></mrow></msub><mo>:</mo><mi>C</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">[</mo><mi>F</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>,</mo><mi>F</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> F_{c_1, c_2} : C(c_1, c_2) \to [F(c_2), F(c_1)] \,. </annotation></semantics></math></div> <p>So, for definiteness, the equalizer we are looking at is that of</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>:</mo><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><msub><mi>c</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>c</mi> <mn>2</mn></msub><mo>∈</mo><mi>C</mi></mrow></munder><msub><mi>ρ</mi> <mrow><msub><mi>c</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>c</mi> <mn>2</mn></msub></mrow></msub><mo>∘</mo><msub><mi>pr</mi> <mrow><mi>F</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex"> \rho := \prod_{c_1, c_2 \in C} \rho_{c_1,c_2}\circ pr_{F(c_1)} </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>λ</mi><mo>:</mo><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><msub><mi>c</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>c</mi> <mn>2</mn></msub><mo>∈</mo><mi>C</mi></mrow></munder><msub><mi>λ</mi> <mrow><msub><mi>c</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>c</mi> <mn>2</mn></msub></mrow></msub><mo>∘</mo><msub><mi>pr</mi> <mrow><mi>F</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex"> \lambda := \prod_{c_1, c_2 \in C} \lambda_{c_1,c_2}\circ pr_{F(c_2)} </annotation></semantics></math></div> <p>This way of writing the <a class="existingWikiWord" href="/nlab/show/limit">limit</a> clearly suggests that it is more natural to have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math> on equal footing. That leads to the following definition.</p> <h5 id="enriched_ends_over_valued_functors_as_equalizers">Enriched ends over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-valued functors as equalizers</h5> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal 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> a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/enriched+category">enriched category</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo lspace="verythinmathspace">:</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>×</mo><mi>C</mi><mo>→</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">F \colon C^{op} \times C \to V</annotation></semantics></math> a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/enriched+functor">enriched functor</a>, the <strong>end</strong> 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 the <a class="existingWikiWord" href="/nlab/show/equalizer">equalizer</a></p> <div class="maruku-equation" id="eq:endeq"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mo>∫</mo> <mrow><mi>c</mi><mo>∈</mo><mi>C</mi></mrow></msub><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>⟶</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>c</mi><mo>∈</mo><mi>Obj</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow></munder><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><munderover><mrow></mrow><munder><mo>⟶</mo><mi>λ</mi></munder><mover><mo>⟶</mo><mi>ρ</mi></mover></munderover><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><msub><mi>c</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>c</mi> <mn>2</mn></msub><mo>∈</mo><mi>Obj</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow></munder><mo stretchy="false">[</mo><mi>C</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>,</mo><mi>F</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> \int_{c \in C} F(c,c) \longrightarrow \prod_{c \in Obj(C)} F(c,c) \underoverset {\underset{\lambda}{\longrightarrow}} {\overset{\rho}{\longrightarrow}} {} \prod_{c_1,c_2 \in Obj(C)} [C(c_1,c_2),F(c_1,c_2)] </annotation></semantics></math></div> <p>with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math> in components given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ρ</mi> <mrow><msub><mi>c</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>c</mi> <mn>2</mn></msub></mrow></msub><mo lspace="verythinmathspace">:</mo><mi>F</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>⟶</mo><mo stretchy="false">[</mo><mi>C</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>,</mo><mi>F</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> \rho_{c_1, c_2} \colon F(c_1,c_1) \longrightarrow [C(c_1,c_2), F(c_1,c_2)] </annotation></semantics></math></div> <p>being the <a class="existingWikiWord" href="/nlab/show/adjunct">adjunct</a> of</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><msub><mi>c</mi> <mn>1</mn></msub><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><mi>C</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>⟶</mo><mo stretchy="false">[</mo><mi>F</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>,</mo><mi>F</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> F(c_1,-) \colon C(c_1, c_2) \longrightarrow [F(c_1,c_1), F(c_1,c_2)] </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>λ</mi> <mrow><msub><mi>c</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>c</mi> <mn>2</mn></msub></mrow></msub><mo lspace="verythinmathspace">:</mo><mi>F</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>⟶</mo><mo stretchy="false">[</mo><mi>C</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>,</mo><mi>F</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> \lambda_{c_1, c_2} \colon F(c_2,c_2) \longrightarrow [C(c_1,c_2), F(c_1,c_2)] </annotation></semantics></math></div> <p>being the <a class="existingWikiWord" href="/nlab/show/adjunct">adjunct</a> of</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>,</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><mi>C</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>⟶</mo><mo