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style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="category_theory">Category theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></strong></p> <h2 id="sidebar_concepts">Concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category">category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cat">Cat</a></p> </li> </ul> <h2 id="sidebar_universal_constructions">Universal constructions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+construction">universal construction</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/representable+functor">representable functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+functor">adjoint functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/limit">limit</a>/<a class="existingWikiWord" href="/nlab/show/colimit">colimit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/weighted+limit">weighted limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/end">end</a>/<a class="existingWikiWord" href="/nlab/show/coend">coend</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kan+extension">Kan extension</a></p> </li> </ul> </li> </ul> <h2 id="sidebar_theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+construction">Grothendieck construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+functor+theorem">adjoint functor theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monadicity+theorem">monadicity theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+lifting+theorem">adjoint lifting theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gabriel-Ulmer+duality">Gabriel-Ulmer duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freyd-Mitchell+embedding+theorem">Freyd-Mitchell embedding theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relation+between+type+theory+and+category+theory">relation between type theory and category theory</a></p> </li> </ul> <h2 id="sidebar_extensions">Extensions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf+and+topos+theory">sheaf and topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a></p> </li> </ul> <h2 id="sidebar_applications">Applications</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/applications+of+%28higher%29+category+theory">applications of (higher) category theory</a></li> </ul> <div> <p> <a href="/nlab/edit/category+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="2category_theory">2-Category theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/2-category+theory">2-category theory</a></strong></p> <p><strong>Definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-category">2-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/strict+2-category">strict 2-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bicategory">bicategory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+bicategory">enriched bicategory</a></p> </li> </ul> <p><strong>Transfors between 2-categories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-functor">2-functor</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/pseudofunctor">pseudofunctor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/lax+functor">lax functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivalence+of+2-categories">equivalence of 2-categories</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-natural+transformation">2-natural transformation</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/lax+natural+transformation">lax natural transformation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/icon">icon</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/modification">modification</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yoneda+lemma+for+bicategories">Yoneda lemma for bicategories</a></p> </li> </ul> <p><strong>Morphisms in 2-categories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fully+faithful+morphism">fully faithful morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/faithful+morphism">faithful morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conservative+morphism">conservative morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pseudomonic+morphism">pseudomonic morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/discrete+morphism">discrete morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/eso+morphism">eso morphism</a></p> </li> </ul> <p><strong>Structures in 2-categories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mate">mate</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monad">monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cartesian+object">cartesian object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fibration+in+a+2-category">fibration in a 2-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/codiscrete+cofibration">codiscrete cofibration</a></p> </li> </ul> <p><strong>Limits in 2-categories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-limit">2-limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-pullback">2-pullback</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/comma+object">comma object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/inserter">inserter</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/inverter">inverter</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equifier">equifier</a></p> </li> </ul> <p><strong>Structures on 2-categories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-monad">2-monad</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/lax-idempotent+2-monad">lax-idempotent 2-monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pseudomonad">pseudomonad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pseudoalgebra+for+a+2-monad">pseudoalgebra for a 2-monad</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+2-category">monoidal 2-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/cartesian+bicategory">cartesian bicategory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gray+tensor+product">Gray tensor product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/proarrow+equipment">proarrow equipment</a></p> </li> </ul> </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='#Definitions'>Definitions</a></li> <ul> <li><a href='#OrdinaryKanExtensions'>Ordinary or weak Kan extensions</a></li> <ul> <li><a href='#GlobalKanExtensions'>Global Kan extensions</a></li> <li><a href='#LocalKanExtensions'>Local Kan extension</a></li> </ul> <li><a href='#Preservation'>Preservation of Kan extensions</a></li> <li><a href='#Pointwise'>Pointwise or strong Kan extensions</a></li> <ul> <li><a href='#PointwiseByWeightedColimits'>In terms of weighted (co)limits</a></li> <li><a href='#PointwiseByCoEnds'>In terms of (co)ends</a></li> <li><a href='#PointwiseByConicalLimits'>In terms of conical (co)limits</a></li> <li><a href='#pointwiseVsWeak'>Comparing the definitions</a></li> </ul> <li><a href='#AbsoluteKanExtension'>Absolute Kan extensions</a></li> <li><a href='#of_functors'>Of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-functors</a></li> <li><a href='#In2cat'>In a general 2-category</a></li> </ul> <li><a href='#existence'>Existence</a></li> <li><a href='#Properties'>Properties</a></li> <ul> <li><a href='#LeftKanOnRepresentables'>Left Kan extension on representables / fully faithfulness</a></li> <li><a href='#LeftKanExtensionPreservingCertainLimits'>Left Kan extensions preserving certain limits</a></li> <li><a href='#AlongFibrations'>Kan extension along (op)fibration</a></li> </ul> <li><a href='#Examples'>Examples</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#nonpointwise_kan_extensions'>Non-pointwise Kan extensions</a></li> <li><a href='#restriction_and_extension_of_sheaves'>Restriction and extension of sheaves</a></li> <li><a href='#ExamplesKanExtensionsInPhysics'>Kan extension in physics</a></li> </ul> <li><a href='#remark_on_terminology_pushforward_vs_pullback'>Remark on terminology: pushforward vs. pullback</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>The <em>Kan extension</em> 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 lspace="verythinmathspace">:</mo><mi>C</mi><mo>→</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">F \colon C \to D</annotation></semantics></math> with respect to a <a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>C</mi></mtd></mtr> <mtr><mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo><mpadded width="0"><mrow><msup><mrow></mrow> <mi>p</mi></msup></mrow></mpadded></mtd></mtr> <mtr><mtd><mi>C</mi><mo>′</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ C \\ \big\downarrow\mathrlap{{}^p} \\ C' } </annotation></semantics></math></div> <p>is, if it exists, a kind of <em>best approximation</em> to the problem of finding a functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>′</mo><mo>→</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">C' \to D</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><mrow><mtable><mtr><mtd><mi>C</mi></mtd> <mtd><mover><mo>→</mo><mi>F</mi></mover></mtd> <mtd><mi>D</mi></mtd></mtr> <mtr><mtd><mpadded width="0" lspace="-100%width"><mrow><msup><mrow></mrow> <mi>p</mi></msup></mrow></mpadded><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd> <mtd><mo>↗</mo></mtd></mtr> <mtr><mtd><mi>C</mi><mo>′</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ C &\stackrel{F}{\to}& D \\ \mathllap{{}^p}\big\downarrow & \nearrow \\ C' } \,, </annotation></semantics></math></div> <p>hence to <a class="existingWikiWord" href="/nlab/show/extension">extending</a> the <a class="existingWikiWord" href="/nlab/show/domain">domain</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> through <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">C'</annotation></semantics></math>.</p> <p>More generally, this notion makes sense not only in <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a> but in any <a class="existingWikiWord" href="/nlab/show/2-category">2-category</a>.</p> <p>Similarly, a <em><a class="existingWikiWord" href="/nlab/show/Kan+lift">Kan lift</a></em> is the best approximation to lifting a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo lspace="verythinmathspace">:</mo><mi>C</mi><mo>→</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">F \colon C \to D</annotation></semantics></math> through a morphism</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>D</mi><mo>′</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>D</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ D' \\ \downarrow \\ D } </annotation></semantics></math></div> <p>to a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>F</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat 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><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>D</mi><mo>′</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mover><mi>F</mi><mo stretchy="false">^</mo></mover></msup><mo>↗</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>C</mi></mtd> <mtd><mover><mo>→</mo><mi>F</mi></mover></mtd> <mtd><mi>D</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && D' \\ & {}^{\hat F}\nearrow & \downarrow \\ C &\stackrel{F}{\to}& D } \,. </annotation></semantics></math></div> <p>Kan extensions are ubiquitous. See the discussion at <em><a href="#Examples">Examples</a></em> below.</p> <h2 id="Definitions">Definitions</h2> <p>There are various slight variants of the definition of <em>Kan extension</em>. In good cases they all exist and all coincide, but in some cases only some of these will actually exist.</p> <p>We (have to) distinguish the following cases:</p> <ol> <li> <p><a href="#OrdinaryKanExtensions">“ordinary” or “weak” Kan extensions</a></p> <p>These define the extension of an entire functor, by an <a class="existingWikiWord" href="/nlab/show/adjunct">adjointness</a> relation.</p> <p>Here we (have to) distinguish further between</p> <ol> <li> <p><a href="#GlobalKanExtensions">global Kan extensions</a>,</p> <p>which define extensions of <em>all</em> possible functors of given domain and codomain (if all of them indeed exist);</p> </li> <li> <p><a href="#LocalKanExtensions">local Kan extensions</a>,</p> <p>which define extensions of single functors only, which may exist even if not every functor has an extension.</p> </li> </ol> </li> <li> <p><a href="#Pointwise">“pointwise” or “strong” Kan extensions</a></p> <p>These define the <em>value</em> of an extended functor on each object (each “point”) by a <a class="existingWikiWord" href="/nlab/show/weighted+limit">weighted (co)limit</a>.</p> <p>Furthermore, a pointwise Kan extension can be <a href="#AbsoluteKanExtension">“absolute”</a>.</p> </li> </ol> <p>If the pointwise version exists, then it coincides with the “ordinary” or “weak” version, but the former may exist without the pointwise version existing. See <a href="#pointwiseVsWeak">below</a> for more.</p> <p>Some authors (such as <a href="Kelly">Kelly</a>) assert that only pointwise Kan extensions deserve the name “Kan extension,” and use the term as “weak Kan extension” for a functor equipped with a universal natural transformation. It is certainly true that most Kan extensions which arise in practice are pointwise. This distinction is even more important in <a class="existingWikiWord" href="/nlab/show/enriched+category">enriched category</a> theory.