stretchy="false">[</mo><mi>F</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>,</mo><mi>F</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> F(-,c_2) \colon C(c_1, c_2) \longrightarrow [F(c_2,c_2), F(c_1,c_2)] \,. </annotation></semantics></math></div> <p>This definition manifestly exhibits the <strong>end as the equalizer of the left and right action</strong> encoded by the <a class="existingWikiWord" href="/nlab/show/distributor">distributor</a> <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> <p>Dually, the coend 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 the <a class="existingWikiWord" href="/nlab/show/coequalizer">coequalizer</a></p> <div class="maruku-equation" id="eq:coendcoeq"><span class="maruku-eq-number">(2)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mrow><msub><mi>c</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>c</mi> <mn>2</mn></msub></mrow></munder><mi>C</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>⊗</mo><mi>F</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>⇉</mo><mspace width="thinmathspace"></mspace><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mi>c</mi></munder><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>→</mo><mspace width="thinmathspace"></mspace><msup><mo>∫</mo> <mi>c</mi></msup><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \coprod_{c_1,c_2} C(c_2,c_1) \otimes F(c_1,c_2)\, \rightrightarrows\, \coprod_c F(c,c)\,\to\, \int^c F(c,c) </annotation></semantics></math></div> <p>with the parallel morphisms again induced by the two actions 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> <h4 id="end_as_a_weighted_limit">End as a weighted limit</h4> <p>The end for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-functors with values in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> serves, among other things, to define <a class="existingWikiWord" href="/nlab/show/weighted+limits">weighted limits</a>, and weighted limits in turn define ends of bifunctors with values in more general <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-categories.</p> <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> and <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> both <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-categories and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><msup><mi>C</mi> <mo lspace="0em" rspace="thinmathspace">op</mo></msup><mo>×</mo><mi>C</mi><mo>→</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">F : C^\op \times C \to D</annotation></semantics></math> an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/enriched+functor">enriched functor</a>, the <strong>end</strong> 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 the <a class="existingWikiWord" href="/nlab/show/weighted+limit">weighted limit</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><mo>∫</mo> <mrow><mi>c</mi><mo>∈</mo><mi>C</mi></mrow></msub><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>≔</mo><mo stretchy="false">{</mo><msub><mi>Hom</mi> <mi>C</mi></msub><mo>,</mo><mi>F</mi><mo stretchy="false">}</mo><mo>=</mo><msup><mi>lim</mi> <mrow><msub><mi>Hom</mi> <mi>C</mi></msub></mrow></msup><mi>F</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \int_{c \in C} F(c,c) \coloneqq \{Hom_C, F\} = lim^{Hom_C} F \,, </annotation></semantics></math></div> <p>with weight <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>:</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>×</mo><mi>C</mi><mo>→</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">Hom_C : C^{op} \times C \to V</annotation></semantics></math>. The <strong>coend</strong> 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 the colimit</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><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>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>≔</mo><msub><mi>Hom</mi> <mrow><msup><mi>C</mi> <mi>op</mi></msup></mrow></msub><mo>*</mo><mi>F</mi><mo>=</mo><msup><mo lspace="0em" rspace="thinmathspace">colim</mo> <mrow><msub><mi>Hom</mi> <mrow><msup><mi>C</mi> <mi>op</mi></msup></mrow></msub></mrow></msup><mi>F</mi></mrow><annotation encoding="application/x-tex"> \int^{c \in C} F(c,c) \coloneqq Hom_{C^{op}} \ast F = \colim^{Hom_{C^{op}}} F </annotation></semantics></math></div> <p>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> weighted by the hom functor of <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>.</p> <h4 id="connecting_the_two_definitions">Connecting the two definitions</h4> <p>If <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 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-category, then the hom functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>×</mo><mi>C</mi><mo>→</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">C(-,-) \colon C^{op} \times C \to V</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/coequalizer">coequalizer</a> in</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mrow><mi>c</mi><mo>,</mo><mi>c</mi><mo>′</mo></mrow></munder><mi>C</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>×</mo><mi>C</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>c</mi><mo>′</mo><mo stretchy="false">)</mo><mo>×</mo><mi>C</mi><mo stretchy="false">(</mo><mi>c</mi><mo>′</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>⇉</mo><mspace width="thinmathspace"></mspace><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mi>c</mi></munder><mi>C</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>×</mo><mi>C</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>→</mo><mspace width="thinmathspace"></mspace><mi>C</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \coprod_{c,c'} C(-,c) \times C(c,c') \times C(c',-) \,\rightrightarrows\, \coprod_c C(-,c) \times C(c,-) \,\to\, C(-,-) </annotation></semantics></math></div> <p>It is also a general fact (see e.g. <a href="#Kelly">Kelly, ch. 3</a>) that weighted (co)limits are cocontinuous in their weight: that is,</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>W</mi><mo>*</mo><mi>V</mi><mo>,</mo><mi>F</mi><mo stretchy="false">}</mo><mo>≅</mo><mo stretchy="false">{</mo><mi>W</mi><mo>,</mo><mo stretchy="false">{</mo><mi>V</mi><mo>−</mo><mo>,</mo><mi>F</mi><mo stretchy="false">}</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex"> \{W \ast V, F\} \cong \{W, \{V-, F\}\}</annotation></semantics></math></div> <p>and</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>W</mi><mo>*</mo><mi>V</mi><mo stretchy="false">)</mo><mo>*</mo><mi>G</mi><mo>≅</mo><mi>W</mi><mo>*</mo><mo stretchy="false">(</mo><mi>V</mi><mo>*</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (W \ast V) \ast G \cong W \ast (V \ast G)</annotation></semantics></math></div> <p>This implies that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>F</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{-,F\}</annotation></semantics></math> takes the coequalizer above to an equalizer, which, after some fiddling with the <a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a>, turns out to be isomorphic to <a class="maruku-eqref" href="#eq:endeq">(1)</a>. Similarly, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>*</mo><mi>F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(- \ast F)</annotation></semantics></math> takes the analogous coequalizer presentation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mi>op</mi></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^{op}(-,-)</annotation></semantics></math> to <a class="maruku-eqref" href="#eq:coendcoeq">(2)</a>.</p> <h2 id="Properties">Properties</h2> <h3 id="SetCoendsAsColimits"><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>-enriched coends as ordinary colimits</h3> <p>Let the enriching category be <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒱</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">\mathcal{V} = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Set">Set</a>. We describe a special way in this case to express ends/coends that give <a class="existingWikiWord" href="/nlab/show/weighted+limits">weighted limits</a>/colimits in terms of ordinary (co)limits over categories of elements.</p> <p>Consider</p> <ul> <li> <p><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 <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>-enriched category/<a class="existingWikiWord" href="/nlab/show/locally+small+category">locally small category</a> <a class="existingWikiWord" href="/nlab/show/tensoring">tensored</a> over <a class="existingWikiWord" href="/nlab/show/Set">Set</a>;</p> </li> <li> <p><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> be a <a class="existingWikiWord" href="/nlab/show/small+category">small category</a>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mi>D</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">F : D \to C</annotation></semantics></math> a functor;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi><mo>:</mo><msup><mi>D</mi> <mi>op</mi></msup><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">W : D^{op} \to Set</annotation></semantics></math> another functor;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>el</mi><mi>W</mi></mrow><annotation encoding="application/x-tex">el W</annotation></semantics></math> 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>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math>.</p> </li> </ul> <p> <div class='num_prop' id='CoendAsColimitOverCategoryOfElements'> <h6>Proposition</h6> <p><strong>(coend as <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a> over <a class="existingWikiWord" href="/nlab/show/category+of+elements">category of elements</a>)</strong> <br /> There is a <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a> 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></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mo>∫</mo> <mrow><mi>d</mi><mo>∈</mo><mi>D</mi></mrow></msup><mi>W</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>F</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo>≃</mo><munder><mi>lim</mi> <mo>→</mo></munder><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>el</mi><mi>W</mi><msup><mo stretchy="false">)</mo> <mi>op</mi></msup><mo>→</mo><mi>D</mi><mover><mo>→</mo><mi>F</mi></mover><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \int^{d \in D} W(d) \cdot F(d) \simeq \lim_{\to}( (el W)^{op} \to D \stackrel{F}{\to} C ) </annotation></semantics></math></div> <p>between the coend, as indicated, and the <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a> over the <a class="existingWikiWord" href="/nlab/show/opposite+category">opposite</a> of 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>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math>.</p> <p></p> </div> </p> <p>This is equation (3.34) in (<a href="#Kelly">Kelly</a>) in view of (3.70).</p> <div class="num_cor" id="ConPres"> <h6 id="corollary">Corollary</h6> <p>Any <a class="existingWikiWord" href="/nlab/show/continuous+functor">continuous functor</a> preserves ends, and any cocontinuous functor preserves coends. In particular, for functors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><msup><mi>D</mi> <mi>op</mi></msup><mo>×</mo><mi>D</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">F: D^{op} \times D \to C</annotation></semantics></math> and <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>, we have the isomorphisms</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><mi>C</mi><mo stretchy="false">(</mo><msup><mo>∫</mo> <mi>x</mi></msup><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>≅</mo><msub><mo>∫</mo> <mi>x</mi></msub><mi>C</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mi>C</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><msub><mo>∫</mo> <mi>x</mi></msub><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mo>≅</mo><msub><mo>∫</mo> <mi>x</mi></msub><mi>C</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>.