</p> <h3 id="OrdinaryKanExtensions">Ordinary or weak Kan extensions</h3> <h4 id="GlobalKanExtensions">Global Kan extensions</h4> <p>Let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>C</mi><mo>′</mo></mrow><annotation encoding="application/x-tex"> p : C \to C' </annotation></semantics></math></div> <p>be a <a class="existingWikiWord" href="/nlab/show/functor">functor</a>. For <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> any other category, write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>p</mi> <mo>*</mo></msup><mo>:</mo><mo stretchy="false">[</mo><mi>C</mi><mo>′</mo><mo>,</mo><mi>D</mi><mo stretchy="false">]</mo><mo>→</mo><mo stretchy="false">[</mo><mi>C</mi><mo>,</mo><mi>D</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> p^* : [C',D] \to [C,D] </annotation></semantics></math></div> <p>for the induced functor on the <a class="existingWikiWord" href="/nlab/show/functor+categories">functor categories</a>: this sends a functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>:</mo><mi>C</mi><mo>′</mo><mo>→</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">h : C' \to D</annotation></semantics></math> to the composite functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>p</mi> <mo>*</mo></msup><mi>h</mi><mo>:</mo><mi>C</mi><mover><mo>→</mo><mi>p</mi></mover><mi>C</mi><mo>′</mo><mover><mo>→</mo><mi>h</mi></mover><mi>D</mi></mrow><annotation encoding="application/x-tex">p^* h : C \stackrel{p}{\to} C' \stackrel{h}{\to} D</annotation></semantics></math>.</p> <div class="num_defn"> <h6 id="definition">Definition</h6> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>p</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">p^*</annotation></semantics></math> has a <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a>, typically denoted</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mo>!</mo></msub><mo>:</mo><mo stretchy="false">[</mo><mi>C</mi><mo>,</mo><mi>D</mi><mo stretchy="false">]</mo><mo>→</mo><mo stretchy="false">[</mo><mi>C</mi><mo>′</mo><mo>,</mo><mi>D</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> p_! : [C,D] \to [C',D] </annotation></semantics></math></div> <p>or</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Lan</mi> <mi>p</mi></msub><mo>:</mo><mo stretchy="false">[</mo><mi>C</mi><mo>,</mo><mi>D</mi><mo stretchy="false">]</mo><mo>→</mo><mo stretchy="false">[</mo><mi>C</mi><mo>′</mo><mo>,</mo><mi>D</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> Lan_p : [C,D] \to [C',D] </annotation></semantics></math></div> <p>then this left adjoint is called the ( <em>ordinary</em> or <em>weak</em> ) <strong>left Kan extension</strong> operation along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>∈</mo><mo stretchy="false">[</mo><mi>C</mi><mo>,</mo><mi>D</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">h \in [C,D]</annotation></semantics></math> we call <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mo>!</mo></msub><mi>h</mi></mrow><annotation encoding="application/x-tex">p_! h</annotation></semantics></math> the <strong>left Kan extension of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math></strong> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>.</p> <p>Similarly, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>p</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">p^*</annotation></semantics></math> has a <a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjoint</a>, this right adjoint is called the <strong>right Kan extension</strong> operation along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>. It is typically denoted</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mo>*</mo></msub><mo>:</mo><mo stretchy="false">[</mo><mi>C</mi><mo>,</mo><mi>D</mi><mo stretchy="false">]</mo><mo>→</mo><mo stretchy="false">[</mo><mi>C</mi><mo>′</mo><mo>,</mo><mi>D</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> p_* : [C,D] \to [C',D] </annotation></semantics></math></div> <p>or</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ran</mi><mo>=</mo><msub><mi>Ran</mi> <mi>p</mi></msub><mo>:</mo><mo stretchy="false">[</mo><mi>C</mi><mo>,</mo><mi>D</mi><mo stretchy="false">]</mo><mo>→</mo><mo stretchy="false">[</mo><mi>C</mi><mo>′</mo><mo>,</mo><mi>D</mi><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Ran = Ran_p: [C,D] \to [C',D] \,. </annotation></semantics></math></div></div> <p>The analogous definition clearly makes sense as stated in other contexts, such as in <a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a>.</p> <p> <div class='num_prop'> <h6>Proposition</h6> <p></p> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>′</mo><mo>=</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">C' = *</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/terminal+category">terminal category</a>, then</p> <ul> <li> <p>the left Kan extension operation forms the <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a> of a functor;</p> </li> <li> <p>the right Kan extension operation forms the <a class="existingWikiWord" href="/nlab/show/limit">limit</a> of a functor.</p> </li> </ul> <p></p> </div> </p> <div class="proof"> <h6 id="proof">Proof</h6> <p>The functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>p</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">p^*</annotation></semantics></math> in this case sends objects <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> of <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> to the <a class="existingWikiWord" href="/nlab/show/constant+functor">constant functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>d</mi></msub></mrow><annotation encoding="application/x-tex">\Delta_d</annotation></semantics></math> 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>. Notice that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>∈</mo><mo stretchy="false">[</mo><mi>C</mi><mo>,</mo><mi>D</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">F \in [C,D]</annotation></semantics></math> any functor,</p> <ul> <li> <p>a <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>d</mi></msub><mo>→</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">\Delta_d \to F</annotation></semantics></math> is the same as a <a class="existingWikiWord" href="/nlab/show/cone">cone</a> 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>;</p> </li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>→</mo><msub><mi>Δ</mi> <mi>d</mi></msub></mrow><annotation encoding="application/x-tex">F \to \Delta_d</annotation></semantics></math> is the same as a <a class="existingWikiWord" href="/nlab/show/cocone">cocone</a> under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>.</p> </li> </ul> <p>Therefore the natural hom-isomorphisms of the <a class="existingWikiWord" href="/nlab/show/adjoint+functor">adjoint functor</a>s <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>p</mi> <mo>!</mo></msub><mo>⊣</mo><msup><mi>p</mi> <mo>*</mo></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(p_! \dashv p^*)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>p</mi> <mo>*</mo></msup><mo>⊣</mo><msub><mi>p</mi> <mo>*</mo></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(p^* \dashv p_*)</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo stretchy="false">(</mo><mi>d</mi><mo>,</mo><msub><mi>p</mi> <mo>*</mo></msub><mi>F</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>Func</mi><mo stretchy="false">(</mo><msub><mi>Δ</mi> <mi>d</mi></msub><mo>,</mo><mi>F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> D(d, p_* F) \simeq Func(\Delta_d, 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><mi>D</mi><mo stretchy="false">(</mo><msub><mi>p</mi> <mo>!</mo></msub><mi>F</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>Func</mi><mo stretchy="false">(</mo><mi>F</mi><mo>,</mo><msub><mi>Δ</mi> <mi>d</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> D(p_! F, d) \simeq Func(F, \Delta_d) </annotation></semantics></math></div> <p>assert that</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mo>*</mo></msub><mi>F</mi></mrow><annotation encoding="application/x-tex">p_* F</annotation></semantics></math> corepresents the <a class="existingWikiWord" href="/nlab/show/cone">cone</a>s 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>: this means by definition that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mo>*</mo></msub><mi>F</mi><mo>=</mo><msub><mi>lim</mi> <mo>←</mo></msub><mi>F</mi></mrow><annotation encoding="application/x-tex">p_* F = \lim_\leftarrow F</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/limit">limit</a> 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>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mo>!</mo></msub><mi>F</mi></mrow><annotation encoding="application/x-tex">p_! F</annotation></semantics></math> represents the <a class="existingWikiWord" href="/nlab/show/cocone">cocone</a>s under <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 by definition that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mo>!</mo></msub><mi>F</mi><mo>=</mo><msub><mi>lim</mi> <mo>→</mo></msub><mi>F</mi></mrow><annotation encoding="application/x-tex">p_! F = \lim_\to F</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/colimit">colimit</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> </li> </ul> </div> <h4 id="LocalKanExtensions">Local Kan extension</h4> <p>There is also a <em>local</em> definition of “the Kan extension of a given functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>” which can exist even if the entire functor defined above does not. This is a generalization of the fact that a <em>particular</em> diagram of shape <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> can have a limit even if not every such diagram does. It is also a special case of the fact discussed at <a class="existingWikiWord" href="/nlab/show/adjoint+functor">adjoint functor</a> that an adjoint functor can fail to exist completely, but may still be partially defined. If the local Kan extension of every single functor exists for some given <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo lspace="verythinmathspace">:</mo><mi>C</mi><mo>→</mo><mi>C</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">p\colon C\to 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>, then these local Kan extensions fit together to define a functor which is the global Kan extension.</p> <p>Thus, by the general notion of “partial adjoints”; we say</p> <div class="num_defn" id="LocalKanExtension"> <h6 id="definition_2">Definition</h6> <p>The local <strong>left Kan extension</strong> of a functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>∈</mo><mo stretchy="false">[</mo><mi>C</mi><mo>,</mo><mi>D</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">F\in [C,D]</annotation></semantics></math> along <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>C</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">p : C \to C'</annotation></semantics></math> is, if it exists, a functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Lan</mi> <mi>p</mi></msub><mspace width="thinmathspace"></mspace><mi>F</mi><mo>:</mo><mi>C</mi><mo>′</mo><mo>→</mo><mi>D</mi></mrow><annotation encoding="application/x-tex"> Lan_p\,F : C'\to D </annotation></semantics></math></div> <p>equipped with 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><msub><mi>Hom</mi> <mrow><mo stretchy="false">[</mo><mi>C</mi><mo>,</mo><mi>D</mi><mo stretchy="false">]</mo></mrow></msub><mo stretchy="false">(</mo><mi>F</mi><mo>,</mo><msup><mi>p</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≅</mo><msub><mi>Hom</mi> <mrow><mo stretchy="false">[</mo><mi>C</mi><mo>′</mo><mo>,</mo><mi>D</mi><mo stretchy="false">]</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>Lan</mi> <mi>p</mi></msub><mspace width="thinmathspace"></mspace><mi>F</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> Hom_{[C,D]}(F,p^*(-))\cong Hom_{[C',D]}(Lan_p\,F,-) \,, </annotation></semantics></math></div> <p>hence a <a class="existingWikiWord" href="/nlab/show/representable+functor">(co)representation</a> of the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mrow><mo stretchy="false">[</mo><mi>C</mi><mo>,</mo><mi>D</mi><mo stretchy="false">]</mo></mrow></msub><mo stretchy="false">(</mo><mi>F</mi><mo>,</mo><msup><mi>p</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Hom_{[C,D]}(F,p^*(-))</annotation></semantics></math>.</p> <p>The local definition of right Kan extensions along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> is dual.</p> </div> <p>As for adjoints and limits, by the usual logic of representable functors this can equivalently be rephrased in terms of <a href="http://ncatlab.org/nlab/show/adjoint%20functor#UniversalArrows">universal morphisms</a>:</p> <div class="num_defn"> <h6 id="definition_3">Definition</h6> <p>The <strong>left Kan extension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Lan</mi><mi>F</mi><mo>=</mo><msub><mi>Lan</mi> <mi>p</mi></msub><mi>F</mi></mrow><annotation encoding="application/x-tex">Lan F = Lan_p F</annotation></semantics></math> of <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 <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>C</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">p :C\to C'</annotation></semantics></math> is a functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Lan</mi><mi>F</mi><mo>:</mo><mi>C</mi><mo>′</mo><mo>→</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">Lan F : C' \to D</annotation></semantics></math> equipped with a <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mi>F</mi></msub><mo>:</mo><mi>F</mi><mo>⇒</mo><msup><mi>p</mi> <mo>*</mo></msup><mi>Lan</mi><mi>F</mi></mrow><annotation encoding="application/x-tex">\eta_F : F \Rightarrow p^* Lan F</annotation></semantics></math>.