</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} C(\int^x F(x, x), c) &\cong \int_x C(F(x, x), c)\\ C(c, \int_x F(x, x)) &\cong \int_x C(c, F(x, x)). \end{aligned} </annotation></semantics></math></div></div> <div class="num_example"> <h6 id="example">Example</h6> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi><mo>=</mo><mi>D</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">W = D(-,e)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/representable+functor">representable functor</a>, then</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>el</mi><mi>W</mi><msup><mo stretchy="false">)</mo> <mi>op</mi></msup><mo>=</mo><mi>D</mi><mo stretchy="false">/</mo><mi>e</mi></mrow><annotation encoding="application/x-tex"> (el W)^{op} = D/e </annotation></semantics></math></div> <p>is the <a class="existingWikiWord" href="/nlab/show/over+category">over category</a> over the representing object <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>. This has a <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a>, namely <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>e</mi><mover><mo>→</mo><mi>Id</mi></mover><mi>e</mi></mrow><annotation encoding="application/x-tex">(e \stackrel{Id}{\to} e</annotation></semantics></math>). Therefore</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mi>lim</mi> <mo>→</mo></munder><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">/</mo><mi>e</mi><mo>→</mo><mi>D</mi><mover><mo>→</mo><mi>F</mi></mover><mi>C</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>F</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \lim_\to( D/e \to D \stackrel{F}{\to} C) \simeq F(e) \,. </annotation></semantics></math></div> <p>Since this is natural 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>, the above proposition asserts a <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</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 stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>≃</mo><msup><mo>∫</mo> <mrow><mi>k</mi><mo>∈</mo><mi>D</mi></mrow></msup><mi>D</mi><mo stretchy="false">(</mo><mi>k</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>⋅</mo><mi>F</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> F(-) \simeq \int^{k \in D} D(k,-) \cdot F(k) \,. </annotation></semantics></math></div> <p>This statement is sometimes called the <a class="existingWikiWord" href="/nlab/show/co-Yoneda+lemma">co-Yoneda lemma</a>.</p> </div> <h3 id="commutativity_of_ends_and_coends">Commutativity of ends and coends</h3> <p>Ordinary <a class="existingWikiWord" href="/nlab/show/limit">limit</a>s commute with each other, if both limits exist separately. The analogous statement does hold for ends and coends. Since it looks like the commutativity of two integrals, this fact is called the <em>Fubini theorem</em> for ends (for instance <a href="http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf#page=35">Kelly, p. 29</a>).</p> <p> <div class='num_prop' id='Fubini'> <h6>Proposition</h6> <p><strong>(Fubini theorem for ends)</strong> <br /> Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒱</mi></mrow><annotation encoding="application/x-tex">\mathcal{V}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a>. Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math> and <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">\mathcal{B}</annotation></semantics></math> be small <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒱</mi></mrow><annotation encoding="application/x-tex">\mathcal{V}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/enriched+categories">enriched categories</a>.</p> <p>Let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo lspace="verythinmathspace">:</mo><mo stretchy="false">(</mo><mi>𝒜</mi><mo>⊗</mo><mi>ℬ</mi><msup><mo stretchy="false">)</mo> <mi>op</mi></msup><mo>⊗</mo><mo stretchy="false">(</mo><mi>𝒜</mi><mo>⊗</mo><mi>ℬ</mi><mo stretchy="false">)</mo><mo>→</mo><mi>𝒱</mi></mrow><annotation encoding="application/x-tex"> T \colon (\mathcal{A} \otimes \mathcal{B})^{op} \otimes (\mathcal{A} \otimes \mathcal{B}) \to \mathcal{V} </annotation></semantics></math></div> <p>be a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒱</mi></mrow><annotation encoding="application/x-tex">\mathcal{V}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/enriched+functor">enriched functor</a>. Then:</p> <p>If for all object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>,</mo><mi>B</mi><mo>′</mo><mo>∈</mo><mi>ℬ</mi></mrow><annotation encoding="application/x-tex">B,B' \in \mathcal{B}</annotation></semantics></math> the end <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∫</mo> <mrow><mi>A</mi><mo>∈</mo><mi>𝒜</mi></mrow></msub><mi>T</mi><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>,</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\int_{A \in \mathcal{A}} T(A,B,A,B')</annotation></semantics></math> exists, then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mo>∫</mo> <mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>𝒜</mi><mo>⊗</mo><mi>ℬ</mi></mrow></msub><mi>T</mi><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>,</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mo>∫</mo> <mrow><mi>A</mi><mo>∈</mo><mi>𝒜</mi></mrow></msub><msub><mo>∫</mo> <mrow><mi>B</mi><mo>∈</mo><mi>ℬ</mi></mrow></msub><mi>T</mi><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>,</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \int_{(A,B) \in \mathcal{A} \otimes \mathcal{B}} T(A,B,A,B) \simeq \int_{A \in \mathcal{A}} \int_{B \in \mathcal{B}} T(A,B,A,B) </annotation></semantics></math></div> <p>if either side exists. In particular, since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi><mo>⊗</mo><mi>ℬ</mi><mo>≃</mo><mi>ℬ</mi><mo>⊗</mo><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A} \otimes \mathcal{B} \simeq \mathcal{B} \otimes \mathcal{A}</annotation></semantics></math>, this implies that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mo>∫</mo> <mrow><mi>B</mi><mo>∈</mo><mi>ℬ</mi></mrow></msub><msub><mo>∫</mo> <mrow><mi>A</mi><mo>∈</mo><mi>𝒜</mi></mrow></msub><mi>T</mi><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>,</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mo>∫</mo> <mrow><mi>A</mi><mo>∈</mo><mi>𝒜</mi></mrow></msub><msub><mo>∫</mo> <mrow><mi>B</mi><mo>∈</mo><mi>ℬ</mi></mrow></msub><mi>T</mi><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>,</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo><mo>,</mo></mrow><annotation encoding="application/x-tex"> \int_{B \in \mathcal{B}} \int_{A \in \mathcal{A}} T(A,B,A,B) \simeq \int_{A \in \mathcal{A}} \int_{B \in \mathcal{B}} T(A,B,A,B), </annotation></semantics></math></div> <p>if either side exists.</p> </div> </p> <h2 id="examples">Examples</h2> <h3 id="natural_transformations">Natural transformations</h3> <div class="num_prop" id="NatTrans"> <h6 id="proposition">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>,</mo><mi>G</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">F, G: C \to D</annotation></semantics></math> be <a class="existingWikiWord" href="/nlab/show/functors">functors</a> between two categories, and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>C</mi><mo>,</mo><mi>D</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mi>F</mi><mo>,</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">[C, D] (F, G)</annotation></semantics></math> be the set of <a class="existingWikiWord" href="/nlab/show/natural+transformations">natural transformations</a> between them. Then, we have</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>C</mi><mo>,</mo><mi>D</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mi>F</mi><mo>,</mo><mi>G</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mo>∫</mo> <mrow><mi>c</mi><mo>∈</mo><mi>C</mi></mrow></msub><mi>D</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>,</mo><mi>G</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex"> [C, D] (F, G) = \int_{c \in C} D(F(c), G(c)). </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof">Proof</h6> <p>An element of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∫</mo> <mrow><mi>c</mi><mo>∈</mo><mi>C</mi></mrow></msub><mi>D</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>,</mo><mi>G</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\int_{c \in C} D(F(c), G(c))</annotation></semantics></math> is, by definition, a collection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mi>c</mi></msub><mo>:</mo><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>→</mo><mi>G</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\tau_c: F(c) \to G(c)</annotation></semantics></math> of morphisms in <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> such that, for any morphism <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>→</mo><mi>d</mi></mrow><annotation encoding="application/x-tex">f: c \to d</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>, the following square commutes:</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>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>F</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mo></mo><mpadded width="0" lspace="-100%width"><mrow><msub><mi>τ</mi> <mi>c</mi></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>τ</mi> <mi>d</mi></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>G</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mtd> <mtd><munder><mo>→</mo><mrow><mi>G</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></munder></mtd> <mtd><mi>G</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ F(c) & \overset{F(f)}{\to} & F(d)\\ ^\mathllap{\tau_c}\downarrow & & \downarrow^\mathrlap{\tau_d}\\ G(c) & \underset{G(f)}{\to} & G(d) } </annotation></semantics></math></div> <p>which is, by definition, a natural transformation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>→</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">F \to G</annotation></semantics></math>.</p> </div> <h3 id="enriched_functor_categories">Enriched functor categories</h3> <p>In light of Proposition <a class="maruku-ref" href="#NatTrans"></a>, we can define the natural transformations object for <a class="existingWikiWord" href="/nlab/show/enriched+functors">enriched functors</a> as an end:</p> <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> and <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> both <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/enriched+category">enriched categories</a>, the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/enriched+functor+category">enriched functor category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>C</mi><mo>,</mo><mi>D</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[C,D]</annotation></semantics></math> is the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/enriched+category">enriched category</a> whose</p> <ul> <li> <p>objects are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/enriched+functors">enriched functors</a> <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>→</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">F : C \to D</annotation></semantics></math>;</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hom-objects">hom-objects</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> are given by the end-formula <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>C</mi><mo>,</mo><mi>D</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mi>F</mi><mo>,</mo><mi>G</mi><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><msub><mo>∫</mo> <mrow><mi>c</mi><mo>∈</mo><mi>C</mi></mrow></msub><mi>D</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>,</mo><mi>G</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">[C,D](F,G) := \int_{c \in C} D(F(c), G(c))</annotation></semantics></math>.