</strong></p> <center><img src="/nlab/files/kan-0.png" alt="" /></center> <p>with the property that every other natural transformation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>⇒</mo><msup><mi>p</mi> <mo>*</mo></msup><mi>G</mi></mrow><annotation encoding="application/x-tex">F \Rightarrow p^* G</annotation></semantics></math> factors uniquely through <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mi>F</mi></msub></mrow><annotation encoding="application/x-tex">\eta_F</annotation></semantics></math> as</p> <center><img src="/nlab/files/kan-1.png" alt="" /></center> <p>Similarly for the right Kan extension, with the direction of the natural transformations reversed:</p> <center><img src="/nlab/files/kan-2.png" alt="" /></center></div> <p>By the usual reasoning (see e.g. <a class="existingWikiWord" href="/nlab/show/Categories+Work">Categories Work</a>, chapter IV, theorem 2), if these representations exist for every <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> then they can be organised into a left (right) adjoint <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Lan</mi> <mi>p</mi></msub></mrow><annotation encoding="application/x-tex">Lan_p</annotation></semantics></math> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Ran</mi> <mi>p</mi></msub></mrow><annotation encoding="application/x-tex">Ran_p</annotation></semantics></math>) to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>p</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">p^*</annotation></semantics></math>.</p> <div class="num_remark" id="InHigherCats"> <h6 id="remark">Remark</h6> <p>The definition in this form makes sense not just in <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a> but in every <a class="existingWikiWord" href="/nlab/show/2-category">2-category</a>. In slightly different terminology, the left Kan extension of a 1-cell <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 1-cell <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>∈</mo><mi>K</mi><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mi>C</mi><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p\in K(C,C')</annotation></semantics></math> in a 2-category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> is a pair <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>Lan</mi> <mi>p</mi></msub><mi>F</mi><mo>,</mo><mi>α</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(Lan_p F,\alpha)</annotation></semantics></math> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>:</mo><mi>F</mi><mo>→</mo><msub><mi>Lan</mi> <mi>p</mi></msub><mi>F</mi><mo>∘</mo><mi>p</mi></mrow><annotation encoding="application/x-tex">\alpha : F\to Lan_p F\circ p</annotation></semantics></math> is a 2-cell which <a class="existingWikiWord" href="/nlab/show/reflections">reflects</a> the object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>∈</mo><mi>K</mi><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mi>D</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F\in K(C,D)</annotation></semantics></math> along the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>p</mi> <mo>*</mo></msup><mo>=</mo><mi>K</mi><mo stretchy="false">(</mo><mi>p</mi><mo>,</mo><mi>D</mi><mo stretchy="false">)</mo><mo>:</mo><mi>K</mi><mo stretchy="false">(</mo><mi>C</mi><mo>′</mo><mo>,</mo><mi>D</mi><mo stretchy="false">)</mo><mo>→</mo><mi>K</mi><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mi>D</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p^* = K(p,D):K(C',D)\to K(C,D)</annotation></semantics></math>. Equivalently, it is such a pair such that for every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo lspace="verythinmathspace">:</mo><mi>C</mi><mo>′</mo><mo>→</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">G\colon C' \to D</annotation></semantics></math>, the function</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>C</mi><mo>′</mo><mo>,</mo><mi>D</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>Lan</mi> <mi>p</mi></msub><mi>F</mi><mo>,</mo><mi>G</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>⋅</mo><mi>α</mi></mrow></mover><mi>K</mi><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>∘</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> K(C',D)(Lan_p F, G) \xrightarrow{- \cdot \alpha} K(C,D)(F, G \circ p) </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/bijection">bijection</a>.</p> <p>In this form, the definition generalizes easily to any <a class="existingWikiWord" href="/nlab/show/n-category">n-category</a> for any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">n\ge 2</annotation></semantics></math>. If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> is an <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, we say that the left Kan extension of a 1-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> along a 1-morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>∈</mo><mi>K</mi><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mi>C</mi><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p\in K(C,C')</annotation></semantics></math> is a pair <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>Lan</mi> <mi>p</mi></msub><mi>F</mi><mo>,</mo><mi>α</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(Lan_p F,\alpha)</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Lan</mi> <mi>p</mi></msub><mi>F</mi><mo lspace="verythinmathspace">:</mo><mi>C</mi><mo>′</mo><mo>→</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">Lan_p F \colon C' \to D</annotation></semantics></math> is a 1-morphism and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>:</mo><mi>F</mi><mo>→</mo><msub><mi>Lan</mi> <mi>p</mi></msub><mi>F</mi><mo>∘</mo><mi>p</mi></mrow><annotation encoding="application/x-tex">\alpha : F\to Lan_p F\circ p</annotation></semantics></math> is a 2-morphism, with the property that for any 1-morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo lspace="verythinmathspace">:</mo><mi>C</mi><mo>′</mo><mo>→</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">G\colon C'\to D</annotation></semantics></math>, the induced functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>C</mi><mo>′</mo><mo>,</mo><mi>D</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>Lan</mi> <mi>p</mi></msub><mi>F</mi><mo>,</mo><mi>G</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>⋅</mo><mi>α</mi></mrow></mover><mi>K</mi><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>∘</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> K(C',D)(Lan_p F, G) \xrightarrow{- \cdot \alpha} K(C,D)(F, G \circ p) </annotation></semantics></math></div> <p>is an <a class="existingWikiWord" href="/nlab/show/equivalence">equivalence</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n-2)</annotation></semantics></math>-categories.</p> </div> <h3 id="Preservation">Preservation of Kan extensions</h3> <p>We say that a Kan extension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Lan</mi> <mi>p</mi></msub><mi>F</mi></mrow><annotation encoding="application/x-tex">Lan_p F</annotation></semantics></math> is <em>preserved</em> by a functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> if the composite <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>∘</mo><msub><mi>Lan</mi> <mi>p</mi></msub><mi>F</mi></mrow><annotation encoding="application/x-tex">G \circ Lan_p F</annotation></semantics></math> is a Kan extension of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mi>F</mi></mrow><annotation encoding="application/x-tex">G F</annotation></semantics></math> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>, and moreover the universal natural transformation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mi>F</mi><mo>→</mo><mi>G</mi><mo stretchy="false">(</mo><msub><mi>Lan</mi> <mi>p</mi></msub><mi>F</mi><mo stretchy="false">)</mo><mi>p</mi></mrow><annotation encoding="application/x-tex">G F \to G(Lan_p F)p</annotation></semantics></math> is the composite of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> with the universal transformation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>→</mo><mo stretchy="false">(</mo><msub><mi>Lan</mi> <mi>p</mi></msub><mi>F</mi><mo stretchy="false">)</mo><mi>p</mi></mrow><annotation encoding="application/x-tex">F\to (Lan_p F)p</annotation></semantics></math>.</p> <h3 id="Pointwise">Pointwise or strong Kan extensions</h3> <p>If the <a class="existingWikiWord" href="/nlab/show/codomain">codomain</a> category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> admits certain <a class="existingWikiWord" href="/nlab/show/limit">(co)limits</a>, then left and right Kan extensions can be constructed, over each object (“point”) of the domain category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">C'</annotation></semantics></math> out of these: Kan extensions that admit this form are called <em>pointwise</em>. (Reviews include (<a href="#Riehl">Riehl, I 1.3</a>)).</p> <p>The notion of pointwise Kan extensions deserves to be discussed in the general context of <a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a>, which we do below. The reader may want to skip ahead to the section</p> <ul> <li><a href="#PointwiseByConicalLimits">pointwise Kan extensions by conical (co)limits</a></li> </ul> <p>which discusses the situation in ordinary (<a class="existingWikiWord" href="/nlab/show/Set">Set</a>-<a class="existingWikiWord" href="/nlab/show/enriched+category">enriched</a>) <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a> in terms of ordinary limits (“conical” limits, defined in terms of <a class="existingWikiWord" href="/nlab/show/cone">cone</a>s, to be distinguished from the more general <a class="existingWikiWord" href="/nlab/show/weighted+limit">weighted limit</a>s). While the formulas in that case are classical and fundamentally useful in practice, they do rely heavily on special properties of the enriching category <a class="existingWikiWord" href="/nlab/show/Set">Set</a>.</p> <p>The general formulation of pointwise Kan extensions in general enriched contexts is</p> <ul> <li><a href="#PointwiseByWeightedColimits">in terms of weighted (co)limits</a>.</li> </ul> <p>In the case that the codomain category is <a class="existingWikiWord" href="/nlab/show/copower">(co)powered</a> these may be expressed equivalently</p> <ul> <li><a href="#PointwiseByCoEnds">in terms of (co)ends</a>.</li> </ul> <p>First, here is a characterization that doesn’t rely on any computational framework:</p> <div class="num_defn"> <h6 id="definition_4">Definition</h6> <p>A Kan extension, def. <a class="maruku-ref" href="#LocalKanExtension"></a>, is called <strong>pointwise</strong> if and only if it is <a href="#Preservation">preserved</a> by all <a class="existingWikiWord" href="/nlab/show/representable+functor">representable functor</a>s.</p> </div> <p>(<a class="existingWikiWord" href="/nlab/show/Categories+Work">Categories Work</a>, theorem X.5.3)</p> <h4 id="PointwiseByWeightedColimits">In terms of weighted (co)limits</h4> <p>Suppose given <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> and <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>C</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">p : C \to C'</annotation></semantics></math> such that for every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>′</mo><mo>∈</mo><mi>C</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">c' \in C'</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/weighted+limit">weighted limit</a></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><msub><mi>Ran</mi> <mi>p</mi></msub><mi>F</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>c</mi><mo>′</mo><mo stretchy="false">)</mo><mo>≔</mo><msup><mi>lim</mi> <mrow><mi>C</mi><mo>′</mo><mo stretchy="false">(</mo><mi>c</mi><mo>′</mo><mo>,</mo><mi>p</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></msup><mi>F</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (Ran_p F)(c') \coloneqq lim^{C'(c',p(-))} F \,. </annotation></semantics></math></div> <p>exists. Then these objects fit together into a functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Ran</mi> <mi>p</mi></msub><mi>F</mi></mrow><annotation encoding="application/x-tex">Ran_p F</annotation></semantics></math> which is a right Kan extension of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>. Dually, if the <a class="existingWikiWord" href="/nlab/show/weighted+limit">weighted colimit</a></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><msub><mi>Lan</mi> <mi>p</mi></msub><mi>F</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>c</mi><mo>′</mo><mo stretchy="false">)</mo><mo>≔</mo><msup><mi>colim</mi> <mrow><mi>C</mi><mo>′</mo><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>,</mo><mi>c</mi><mo>′</mo><mo stretchy="false">)</mo></mrow></msup><mi>F</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (Lan_p F)(c') \coloneqq colim^{C'(p(-),c')} F \,. </annotation></semantics></math></div> <p>exists for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">c'</annotation></semantics></math>, then they fit together into a left Kan extension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Lan</mi> <mi>p</mi></msub><mi>F</mi></mrow><annotation encoding="application/x-tex">Lan_p F</annotation></semantics></math>. These definitions evidently make sense in the generality 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>-<a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a> 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/closed+monoidal+category">closed</a> <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a>. (In fact, they can be modified slightly to make sense in the full generality of a <a class="existingWikiWord" href="/nlab/show/2-category+equipped+with+proarrows">2-category equipped with proarrows</a>.)</p> <p>In particular, this means that 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 class="existingWikiWord" href="/nlab/show/small+category">small</a> 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> is <a class="existingWikiWord" href="/nlab/show/complete+category">complete</a> (resp. cocomplete), then all right (resp. left) Kan extensions of functors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo lspace="verythinmathspace">:</mo><mi>C</mi><mo>→</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">F\colon C\to D</annotation></semantics></math> exist along any functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo lspace="verythinmathspace">:</mo><mi>C</mi><mo>→</mo><mi>C</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">p\colon C\to C'</annotation></semantics></math>.