</p> </li> </ul> <div class="num_example"> <h6 id="example_2">Example</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>=</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">V = Set</annotation></semantics></math> this reproduces of course the ordinary <a class="existingWikiWord" href="/nlab/show/functor+category">functor category</a>.</p> </div> <div class="num_example"> <h6 id="example_3">Example</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>=</mo><msub><mi>ℝ</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msub><mo>∪</mo><mo stretchy="false">{</mo><mn>∞</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">V = \mathbb{R}_{\geq 0}\cup \{\infty\}</annotation></semantics></math> with the monoidal product given by addition, a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-enriched category <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 <a class="existingWikiWord" href="/nlab/show/metric+space">metric space</a>, with the distance between points <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x, y \in X</annotation></semantics></math> given by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X(x, y)</annotation></semantics></math>. Given two metric spaces <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>,</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X, Y</annotation></semantics></math> and maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f, g: X \to Y</annotation></semantics></math>, the distance between the maps is</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>,</mo><mi>Y</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mo>∫</mo> <mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow></msub><mi>Y</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><munder><mi>sup</mi> <mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow></munder><mi>Y</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex"> [X, Y](f, g) = \int_{x \in X} Y(f(x), g(x)) = \sup_{x \in X} Y(f(x), g(x)). </annotation></semantics></math></div></div> <h3 id="kan_extension">Kan extension</h3> <p>If the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/enriched+category">enriched category</a> <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> is <a class="existingWikiWord" href="/nlab/show/copower">tensor</a>ed over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>, then the (left) <a class="existingWikiWord" href="/nlab/show/Kan+extension">Kan extension</a> of a <a class="existingWikiWord" href="/nlab/show/functor">functor</a> <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>→</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">F : C \to D</annotation></semantics></math> along a functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">p : C \to B</annotation></semantics></math> is given by the coend</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Lan</mi><mi>F</mi><mo>:</mo><mi>b</mi><mo>↦</mo><msup><mo>∫</mo> <mrow><mi>c</mi><mo>∈</mo><mi>C</mi></mrow></msup><mi>hom</mi><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Lan F : b \mapsto \int^{c \in C} hom(p(c),b) \cdot F(c) \,. </annotation></semantics></math></div> <p>See <a class="existingWikiWord" href="/nlab/show/Kan+extension">Kan extension</a> for more details.</p> <h3 id="geometric_realization">Geometric realization</h3> <p>A special case of the example of Kan extension is that of <a class="existingWikiWord" href="/nlab/show/geometric+realization">geometric realization</a>.</p> <p>Very generally, geometric realization is the left Kan extension of a functor <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>→</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">F : C \to D</annotation></semantics></math> along the <a class="existingWikiWord" href="/nlab/show/Yoneda+embedding">Yoneda embedding</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>:</mo><mi>C</mi><mo>→</mo><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>V</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">Y : C \to [C^{op},V]</annotation></semantics></math>.</p> <p>The “geometric realization” of an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>V</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">X \in [C^{op},V]</annotation></semantics></math> with respect to <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 then the coend</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 stretchy="false">|</mo><mo>:</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><mi>hom</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>,</mo><mi>X</mi><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><mi>X</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> |X| := \int^{c \in C} F(c) \cdot hom(Y(c),X) \simeq \int^{c \in C} F(c) \cdot X(c) \,, </annotation></semantics></math></div> <p>where the last step on the right uses the <a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a>.</p> <p>More specifically, this is traditionally thought of as applying to the case where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>=</mo><mi>Δ</mi></mrow><annotation encoding="application/x-tex">C = \Delta</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/simplex+category">simplex category</a> and where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mi>Δ</mi><mo>→</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex">F : \Delta \to Top</annotation></semantics></math> regards the abstract <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>-<a class="existingWikiWord" href="/nlab/show/simplex">simplex</a> as the standard simplex as a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>.