</p> <p>One can prove that any Kan extension constructed in this way must be pointwise, in the sense of being preserved by all representables as above. Moreover, conversely, if a Kan extension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Lan</mi> <mi>p</mi></msub><mi>F</mi></mrow><annotation encoding="application/x-tex">Lan_p F</annotation></semantics></math> is pointwise, then one can prove that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>Lan</mi> <mi>p</mi></msub><mi>F</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>c</mi><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(Lan_p F)(c')</annotation></semantics></math> must be in fact a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>′</mo><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>,</mo><mi>c</mi><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C'(p(-),c')</annotation></semantics></math>-weighted colimit 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>, and dually; thus the two notions are equivalent.</p> <p>Unfolding the definitions of weighted (co)limits, these can be defined as representing objects</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>d</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">(</mo><msub><mi>Ran</mi> <mi>p</mi></msub><mi>F</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>c</mi><mo>′</mo><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><msup><mi>Set</mi> <mi>C</mi></msup><mo maxsize="1.8em" minsize="1.8em">(</mo><mi>C</mi><mo>′</mo><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>c</mi><mo>′</mo><mo>,</mo><mi>p</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mi>D</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>d</mi><mo>,</mo><mi>F</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mo maxsize="1.8em" minsize="1.8em">)</mo></mrow><annotation encoding="application/x-tex"> D\big( d ,\, (Ran_p F)(c') \big) \;\simeq\; Set^C\Big( C'\big(c', p(-)\big) ,\, D\big(d, F(-)\big) \Big) </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>D</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mo stretchy="false">(</mo><msub><mi>Lan</mi> <mi>p</mi></msub><mi>F</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>c</mi><mo>′</mo><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mi>d</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><msup><mi>Set</mi> <mrow><msup><mi>C</mi> <mi>op</mi></msup></mrow></msup><mo maxsize="1.8em" minsize="1.8em">(</mo><mi>C</mi><mo>′</mo><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>p</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>,</mo><mi>c</mi><mo>′</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mi>D</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>F</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>,</mo><mi>d</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mo maxsize="1.8em" minsize="1.8em">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> D\big( (Lan_p F)(c') ,\, d \big) \;\simeq\; Set^{C^{op}}\Big( C'\big(p(-), c'\big) ,\, D\big(F(-), d\big) \Big) \,. </annotation></semantics></math></div> <p>Similarly, 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 class="existingWikiWord" href="/nlab/show/enriched+categories">enriched categories</a>, replace <a class="existingWikiWord" href="/nlab/show/Set">Set</a> here with the <a class="existingWikiWord" href="/nlab/show/cosmos+for+enrichment">cosmos for enrichment</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>.</p> <h4 id="PointwiseByCoEnds">In terms of (co)ends</h4> <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/power">power</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 above weighted limit may be re-expressed in terms of an <a class="existingWikiWord" href="/nlab/show/end">end</a> as</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><msub><mi>Ran</mi> <mi>p</mi></msub><mi>F</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>c</mi><mo>′</mo><mo stretchy="false">)</mo><mo>≃</mo><msub><mo>∫</mo> <mrow><mi>c</mi><mo>∈</mo><mi>C</mi></mrow></msub><mi>C</mi><mo>′</mo><mo stretchy="false">(</mo><mi>c</mi><mo>′</mo><mo>,</mo><mi>p</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⋔</mo><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (Ran_p F)(c') \simeq \int_{c \in C} C'(c',p(c))\pitchfork F(c) </annotation></semantics></math></div> <p>(where again <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> and <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>C</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">p : C \to C'</annotation></semantics></math>).</p> <p>So in particular when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>=</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">D = V</annotation></semantics></math> this is</p> <div class="maruku-equation" id="eq:RightKanExtensionViaEndFormula"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>Ran</mi> <mi>p</mi></msub><mi>F</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>c</mi><mo>′</mo><mo stretchy="false">)</mo><mo>≃</mo><msub><mo>∫</mo> <mrow><mi>c</mi><mo>∈</mo><mi>C</mi></mrow></msub><mo stretchy="false">[</mo><mi>C</mi><mo>′</mo><mo stretchy="false">(</mo><mi>c</mi><mo>′</mo><mo>,</mo><mi>p</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (Ran_p F)(c') \simeq \int_{c \in C} [C'(c',p(c)),F(c)] \,. </annotation></semantics></math></div> <p>(<a href="#Kelly">Kelly (4.24)</a>)</p> <p>Similarly, if <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">copowered</a> 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 Kan extension is given by a <a class="existingWikiWord" href="/nlab/show/coend">coend</a>.</p> <div class="maruku-equation" id="eq:LeftKanExtensionViaCoendFormula"><span class="maruku-eq-number">(2)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>Lan</mi> <mi>p</mi></msub><mi>F</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>c</mi><mo>′</mo><mo stretchy="false">)</mo><mo>≃</mo><msup><mo>∫</mo> <mrow><mi>c</mi><mo>∈</mo><mi>C</mi></mrow></msup><mi>C</mi><mo>′</mo><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>,</mo><mi>c</mi><mo>′</mo><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_p F)(c') \simeq \int^{c \in C} C'(p(c),c') \cdot F(c) \,. </annotation></semantics></math></div> <p>(<a href="#Kelly">Kelly (4.25)</a>)</p> <div class="num_example" id="CoendFormulaForPresheavesOfSets"> <h6 id="example">Example</h6> <p><strong>(coend formula for left Kan extension of presheaves)</strong></p> <p>The coend formula for the left Kan extension is nicely understood when thinking 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> 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> above as <a class="existingWikiWord" href="/nlab/show/opposite+categories">opposite categories</a> and 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">\mathcal{V} = Set</annotation></semantics></math>, so that it takes <a class="existingWikiWord" href="/nlab/show/presheaves">presheaves</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> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo lspace="verythinmathspace">:</mo><mi>C</mi><mo>→</mo><mi>C</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">p \colon C \to C'</annotation></semantics></math> to presheaves <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Lan</mi> <mi>p</mi></msub><mi>F</mi></mrow><annotation encoding="application/x-tex">Lan_p F</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">C'</annotation></semantics></math>, by the formula</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><msub><mi>Lan</mi> <mi>p</mi></msub><mi>F</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>c</mi><mo>′</mo><mo stretchy="false">)</mo><mo>≃</mo><msup><mo>∫</mo> <mrow><mi>c</mi><mo>∈</mo><mi>C</mi></mrow></msup><mi>C</mi><mo>′</mo><mo stretchy="false">(</mo><mi>c</mi><mo>′</mo><mo>,</mo><mi>p</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><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_p F)(c') \simeq \int^{c \in C} C'(c', p(c)) \times F(c) \,. </annotation></semantics></math></div> <p>Using the <a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a> to rewrite <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 stretchy="false">)</mo><mo>≃</mo><msub><mi>Hom</mi> <mrow><mi>PSh</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(c) \simeq Hom_{PSh(C)}(c,F)</annotation></semantics></math>, this 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><msub><mi>Lan</mi> <mi>p</mi></msub><mi>F</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>c</mi><mo>′</mo><mo stretchy="false">)</mo><mo>≃</mo><msup><mo>∫</mo> <mrow><mi>c</mi><mo>∈</mo><mi>C</mi></mrow></msup><msub><mi>Hom</mi> <mrow><mi>C</mi><mo>′</mo></mrow></msub><mo stretchy="false">(</mo><mi>c</mi><mo>′</mo><mo>,</mo><mi>p</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>×</mo><msub><mi>Hom</mi> <mrow><mi>PSh</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>F</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (Lan_p F)(c') \simeq \int^{c \in C} Hom_{C'}(c', p(c)) \times Hom_{PSh(C)}(c,F) \,. </annotation></semantics></math></div> <p>In this form one sees that the coend produces the set whose elements are <a class="existingWikiWord" href="/nlab/show/equivalence+classes">equivalence classes</a> of pairs of morphisms</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><mo>→</mo><mi>p</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>,</mo><mi>c</mi><mo>→</mo><mi>F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (c' \to p(c), c \to F) </annotation></semantics></math></div> <p>where two such are regarded as equivalent whenever there is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><msub><mi>c</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>c</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">f \colon c_1 \to c_2</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><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>c</mi><mo>′</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><mi>p</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd></mtd> <mtd><mi>p</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msub><mi>c</mi> <mn>1</mn></msub></mtd> <mtd></mtd> <mtd><mover><mo>⟶</mo><mi>f</mi></mover></mtd> <mtd></mtd> <mtd><msub><mi>c</mi> <mn>2</mn></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><mo>↙</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>F</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && c' \\ & \swarrow && \searrow \\ p(c_1) && \stackrel{p(f)}{\longrightarrow} && p(c_2) \\ c_1 && \stackrel{f}{\longrightarrow} && c_2 \\ & \searrow && \swarrow \\ && F } \,. </annotation></semantics></math></div> <p>This is particularly suggestive in cases when we may think of the objects 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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">C'</annotation></semantics></math> on the same footing, notably when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> inclusion. For in that case we may imagine that a representative pair <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><mo>→</mo><mi>p</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>,</mo><mi>c</mi><mo>→</mo><mi>F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(c' \to p(c), c \to F)</annotation></semantics></math> is a stand-in for the actual pullback of elements 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> via forming the composite “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>′</mo><mo>→</mo><mi>c</mi><mo>→</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">c'\to c \to F</annotation></semantics></math>”, only that this composite is not defined. But the above equivalence relation is precisely that under which this composite would be invariant.</p> </div> <h4 id="PointwiseByConicalLimits">In terms of conical (co)limits</h4> <p>In the case of functors between ordinary <a class="existingWikiWord" href="/nlab/show/locally+small+categories">locally small categories</a>, hence in the special case 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>-<a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a> for <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">V = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Set">Set</a>, there is an expression of a weighted (co)limit and hence a pointwise Kan extension as an ordinary (“conical”, meaning: in terms of <a class="existingWikiWord" href="/nlab/show/cone">cone</a>s) (co)limit over a <a class="existingWikiWord" href="/nlab/show/comma+category">comma category</a>:</p> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>Let</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> 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>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> have all small <a class="existingWikiWord" href="/nlab/show/limit">limit</a>s.</p> </li> </ul> <p>Then the right 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> of locally small categories 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>C</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">p : C \to C'</annotation></semantics></math> exists and its value on an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>′</mo><mo>∈</mo><mi>C</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">c' \in C'</annotation></semantics></math> is given by the <a class="existingWikiWord" href="/nlab/show/limit">limit</a></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><msub><mi>Ran</mi> <mi>p</mi></msub><mi>F</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>c</mi><mo>′</mo><mo stretchy="false">)</mo><mo>≃</mo><munder><mi>lim</mi> <mo>←</mo></munder><mrow><mo>(</mo><mo stretchy="false">(</mo><msub><mi>Δ</mi> <mrow><mi>c</mi><mo>′</mo></mrow></msub><mo stretchy="false">/</mo><mi>p</mi><mo stretchy="false">)</mo><mo>→</mo><mi>C</mi><mover><mo>→</mo><mi>F</mi></mover><mi>D</mi><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> (Ran_p F)(c') \simeq \lim_\leftarrow \left((\Delta_{c'}/p) \to C \stackrel{F}{\to} D\right) \,, </annotation></semantics></math></div> <p>where</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mrow><mi>c</mi><mo>′</mo></mrow></msub><mo stretchy="false">/</mo><mi>p</mi></mrow><annotation encoding="application/x-tex">\Delta_{c'}/p</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/comma+category">comma category</a>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mrow><mi>c</mi><mo>′</mo></mrow></msub><mo stretchy="false">/</mo><mi>p</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">\Delta_{c'}/p \to C</annotation></semantics></math> is the canonical <a class="existingWikiWord" href="/nlab/show/forgetful+functor">forgetful functor</a>.</p> </li> </ul> <p>Likewise, if <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> admits small <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a>s, the left Kan extension of a functor exists and is pointwise given by the <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a></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><msub><mi>Lan</mi> <mi>p</mi></msub><mi>F</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>c</mi><mo>′</mo><mo stretchy="false">)</mo><mo>≃</mo><munder><mi>lim</mi> <mo>→</mo></munder><mrow><mo>(</mo><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">/</mo><msub><mi>Δ</mi> <mrow><mi>c</mi><mo>′</mo></mrow></msub><mo stretchy="false">)</mo><mo>→</mo><mi>C</mi><mover><mo>→</mo><mi>F</mi></mover><mi>D</mi><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (Lan_p F)(c') \simeq \lim_\to \left((p/\Delta_{c'}) \to C \stackrel{F}{\to} D\right) \,. </annotation></semantics></math></div></div> <p>This appears for instance as (<a href="#Borceux">Borceux, I, thm 3.7.2</a>). Discussion in the context of <a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a> is in (<a href="#Kelly">Kelly, section 3.4</a>).</p> <p>A cartoon picture of the forgetful functor out of the <a class="existingWikiWord" href="/nlab/show/comma+category">comma category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo stretchy="false">/</mo><msub><mi>Δ</mi> <mrow><mi>c</mi><mo>′</mo></mrow></msub><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">p/\Delta_{c'} \to C</annotation></semantics></math>, useful to keep in mind, is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mi>p</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd></mtd> <mtd><mi>p</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><mo>↙</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>c</mi><mo>′</mo></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>↦</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><msub><mi>c</mi> <mn>1</mn></msub><mover><mo>→</mo><mi>ϕ</mi></mover><msub><mi>c</mi> <mn>2</mn></msub><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \left( \array{ p(c_1) &&\stackrel{p(\phi)}{\to}&& p(c_2) \\ & \searrow && \swarrow \\ && c' } \right) \;\; \mapsto \;\; \left( c_1 \stackrel{\phi}{\to} c_2 \right) \,. </annotation></semantics></math></div> <p>The <a class="existingWikiWord" href="/nlab/show/comma+category">comma category</a> here is equivalently the <a class="existingWikiWord" href="/nlab/show/category+of+elements">category of elements</a> of the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>′</mo><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>,</mo><mi>c</mi><mo>′</mo><mo stretchy="false">)</mo><mo>:</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">C'(p(-), c') : C^{op} \to Set</annotation></semantics></math></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>p</mi><mo stretchy="false">/</mo><msub><mi>Δ</mi> <mrow><mi>c</mi><mo>′</mo></mrow></msub><mo stretchy="false">)</mo><mo>≃</mo><mi>el</mi><mo stretchy="false">(</mo><mi>C</mi><mo>′</mo><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>,</mo><mi>c</mi><mo>′</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (p/\Delta_{c'}) \simeq el( C'(p(-), c') ) \,. </annotation></semantics></math></div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>Consider the case of the left Kan extension, the other case works analogously, but dually.</p> <p>First notice that the above pointwise definition of values of a functor canonically extends to an actual functor:</p> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>:</mo><mi>c</mi><msub><mo>′</mo> <mn>1</mn></msub><mo>→</mo><mi>c</mi><msub><mo>′</mo> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\phi : c'_1 \to c'_2</annotation></semantics></math> any morphism in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">C'</annotation></semantics></math> we get a functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mo>*</mo></msub><mo>:</mo><mi>p</mi><mo stretchy="false">/</mo><msub><mi>Δ</mi> <mrow><mi>c</mi><msub><mo>′</mo> <mn>1</mn></msub></mrow></msub><mo>→</mo><mi>p</mi><mo stretchy="false">/</mo><msub><mi>Δ</mi> <mrow><mi>c</mi><msub><mo>′</mo> <mn>2</mn></msub></mrow></msub></mrow><annotation encoding="application/x-tex"> \phi_* : p/\Delta_{c'_1} \to p/\Delta_{c'_2} </annotation></semantics></math></div> <p>of comma categories, by postcomposition. This morphism of <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a>s induces canonically a corresponding morphism of <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a>s</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><msub><mi>Lan</mi> <mi>p</mi></msub><mi>F</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>c</mi><msub><mo>′</mo> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><msub><mi>Lan</mi> <mi>p</mi></msub><mi>F</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>c</mi><msub><mo>′</mo> <mn>2</mn></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (Lan_p F)(c'_1) \to (Lan_p F)(c'_2) \,. </annotation></semantics></math></div> <p>Now for the universal property of the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Lan</mi> <mi>p</mi></msub><mi>F</mi></mrow><annotation encoding="application/x-tex">Lan_p F</annotation></semantics></math> defined this way. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>′</mo><mo>∈</mo><mi>C</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">c' \in C'</annotation></semantics></math> denote the components of the colimiting <a class="existingWikiWord" href="/nlab/show/cocone">cocone</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>Lan</mi> <mi>p</mi></msub><mi>F</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>c</mi><mo>′</mo><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><msub><mi>lim</mi> <mo>→</mo></msub><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">/</mo><msub><mi>Δ</mi> <mrow><mi>c</mi><mo>′</mo></mrow></msub><mo>→</mo><mi>C</mi><mover><mo>→</mo><mi>F</mi></mover><mi>D</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(Lan_p F)(c') := \lim_{\to}( p/\Delta_{c'} \to C \stackrel{F}{\to} D) </annotation></semantics></math> by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>s</mi> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">s_{(-)}</annotation></semantics></math>, as in</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><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mover><mo>→</mo><mi>ϕ</mi></mover><mi>c</mi><mo>′</mo><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd></mtd> <mtd><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mover><mo>→</mo><mi>λ</mi></mover><mi>c</mi><mo>′</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd></mtr> <mtr><mtd></mtd></mtr> <mtr><mtd><mi>F</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd></mtd> <mtd><mi>F</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>s</mi> <mi>ϕ</mi></msub></mrow></mpadded></msub><mo>↘</mo></mtd> <mtd></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mrow><msub><mi>s</mi> <mi>λ</mi></msub></mrow></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">(</mo><msub><mi>Lan</mi> <mi>p</mi></msub><mi>F</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>c</mi><mo>′</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ (p(c_1)\stackrel{\phi}{\to} c') &&\stackrel{p(h)}{\to}&& (p(c_2)\stackrel{\lambda}{\to} c') \\ \\ \\ F(c_1) &&\stackrel{F(h)}{\to}&& F(c_2) \\ & {}_{\mathllap{s_\phi}}\searrow && \swarrow_{\mathrlap{s_{\lambda}}} \\ && (Lan_p F)(c') } \,. </annotation></semantics></math></div> <p>We now construct in components a natural transformation</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mi>F</mi></msub><mo>:</mo><mi>F</mi><mo>→</mo><msup><mi>p</mi> <mo>*</mo></msup><msub><mi>Lan</mi> <mi>p</mi></msub><mi>F</mi></mrow><annotation encoding="application/x-tex"> \eta_F : F \to p^* Lan_p F </annotation></semantics></math></div> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Lan</mi> <mi>p</mi></msub><mi>F</mi></mrow><annotation encoding="application/x-tex">Lan_p F</annotation></semantics></math> defined as above, and show that it satisfies the required universal property. The components of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mi>F</mi></msub></mrow><annotation encoding="application/x-tex">\eta_F</annotation></semantics></math> over <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> are morphisms</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mi>F</mi></msub><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>:</mo><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><msub><mi>Lan</mi> <mi>p</mi></msub><mi>F</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \eta_F(c) : F(c) \to (Lan_p F)(p (c)) \,. </annotation></semantics></math></div> <p>Take these to be 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>η</mi> <mi>F</mi></msub><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><msub><mi>s</mi> <mrow><msub><mi>Id</mi> <mrow><mi>p</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></msub></mrow></msub></mrow><annotation encoding="application/x-tex"> \eta_F(c) := s_{Id_{p(c)}} </annotation></semantics></math></div> <p>(this is similar to what happens in the proof of the <a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a>, all of these arguments are variants of the argument for the Yoneda lemma, and vice versa). It is straightforward, if somewhat tedious, to check that these are natural, and that the natural transformation defined this way has the required universal property.</p> </div> <h4 id="pointwiseVsWeak">Comparing the definitions</h4> <p>We have seen that if <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> has enough limits or colimits, then a pointwise Kan extension can be defined in terms of these limits, and will necessarily satisfy the universal property described first. However, not all Kan extensions are pointwise: that is, having a universal transformation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>→</mo><mo stretchy="false">(</mo><msub><mi>Lan</mi> <mi>p</mi></msub><mi>F</mi><mo stretchy="false">)</mo><mi>p</mi></mrow><annotation encoding="application/x-tex">F \to (Lan_p F)p</annotation></semantics></math> does not necessarily imply that the individual values of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Lan</mi> <mi>p</mi></msub><mi>F</mi></mrow><annotation encoding="application/x-tex">Lan_p F</annotation></semantics></math> are limits or colimits in its codomain. Non-pointwise Kan extensions can exist even when <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> does not admit very many limits.</p> <p>It should be noted, though, that pointwise Kan extensions can still exist, and hence the particular requisite limits/colimits exist, even if <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 not (co)complete. For instance, the Kan extensions that arise in the study of <a class="existingWikiWord" href="/nlab/show/derived+functor">derived functor</a>s are pointwise, and in fact <a class="existingWikiWord" href="/nlab/show/absolute+colimit">absolute</a> (preserved by <em>all</em> functors), even though their codomains are <a class="existingWikiWord" href="/nlab/show/homotopy+categories">homotopy categories</a> which generally do not admit all limits and colimits.</p> <p>Non-pointwise Kan extensions seem to be very rare in practice. However, the abstract notion of Kan extension (sometimes called simply “extension”) in a 2-category, and its dual notion of lifting, can be useful in <a class="existingWikiWord" href="/nlab/show/2-category+theory">2-category theory</a>. For instance, <a class="existingWikiWord" href="/nlab/show/bicategories">bicategories</a> such as <a class="existingWikiWord" href="/nlab/show/Prof">Prof</a> admit all right extensions and right liftings; a bicategory with this property may be considered a <a class="existingWikiWord" href="/nlab/show/horizontal+categorification">horizontal categorification</a> of a <a class="existingWikiWord" href="/nlab/show/closed+monoidal+category">closed monoidal category</a>.