</p> <h3 id="tensor_product_of_functors">Tensor product of functors</h3> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>:</mo><msup><mi>C</mi> <mo lspace="0em" rspace="thinmathspace">op</mo></msup><mo>→</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">S : C^\op \to D</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">T : C \to D</annotation></semantics></math> are functors, their <a class="existingWikiWord" href="/nlab/show/tensor+product+of+functors">tensor product</a> is the coend</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>S</mi><msub><mo>⊗</mo> <mi>C</mi></msub><mi>T</mi><mo>=</mo><msup><mo>∫</mo> <mi>c</mi></msup><mi>S</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>T</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>,</mo></mrow><annotation encoding="application/x-tex"> S \otimes_C T = \int^c S(c) \otimes T(c), </annotation></semantics></math></div> <p>where the tensor product on the right hand side refers to some <a class="existingWikiWord" href="/nlab/show/monoidal+structure">monoidal structure</a> 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>.</p> <h2 id="coend_calculus">(Co)end calculus</h2> <p>The formal properties of (co)ends in Propositions <a class="maruku-ref" href="#ConPres"></a>, <a class="maruku-ref" href="#Fubini"></a> and <a class="maruku-ref" href="#NatTrans"></a> allow us to prove certain results by <a class="existingWikiWord" href="/nlab/show/abstract+nonsense">abstract nonsense</a>.</p> <div class="num_example"> <h6 id="example_4">Example</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><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> be a functor. We prove the <a class="existingWikiWord" href="/nlab/show/co-Yoneda+lemma">co-Yoneda lemma</a>, that</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><mi>c</mi><mo stretchy="false">)</mo><mo>≃</mo><msup><mo>∫</mo> <mrow><mi>c</mi><mo>′</mo><mo>∈</mo><mi>C</mi></mrow></msup><mi>C</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>c</mi><mo>′</mo><mo stretchy="false">)</mo><mo>×</mo><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> F(c) \simeq \int^{c' \in C} C(c,c')\times F(c') </annotation></semantics></math></div> <p>We perform the following manipulations, where each isomorphism is natural:</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><mi>Set</mi><mo stretchy="false">(</mo><msup><mo>∫</mo> <mrow><mi>c</mi><mo>′</mo><mo>∈</mo><mi>C</mi></mrow></msup><mi>C</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>c</mi><mo>′</mo><mo stretchy="false">)</mo><mo>×</mo><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo>′</mo><mo stretchy="false">)</mo><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>≃</mo><msub><mo>∫</mo> <mrow><mi>c</mi><mo>′</mo><mo>∈</mo><mi>C</mi></mrow></msub><mi>Set</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>c</mi><mo>′</mo><mo stretchy="false">)</mo><mo>×</mo><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo>′</mo><mo stretchy="false">)</mo><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><msub><mo>∫</mo> <mrow><mi>c</mi><mo>′</mo><mo>∈</mo><mi>C</mi></mrow></msub><mi>Set</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>c</mi><mo>′</mo><mo stretchy="false">)</mo><mo>,</mo><mi>Set</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo>′</mo><mo stretchy="false">)</mo><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mo stretchy="false">[</mo><mi>C</mi><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>,</mo><mi>Set</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mi>Set</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>.</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} Set (\int^{c' \in C} C(c,c')\times F(c'), y) &\simeq \int_{c' \in C} Set (C(c,c')\times F(c'), y)\\ &\simeq \int_{c' \in C} Set (C(c, c'), Set(F(c'), y))\\ &\simeq [C, Set] (C(c, -), Set(F(-), y))\\ &\simeq Set(F(c), y). \end{aligned} </annotation></semantics></math></div> <p>So by the <a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a>, we have</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><mi>c</mi><mo stretchy="false">)</mo><mo>≃</mo><msup><mo>∫</mo> <mrow><mi>c</mi><mo>′</mo><mo>∈</mo><mi>C</mi></mrow></msup><mi>C</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>c</mi><mo>′</mo><mo stretchy="false">)</mo><mo>×</mo><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo>′</mo><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex"> F(c) \simeq \int^{c' \in C} C(c,c')\times F(c'). </annotation></semantics></math></div></div> <p>For more examples see e.g. <a href="#Loregian21">Loregian (2021)</a>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/lax+biend">lax biend</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-end">(∞,1)-end</a></p> </li> </ul> <div> <table><thead><tr><th></th><th><a class="existingWikiWord" href="/nlab/show/homotopy+%28as+an+operation%29">homotopy</a></th><th><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></th><th><a class="existingWikiWord" href="/nlab/show/homology">homology</a></th></tr></thead><tbody><tr><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 stretchy="false">[</mo><msup><mi>S</mi> <mi>n</mi></msup><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[S^n,-]</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>A</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[-,A]</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>⊗</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">(-) \otimes A</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/covariant+functor">covariant</a> <a