</p> <h3 id="AbsoluteKanExtension">Absolute Kan extensions</h3> <p>An <em>absolute</em> Kan extension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Lan</mi> <mi>p</mi></msub><mi>F</mi></mrow><annotation encoding="application/x-tex">Lan_p F</annotation></semantics></math> is one which is <a href="#Preservation">preserved</a> by all functors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> out of the codomain 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><mi>G</mi><mo stretchy="false">(</mo><msub><mi>Lan</mi> <mi>p</mi></msub><mi>F</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>Lan</mi> <mi>p</mi></msub><mo stretchy="false">(</mo><mi>G</mi><mi>F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> G (Lan_p F) \simeq Lan_p(G F) </annotation></semantics></math></div> <p>(same for right Kan extensions).</p> <p>The most prominent example of absolute Kan extensions is given by <a class="existingWikiWord" href="/nlab/show/adjoint+functor">adjoint functors</a>; in fact they can be defined as certain absolute Kan extensions. See there for the precise statement.</p> <p> <div class='num_remark'> <h6>Remark</h6> <p><strong>(absolute vs pointwise)</strong> <br /> Absolute Kan extensions are always pointwise, as the latter can be defined as those preserved by representables; there are (lots of) examples of pointwise Kan extensions which are not absolute.</p> <p>Note that in a general 2-category, absolute Kan extensions make perfect sense, while for defining pointwise ones more structure is needed: <a class="existingWikiWord" href="/nlab/show/comma+object">comma objects</a> and/or some structure which would let us work with (co)limits <em>inside</em> that 2-category (such as a <a class="existingWikiWord" href="/nlab/show/Yoneda+structure">(co)Yoneda structure</a> or a <a class="existingWikiWord" href="/nlab/show/2-category+equipped+with+proarrows">proarrow equipment</a>).</p> </div> </p> <h3 id="of_functors">Of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-functors</h3> <p>The global definition of Kan extensions for functors in terms of left/right adjoints to pullbacks may be interpreted essentially verbatim in the context of <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-categories">(∞,1)-categories</a></p> <p>See at <em><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Kan+extension">(∞,1)-Kan extension</a></em>.</p> <h3 id="In2cat">In a general 2-category</h3> <p>The Kan extension of a functor may be regarded more abstractly as an extension-problem in the <a class="existingWikiWord" href="/nlab/show/2-category">2-category</a> <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a> of categories. The same extension problem can be stated verbatim in any <a class="existingWikiWord" href="/nlab/show/2-category">2-category</a> and hence there is a corresponding more general notion of Kan extensions of <a class="existingWikiWord" href="/nlab/show/1-morphisms">1-morphisms</a> in <a class="existingWikiWord" href="/nlab/show/2-categories">2-categories</a>. This is discussed in (<a href="#Lack09">Lack 09, section 2.2</a>).</p> <p>The question of defining a <em>pointwise</em> Kan extension in a general 2-category is more subtle, and there are at least two distinct approaches. If the 2-category has <a class="existingWikiWord" href="/nlab/show/comma+objects">comma objects</a>, then we can define a Kan extension to be pointwise if it remains a Kan extension upon pasting with any comma object; this is an “internalization” of the above definition in terms of <em>conical</em> colimits. On the other hand, in a <a class="existingWikiWord" href="/nlab/show/2-category+equipped+with+proarrows">2-category equipped with proarrows</a> we can define pointwise Kan extensions as particular weighted (co)limits using a representable weight; this generalizes the above definition as a weighted (co)limit.</p> <p>In some 2-categories such as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cat</mi></mrow><annotation encoding="application/x-tex">Cat</annotation></semantics></math>, both definitions agree; but in others they do not, and in general in this case it is the equipment-theoretic version that is “correct”. For instance, in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mi>Cat</mi></mrow><annotation encoding="application/x-tex">V Cat</annotation></semantics></math> the equipment-theoretic version gives the right notion of pointwise Kan extension, whereas the comma-object one is too strong.</p> <p>As a concrete example, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>=</mo><mi>Cat</mi></mrow><annotation encoding="application/x-tex">V=Cat</annotation></semantics></math>, so that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mi>Cat</mi><mo>=</mo><mn>2</mn><mi>Cat</mi></mrow><annotation encoding="application/x-tex">V Cat = 2 Cat</annotation></semantics></math>; then comma objects are not informative enough because they “don’t see the 2-cells”. In even more specificity, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/walking">walking</a> 2-cell and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> the walking pair of parallel 1-morphisms, with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mn>1</mn><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">f:1\to B</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>:</mo><mn>1</mn><mo>→</mo><mi>M</mi></mrow><annotation encoding="application/x-tex">g:1\to M</annotation></semantics></math> the inclusions of the common domain of the parallel 1-morphisms; then the equipment-theoretic-pointwise <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Lan</mi> <mi>f</mi></msub><mi>g</mi></mrow><annotation encoding="application/x-tex">Lan_f g</annotation></semantics></math> is constant at the domain object, whereas the comma-object-pointwise <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Lan</mi> <mi>f</mi></msub><mi>g</mi></mrow><annotation encoding="application/x-tex">Lan_f g</annotation></semantics></math> does not exist. See <a href="#Roald13">(Roald, Example 2.24)</a> for details.</p> <h2 id="existence">Existence</h2> <p>The following reproduces a <a href="https://mathoverflow.net/questions/365947/when-size-matters-in-category-theory-for-the-working-mathematician/365951#365951">MathOverflow answer</a> by <a class="existingWikiWord" href="/nlab/show/Ivan+Di+Liberti">Ivan Di Liberti</a>:</p> <p> <div class='num_lemma'> <h6>Lemma</h6> <p>(Kan). Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="sans-serif"><mi>B</mi></mstyle><mover><mo>←</mo><mi>f</mi></mover><mstyle mathvariant="sans-serif"><mi>A</mi></mstyle><mover><mo>→</mo><mi>g</mi></mover><mstyle mathvariant="sans-serif"><mi>C</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathsf{B} \stackrel{f}{\leftarrow} \mathsf{A} \stackrel{g}{\to} \mathsf{C}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/span">span</a> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="sans-serif"><mi>A</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathsf{A}</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/small+category">small</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="sans-serif"><mi>C</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathsf{C}</annotation></semantics></math> is (small) <a class="existingWikiWord" href="/nlab/show/cocomplete+category">cocomplete</a>. Then the <a class="existingWikiWord" href="/nlab/show/left+Kan+extension">left Kan extension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="sans-serif"><mi>lan</mi></mstyle> <mi>f</mi></msub><mi>g</mi></mrow><annotation encoding="application/x-tex">\mathsf{lan}_f g</annotation></semantics></math> exists.</p> </div> </p> <p>Kan extensions are a useful tool in everyday practice, with applications in many different topics of <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a>. In this lemma (which is one of the most used in this topic) the set-theoretic issue is far from being hidden: <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="sans-serif"><mi>A</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathsf{A}</annotation></semantics></math> needs to be small (with respect to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ob</mi><mo stretchy="false">(</mo><mstyle mathvariant="sans-serif"><mi>C</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ob(\mathsf{C})</annotation></semantics></math>!</strong> There is no chance that the lemma is true when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="sans-serif"><mi>A</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathsf{A}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/large+category">large category</a>. Indeed since <a class="existingWikiWord" href="/nlab/show/colimits">colimits</a> can be computed via Kan extensions, the lemma would imply that every (small) <a class="existingWikiWord" href="/nlab/show/cocomplete+category">cocomplete category</a> is large cocomplete, which is not allowed because <a href="complete+small+category#CompleteSmallCategoriesArePosets">cocomplete small categories are posets</a>. Also, there is no chance to solve the problem by saying: <em>well, let’s just consider <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="sans-serif"><mi>C</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathsf{C}</annotation></semantics></math> to be large-cocomplete</em>, again because <a href="complete+small+category#CompleteSmallCategoriesArePosets">cocomplete small categories are posets</a>.</p> <p>This problem is hard to avoid because the size of the categories of our interest is <em>as a fact</em> always larger than the size of their inhabitants (this just means that most of the time <strong>Ob</strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="sans-serif"><mi>C</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathsf{C}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/proper+class">proper class</a>, as big as the size of the <a class="existingWikiWord" href="/nlab/show/enriched+category">enrichment</a>).</p> <p>Notice that the Kan extension problem <strong>recovers the <a class="existingWikiWord" href="/nlab/show/adjoint+functor+theorem">adjoint functor theorem</a> one,</strong> because adjoints are computed via Kan extensions of identities of large categories. Indeed, in that case, the <a class="existingWikiWord" href="/nlab/show/solution+set+condition">solution set condition</a> is precisely what is needed in order to cut down the size of some colimits that otherwise would be too large to compute, as can be synthesized by the sharp version of the Kan lemma.</p> <p> <div class='num_lemma'> <h6>Lemma</h6> <p><strong>Sharp Kan lemma.</strong> Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="sans-serif"><mi>B</mi></mstyle><mover><mo>←</mo><mi>f</mi></mover><mstyle mathvariant="sans-serif"><mi>A</mi></mstyle><mover><mo>→</mo><mi>g</mi></mover><mstyle mathvariant="sans-serif"><mi>C</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathsf{B} \stackrel{f}{\leftarrow} \mathsf{A} \stackrel{g}{\to} \mathsf{C}</annotation></semantics></math> be a span where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="sans-serif"><mi>B</mi></mstyle><mo stretchy="false">(</mo><mi>f</mi><mo>−</mo><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathsf{B}(f-,b)</annotation></semantics></math> is a small presheaf for every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><mstyle mathvariant="sans-serif"><mi>B</mi></mstyle></mrow><annotation encoding="application/x-tex">b \in \mathsf{B}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="sans-serif"><mi>C</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathsf{C}</annotation></semantics></math> is (small) cocomplete. Then the left Kan extension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="sans-serif"><mi>lan</mi></mstyle> <mi>f</mi></msub><mi>g</mi></mrow><annotation encoding="application/x-tex">\mathsf{lan}_f g</annotation></semantics></math> exists.</p> </div> </p> <p>Indeed this lemma allows <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="sans-serif"><mi>A</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathsf{A}</annotation></semantics></math> to be large, but we must pay a tribute to its presheaf category: <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> needs to be somehow <em>locally small</em> (with respect to <strong>Ob</strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="sans-serif"><mi>C</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathsf{C}</annotation></semantics></math>).</p> <p> <div class='num_lemma'> <h6>Lemma</h6> <p><strong>Kan lemma Fortissimo.</strong> Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="sans-serif"><mi>A</mi></mstyle><mover><mo>→</mo><mi>f</mi></mover><mstyle mathvariant="sans-serif"><mi>B</mi></mstyle></mrow><annotation encoding="application/x-tex"> \mathsf{A} \stackrel{f}{\to} \mathsf{B} </annotation></semantics></math> be a functor. The following are equivalent:</p> <ul> <li>for every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>:</mo><mstyle mathvariant="sans-serif"><mi>A</mi></mstyle><mo>→</mo><mstyle mathvariant="sans-serif"><mi>C</mi></mstyle></mrow><annotation encoding="application/x-tex">g :\mathsf{A} \to \mathsf{C}</annotation></semantics></math> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="sans-serif"><mi>C</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathsf{C}</annotation></semantics></math> is a small-cocomplete category, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="sans-serif"><mi>lan</mi></mstyle> <mi>f</mi></msub><mi>g</mi></mrow><annotation encoding="application/x-tex">\mathsf{lan}_f g</annotation></semantics></math> exists.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="sans-serif"><mi>lan</mi></mstyle> <mi>f</mi></msub><mi>y</mi></mrow><annotation encoding="application/x-tex">\mathsf{lan}_f y</annotation></semantics></math> exists, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math> is the Yoneda embedding in the category of small presheaves <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>:</mo><mstyle mathvariant="sans-serif"><mi>A</mi></mstyle><mo>→</mo><mi>𝒫</mi><mo stretchy="false">(</mo><mstyle mathvariant="sans-serif"><mi>A</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">y: \mathsf{A} \to \mathcal{P}(\mathsf{A})</annotation></semantics></math>.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="sans-serif"><mi>B</mi></mstyle><mo stretchy="false">(</mo><mi>f</mi><mo>−</mo><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathsf{B}(f-,b)</annotation></semantics></math> is a is small presheaf for every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><mstyle mathvariant="sans-serif"><mi>B</mi></mstyle></mrow><annotation encoding="application/x-tex">b \in \mathsf{B}</annotation></semantics></math>.</li> </ul> <p></p> </div> </p> <p>Even unconsciously, the previous discussion is one of the reasons of the popularity of <a class="existingWikiWord" href="/nlab/show/locally+presentable+categories">locally presentable categories</a>. Indeed, having a dense generator is a good compromise between <em>generality and tameness</em>. As an evidence of this, in the context of <a class="existingWikiWord" href="/nlab/show/accessible+categories">accessible categories</a> the sharp Kan lemma can be simplified.</p> <p> <div class='num_lemma'> <h6>Lemma</h6> <p><strong>Tame Kan lemma.</strong> Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="sans-serif"><mi>B</mi></mstyle><mover><mo>←</mo><mi>f</mi></mover><mstyle mathvariant="sans-serif"><mi>A</mi></mstyle><mover><mo>→</mo><mi>g</mi></mover><mstyle mathvariant="sans-serif"><mi>C</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathsf{B} \stackrel{f}{\leftarrow} \mathsf{A} \stackrel{g}{\to} \mathsf{C}</annotation></semantics></math> be a span of accessible categories, where <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 accessible functor and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="sans-serif"><mi>C</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathsf{C}</annotation></semantics></math> is (small) cocomplete. Then the left Kan extension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="sans-serif"><mi>lan</mi></mstyle> <mi>f</mi></msub><mi>g</mi></mrow><annotation encoding="application/x-tex">\mathsf{lan}_f g</annotation></semantics></math> exists.</p> </div> </p> <p><em>References for Sharp.</em> I am not aware of a reference for this result. It can follow from a careful analysis of <strong>Prop. A.7</strong> in my paper <strong>Codensity: Isbell duality, pro-objects, compactness and accessibility</strong>. The structure of the proof remains the same, presheaves must be replaced by small presheaves.</p> <p><em>References for Tame.</em> This is an exercise, it can follow directly from the sharp Kan lemma, but it’s enough to properly combine the usual Kan lemma, <strong>Prop A.1&2</strong> of the above-mentioned paper, and the fact that <a class="existingWikiWord" href="/nlab/show/accessible+functors">accessible functors</a> have arity.</p> <h2 id="Properties">Properties</h2> <h3 id="LeftKanOnRepresentables">Left Kan extension on representables / fully faithfulness</h3> <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">\mathcal{V}</annotation></semantics></math> be a suitable enriching category (a <a class="existingWikiWord" href="/nlab/show/cosmos">cosmos</a>). Notably <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> may be <a class="existingWikiWord" href="/nlab/show/Set">Set</a>.</p> <div class="num_prop" id="LeftKanExtensionBasicProp"> <h6 id="proposition_2">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">F : C \to D</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">\mathcal{V}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/enriched+functor">enriched functor</a> between <a class="existingWikiWord" href="/nlab/show/small+category">small</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+categories">enriched categories</a> we have</p> <ol> <li> <p>the left Kan extension along <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> takes <a class="existingWikiWord" href="/nlab/show/representable+functor">representable</a> <a class="existingWikiWord" href="/nlab/show/presheaves">presheaves</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>:</mo><mi>C</mi><mo>→</mo><mi>𝒱</mi></mrow><annotation encoding="application/x-tex">C(c,-) : C \to \mathcal{V}</annotation></semantics></math> to their image under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Lan</mi> <mi>F</mi></msub><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>D</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Lan_F C(c, -) \simeq D(F(c), -) </annotation></semantics></math></div> <p>for all <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>.</p> </li> <li> <p>if <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 <a class="existingWikiWord" href="/nlab/show/full+and+faithful+functor">full and faithful functor</a> then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>F</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><msub><mi>Lan</mi> <mi>F</mi></msub><mi>H</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">F^* (Lan_F H) \simeq H</annotation></semantics></math> and in fact the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>Lan</mi> <mi>F</mi></msub><mo>⊣</mo><msup><mi>F</mi> <mo>*</mo></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(Lan_F \dashv F^*)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/unit+of+an+adjunction">unit of an adjunction</a> is 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>Id</mi><mover><mo>→</mo><mo>≃</mo></mover><msup><mi>F</mi> <mo>*</mo></msup><mo>∘</mo><msub><mi>Lan</mi> <mi>F</mi></msub></mrow><annotation encoding="application/x-tex"> Id \stackrel{\simeq}{\to} F^* \circ Lan_{F} </annotation></semantics></math></div> <p>whence it follows (by <a href="adjoint+functor#FullyFaithfulAndInvertibleAdjoints">this property</a> of <a class="existingWikiWord" href="/nlab/show/adjoint+functor">adjoint functor</a>s) that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Lan</mi> <mi>F</mi></msub><mo>:</mo><mo stretchy="false">[</mo><mi>C</mi><mo>,</mo><mi>𝒱</mi><mo stretchy="false">]</mo><mo>→</mo><mo stretchy="false">[</mo><mi>D</mi><mo>,</mo><mi>𝒱</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">Lan_F : [C,\mathcal{V}] \to [D,\mathcal{V}]</annotation></semantics></math> is itself a <a class="existingWikiWord" href="/nlab/show/full+and+faithful+functor">full and faithful functor</a>.</p> </li> </ol> </div> <p>The second statement appears for instance as (<a href="#Kelly">Kelly, prop. 4.23</a>).</p> <div class="proof"> <h6 id="proof_3">Proof</h6> <p>For the first statement, using the <a class="existingWikiWord" href="/nlab/show/coend">coend</a> formula for the left Kan extension <a href="#PointwiseByCoEnds">above</a> we have naturally in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>′</mo><mo>∈</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">d' \in D</annotation></semantics></math> the expression</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mi>Lan</mi> <mi>F</mi></msub><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>d</mi><mo>′</mo><mo>↦</mo></mtd> <mtd><msup><mo>∫</mo> <mrow><mi>c</mi><mo>′</mo><mo>∈</mo><mi>C</mi></mrow></msup><mi>D</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo>′</mo><mo stretchy="false">)</mo><mo>,</mo><mi>d</mi><mo>′</mo><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><mo stretchy="false">(</mo><mi>c</mi><mo>′</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><msup><mo>∫</mo> <mrow><mi>c</mi><mo>′</mo><mo>∈</mo><mi>C</mi></mrow></msup><mi>D</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo>′</mo><mo stretchy="false">)</mo><mo>,</mo><mi>d</mi><mo>′</mo><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></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mi>D</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>,</mo><mi>d</mi><mo>′</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} Lan_F C(c,-) : d' \mapsto & \int^{c' \in C} D(F(c'), d') \cdot C(c,-)(c') \\ & \simeq \int^{c' \in C} D(F(c'), d') \cdot C(c,c') \\ & \simeq D(F(c), d') \end{aligned} \,. </annotation></semantics></math></div> <p>Here the last step is called sometimes the <a class="existingWikiWord" href="/nlab/show/co-Yoneda+lemma">co-Yoneda lemma</a>. It follows for instance by observing that <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><mo>∈</mo><mi>C</mi></mrow></msup><mi>D</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo>′</mo><mo stretchy="false">)</mo><mo>,</mo><mi>d</mi><mo>′</mo><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></mrow><annotation encoding="application/x-tex">\int^{c' \in C} D(F(c'), d') \cdot C(c,c')</annotation></semantics></math> is equivalently dually the expression for the left Kan extension of the non-representable <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</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>d</mi><mo>′</mo><mo stretchy="false">)</mo><mo>:</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>→</mo><mi>𝒱</mi></mrow><annotation encoding="application/x-tex">D(F(-),d') : C^{op} \to \mathcal{V}</annotation></semantics></math> along the <em>identity</em> functor.</p> <p>Similarly for the second, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>:</mo><mi>D</mi><mo>→</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">H : D \to E</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">\mathcal{V}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/enriched+functor">enriched functor</a> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/copower">copowered</a> 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">\mathcal{V}</annotation></semantics></math>, then its left Kan extension evaluated on the image 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</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mi>Lan</mi> <mi>F</mi></msub><mi>H</mi><mo>:</mo><mi>F</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo>↦</mo></mtd> <mtd><msup><mo>∫</mo> <mrow><mi>c</mi><mo>∈</mo><mi>C</mi></mrow></msup><mi>D</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>,</mo><mi>F</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⋅</mo><mi>H</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><msup><mo>∫</mo> <mrow><mi>c</mi><mo>∈</mo><mi>C</mi></mrow></msup><mi>C</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>H</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mi>H</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} Lan_F H : F(d) \mapsto & \int^{c \in C} D(F(c), F(d)) \cdot H(c) \\ & \simeq \int^{c \in C} C(c, d) \cdot H(c) \\ & \simeq H(d) \end{aligned} \,. </annotation></semantics></math></div></div> <h3 id="LeftKanExtensionPreservingCertainLimits">Left Kan extensions preserving certain limits</h3> <p>The following statement says that <a class="existingWikiWord" href="/nlab/show/left+exact+functors">left exact functors</a> into <a class="existingWikiWord" href="/nlab/show/toposes">toposes</a> have left exact left Kan extension along the <a class="existingWikiWord" href="/nlab/show/Yoneda+embedding">Yoneda embedding</a> (<a class="existingWikiWord" href="/nlab/show/Yoneda+extension">Yoneda extension</a>) and that this is the <a class="existingWikiWord" href="/nlab/show/inverse+image">inverse image</a> of a <a class="existingWikiWord" href="/nlab/show/geometric+morphism">geometric morphism</a> of <a class="existingWikiWord" href="/nlab/show/sheaf+toposes">sheaf toposes</a> if the original functor preserves <a class="existingWikiWord" href="/nlab/show/covers">covers</a>.</p> <p>(We state this in <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+theory">(∞,1)-category theory</a>, the same statement holds true in plain <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a> by just disregarding all occurrences of “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>”.)</p> <div class="num_prop" id="YonedaExtensionOfLeftExact"> <h6 id="proposition_3">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> be an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</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">\mathcal{C}</annotation></semantics></math> be an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-site">(∞,1)-site</a> with <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-sheaf+%28%E2%88%9E%2C1%29-category">(∞,1)-sheaf (∞,1)-category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sh</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh(\mathcal{C})</annotation></semantics></math>. Then the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Top</mi><mo stretchy="false">(</mo><mi>𝒳</mi><mo>,</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>Func</mi> <mrow><mi>lex</mi><mo>,</mo><mi>leftadj</mi></mrow></msup><mo stretchy="false">(</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><mo>,</mo><mi>𝒳</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>∘</mo><mi>L</mi></mrow></mover><mi>Func</mi><mo stretchy="false">(</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><mo>,</mo><mi>𝒳</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>∘</mo><mi>Y</mi></mrow></mover><mi>Func</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mi>𝒳</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Top(\mathcal{X}, Sh(\mathcal{C})) \simeq Func^{lex, leftadj}(Sh(\mathcal{C}), \mathcal{X}) \stackrel{(-)\circ L }{\longrightarrow} Func(PSh(\mathcal{C}), \mathcal{X}) \stackrel{(-)\circ Y }{\longrightarrow} Func(\mathcal{C}, \mathcal{X}) </annotation></semantics></math></div> <p>given by precomposition with <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stackification">∞-stackification</a>/<a class="existingWikiWord" href="/nlab/show/sheafification">sheafification</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> and with the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Yoneda+embedding">(∞,1)-Yoneda embedding</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/full+and+faithful+%28%E2%88%9E%2C1%29-functor">full and faithful (∞,1)-functor</a>. Moreover, its <a class="existingWikiWord" href="/nlab/show/essential+image">essential image</a> consisist of those <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functors">(∞,1)-functors</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>𝒞</mi><mo>⟶</mo><mi>𝒳</mi></mrow><annotation encoding="application/x-tex">f \colon \mathcal{C} \longrightarrow \mathcal{X}</annotation></semantics></math> which are <a class="existingWikiWord" href="/nlab/show/left+exact+%28%E2%88%9E%2C1%29-functor">left exact</a> and which preserve covers in that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><mi>X</mi><msub><mo stretchy="false">}</mo> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\{U_i \to X\}_i</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/covering">covering</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">\mathcal{C}</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mi>i</mi></msub><mi>f</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>→</mo><mi>f</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\coprod_i f(U_i) \to f(X)</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/effective+epimorphism">effective epimorphism</a> 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">\mathcal{X}</annotation></semantics></math>.