class="existingWikiWord" href="/nlab/show/hom+functor">hom</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/contravariant+functor">contravariant</a> <a class="existingWikiWord" href="/nlab/show/hom+functor">hom</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Ext">Ext</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Ext">Ext</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Tor">Tor</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/end">end</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/end">end</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/coend">coend</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/derived+hom+space">derived hom space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi><mi>Hom</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mi>n</mi></msup><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{R}Hom(S^n,-)</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cocycles">cocycles</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi><mi>Hom</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{R}Hom(-,A)</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/derived+tensor+product">derived tensor product</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><msup><mo>⊗</mo> <mi>𝕃</mi></msup><mi>A</mi></mrow><annotation encoding="application/x-tex">(-) \otimes^{\mathbb{L}} A</annotation></semantics></math></td></tr> </tbody></table> </div> <h2 id="references">References</h2> <p>The notion of (co)ends as introduced in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Nobuo+Yoneda">Nobuo Yoneda</a>, §4 of: <em>On ext and exact sequences</em> (PhD thesis), Journal of the Faculty of Science, Section I. <strong>8</strong> University of Tokyo (1960) 507–576 [<a href="http://alpha.math.uga.edu/~lorenz/YonedaExtExactSequences.pdf">pdf</a>, <a href="https://ci.nii.ac.jp/naid/500000325773">CiNii:naid/500000325773</a>]</li> </ul> <p>An early account with an eye towards application in <a class="existingWikiWord" href="/nlab/show/geometric+realization+of+simplicial+topological+spaces">geometric realization of simplicial topological spaces</a>:</p> <ul> <li id="MacLane70"><a class="existingWikiWord" href="/nlab/show/Saunders+MacLane">Saunders MacLane</a>, Section 2 of: <em>The Milgram bar construction as a tensor product of functors</em>, In: F.P. Peterson (eds.) <em>The Steenrod Algebra and Its Applications: A Conference to Celebrate N.E. Steenrod’s Sixtieth Birthday</em>, Lecture Notes in Mathematics <em>168</em>, Springer 1970 (<a href="https://doi.org/10.1007/BFb0058523">doi:10.1007/BFb0058523</a>, <a href="https://link.springer.com/content/pdf/10.1007/BFb0058523.pdf">pdf</a>)</li> </ul> <p>Textbook accounts:</p> <ul> <li id="Kelly82"> <p><a class="existingWikiWord" href="/nlab/show/Max+Kelly">Max Kelly</a>, <em>Basic concepts of enriched category theory</em>, London Math. Soc. Lec. Note Series <strong>64</strong>, Cambridge Univ. Press (1982), Reprints in Theory and Applications of Categories <strong>10</strong> (2005) 1-136 [<a href="https://www.cambridge.org/de/academic/subjects/mathematics/logic-categories-and-sets/basic-concepts-enriched-category-theory?format=PB&isbn=9780521287029">ISBN:9780521287029</a>, <a href="http://www.tac.mta.ca/tac/reprints/articles/10/tr10abs.html">tac:tr10</a>, <a href="http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf">pdf</a>]</p> <ul> <li> <p>ends of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-valued bifunctors are discussed in section 2.1</p> </li> <li> <p>the enriched functor category that they give rise to is discussed in section 2.2;</p> </li> <li> <p>enriched <a class="existingWikiWord" href="/nlab/show/weighted+limits">weighted limits</a> in terms of enriched functor categories are in section 3.1</p> </li> <li> <p>the end of general <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-enriched functors in terms of weighted limits is in section 3.10 .</p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Francis+Borceux">Francis Borceux</a>, Def. 6.6.8 in: <em><a class="existingWikiWord" href="/nlab/show/Handbook+of+Categorical+Algebra">Handbook of Categorical Algebra</a></em>, Vol. 2: <em>Categories and Structures</em>, Encyclopedia of Mathematics and its Applications <strong>50</strong> Cambridge University Press (1994) (<a href="https://doi.org/10.1017/CBO9780511525865">doi:10.1017/CBO9780511525865</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Emily+Riehl">Emily Riehl</a>, §4.1 in: <em><a class="existingWikiWord" href="/nlab/show/Categorical+Homotopy+Theory">Categorical Homotopy Theory</a></em>, Cambridge University Press (2014) [<a href="https://doi.org/10.1017/CBO9781107261457">doi:10.1017/CBO9781107261457</a>, <a href="http://www.math.jhu.edu/~eriehl/cathtpy.pdf">pdf</a>]</p> </li> <li id="Loregian21"> <p><a class="existingWikiWord" href="/nlab/show/Fosco+Loregian">Fosco Loregian</a>, <em>Coend calculus</em>, Cambridge University Press (2021) [<a href="http://arxiv.org/abs/1501.02503">arXiv:1501.02503</a>, <a href="https://doi.org/10.1017/9781108778657">doi:10.1017/9781108778657</a>, ISBN:9781108778657)]</p> </li> </ul> <p>See also:</p> <ul> <li><a href="http://golem.ph.utexas.edu/category/2014/01/ends.html">Ends</a>, <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>-Category Café discussion.</li> </ul> <p>Application in <a class="existingWikiWord" href="/nlab/show/conformal+field+theory">conformal field theory</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/J%C3%BCrgen+Fuchs">Jürgen Fuchs</a>, <a class="existingWikiWord" href="/nlab/show/Christoph+Schweigert">Christoph Schweigert</a>, <em>Coends in conformal field theory</em> (<a href="https://arxiv.org/abs/1604.01670">arXiv:1604.01670</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on February 5, 2025 at 01:11:23. 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