</p> </div> <p>This appears as <a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">Lurie, HTT, prop. 6.2.3.20</a>.</p> <div class="num_remark"> <h6 id="remark_2">Remark</h6> <p>Prop. <a class="maruku-ref" href="#YonedaExtensionOfLeftExact"></a> is a central statement in the theory of <a class="existingWikiWord" href="/nlab/show/classifying+toposes">classifying toposes</a>. See there for more.</p> </div> <p>For more discussion of left exactness properties preserved by left Kan extension see also (<a href="#BorceuxDay">Borceux-Day</a>, <a href="#KarazerisProtsonis">Karazeris-Protsonis</a>).</p> <h3 id="AlongFibrations">Kan extension along (op)fibration</h3> <div class="num_prop"> <h6 id="proposition_4">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>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> be a small <a class="existingWikiWord" href="/nlab/show/opfibration">opfibration</a> of categories, 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">\mathcal{C}</annotation></semantics></math> be a category with all small <a class="existingWikiWord" href="/nlab/show/colimits">colimits</a>. Then for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>∈</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">d \in D</annotation></semantics></math> the inclusion</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo>→</mo><mi>f</mi><mo stretchy="false">/</mo><mi>d</mi></mrow><annotation encoding="application/x-tex"> f^{-1}(d) \to f/d </annotation></semantics></math></div> <p>of the <a class="existingWikiWord" href="/nlab/show/fiber">fiber</a> over <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> into the <a class="existingWikiWord" href="/nlab/show/comma+category">comma category</a> given by</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>↦</mo><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><msub><mi>Id</mi> <mi>d</mi></msub><mo>=</mo><msub><mi>Id</mi> <mrow><mi>f</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> c \mapsto (c, Id_{d} = Id_{f(c)}) </annotation></semantics></math></div> <p>has a <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a>. given by</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>f</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>→</mo><mi>d</mi><mo stretchy="false">)</mo><mo>↦</mo><mi>c</mi><mo>′</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> (c, f(c) \to d) \mapsto c' \,, </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>→</mo><mi>c</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">c \to c'</annotation></semantics></math> is a coCartesian lift of <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 stretchy="false">)</mo><mo>→</mo><mi>d</mi></mrow><annotation encoding="application/x-tex">f(c) \to d</annotation></semantics></math>.</p> <p>Therefore (by the discussion <em><a href="final+functor#Examples">here</a></em>) it is a <a class="existingWikiWord" href="/nlab/show/cofinal+functor">cofinal functor</a>. Accordingly, the local formula for the left Kan extension</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> <mo>!</mo></msub><mo>:</mo><mo stretchy="false">[</mo><mi>C</mi><mo>,</mo><mi>𝒟</mi><mo stretchy="false">]</mo><mo>→</mo><mo stretchy="false">[</mo><mi>D</mi><mo>,</mo><mi>𝒟</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> f_! : [C, \mathcal{D}] \to [D, \mathcal{D}] </annotation></semantics></math></div> <p>is equivalently given by taking the colimit over the <a class="existingWikiWord" href="/nlab/show/fiber">fiber</a>:</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> <mo>!</mo></msub><mi>X</mi><mo>:</mo><mi>d</mi><mo>↦</mo><munder><mi>lim</mi> <munder><mo>→</mo><mrow><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo></mrow></munder></munder><mi>X</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> f_! X : d \mapsto \lim_{\underset{f^{-1}(d)}{\to}} X \,. </annotation></semantics></math></div></div> <p>A similar result holds for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categories. See <a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">Lurie, HTT, prop. 4.3.3.10</a>, set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>=</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">S=Y</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mo>=</mo><msub><mo lspace="0em" rspace="thinmathspace">id</mo> <mi>Y</mi></msub></mrow><annotation encoding="application/x-tex">q = \id_Y</annotation></semantics></math>.</p> <h2 id="Examples">Examples</h2> <p>The central point about examples of Kan extensions is:</p> <p><em>Kan extensions are ubiquitous</em>.</p> <p>To a fair extent, <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a> is all about Kan extensions and the other <a class="existingWikiWord" href="/nlab/show/universal+construction">universal construction</a>s: <a class="existingWikiWord" href="/nlab/show/limit">limit</a>s, <a class="existingWikiWord" href="/nlab/show/adjoint+functor">adjoint functor</a>s, <a class="existingWikiWord" href="/nlab/show/representable+functor">representable functor</a>s, which are all special cases of Kan extensions – and Kan extensions are special cases of these.</p> <p>Listing examples of Kan extensions in <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a> is much like listing examples of <a class="existingWikiWord" href="/nlab/show/integral">integral</a>s in <a class="existingWikiWord" href="/nlab/show/analysis">analysis</a>: one can and does fill books with these. (In fact, that analogy has more to it than meets the casual eye: see <a class="existingWikiWord" href="/nlab/show/coend">coend</a> for more).</p> <p>Keeping that in mind, we do list some special cases and special classes of examples that are useful to know. But any list is necessarily wildly incomplete.</p> <h3 id="general">General</h3> <ul> <li> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>′</mo><mo>=</mo></mrow><annotation encoding="application/x-tex">C' = </annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/point">point</a>, the right Kan extension 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/limit">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>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ran</mi><mi>F</mi><mo>≃</mo><mi>lim</mi><mi>F</mi></mrow><annotation encoding="application/x-tex">Ran F \simeq \lim F</annotation></semantics></math> and the left Kan extension is the <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Lan</mi><mi>F</mi><mo>≃</mo><mi>colim</mi><mi>F</mi></mrow><annotation encoding="application/x-tex">Lan F \simeq colim F</annotation></semantics></math>.</p> </li> <li> <p>For <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>Y</mi></mrow><annotation encoding="application/x-tex">f : X \to Y</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/site">morphism of sites</a> coming from a functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mi>t</mi></msup><mo>:</mo><msub><mi>S</mi> <mi>Y</mi></msub><mo>→</mo><msub><mi>S</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">f^t : S_Y \to S_X</annotation></semantics></math> of the underlying categories, the left Kan extension of functors along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mi>t</mi></msup></mrow><annotation encoding="application/x-tex">f^t</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/inverse+image">inverse image</a> operation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>:</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>→</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f^{-1} : PSh(Y) \to PSh(X)</annotation></semantics></math>.</p> </li> <li> <p>see also <a class="existingWikiWord" href="/nlab/show/examples+of+Kan+extensions">examples of Kan extensions</a></p> </li> </ul> <h3 id="nonpointwise_kan_extensions">Non-pointwise Kan extensions</h3> <p>Examples of Kan extensions that are not point-wise are discussed in <a href="#Borceux">Borceux, exercise 3.9.7</a>.</p> <h3 id="restriction_and_extension_of_sheaves">Restriction and extension of sheaves</h3> <p>For more on the following see also</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/restriction+and+extension+of+sheaves">restriction and extension of sheaves</a></li> </ul> <p>The basic example for left Kan extensions using the above pointwise formula, is in the construction of the pullback of sheaves along a morphism of topological spaces. Let <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>Y</mi></mrow><annotation encoding="application/x-tex">f:X\to Y</annotation></semantics></math> be a continuous map and <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> a presheaf over <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>. Then the formula <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>f</mi> <mo>*</mo></msub><mi>F</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo>=</mo><mi>F</mi><mo stretchy="false">(</mo><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(f_* F)(U) = F(f^{-1}(U))</annotation></semantics></math> clearly defines a presheaf <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mo>*</mo></msub><mi>F</mi></mrow><annotation encoding="application/x-tex">f_* F</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math>, which is in fact a sheaf if <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. On the other hand, given a presheaf <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> we can not define pullback presheaf <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>G</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo>=</mo><mi>G</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(f^{-1} G)(V)=G(f(V))</annotation></semantics></math> because <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(V)</annotation></semantics></math> might not be open in general (unless <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 open map). For Grothendieck sites such <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(V)</annotation></semantics></math> would not make even sense. But one can consider approximating from above by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo stretchy="false">(</mo><mi>W</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G(W)</annotation></semantics></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi><mo>⊃</mo><mi>f</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">W\supset f(V)</annotation></semantics></math> which are open and take a colimit of this diagram of inclusions (all <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> are bigger, so getting down to the lower bound means going reverse to the direction of inclusions). But inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo>⊂</mo><mi>W</mi></mrow><annotation encoding="application/x-tex">f(V)\subset W</annotation></semantics></math> implies <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>⊂</mo><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⊂</mo><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>W</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">V\subset f^{-1}(f(V))\subset f^{-1}(W)</annotation></semantics></math>. The latter identity <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>⊂</mo><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>W</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">V\subset f^{-1}(W)</annotation></semantics></math> involves <em>only open sets</em>. Thus we take a colimit over the comma category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">↓</mo><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(V\downarrow f^{-1})</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>. If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> is a sheaf, the colimit <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G(V)</annotation></semantics></math> understood as a rule <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>↦</mo><mi>G</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">V\mapsto G(V)</annotation></semantics></math> is still not a sheaf, we need to <a class="existingWikiWord" href="/nlab/show/sheafification">sheafify</a>. The result is sheaf-theoretic pullback</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>G</mi><mo>=</mo><mi mathvariant="normal">sheafify</mi><mo stretchy="false">(</mo><mi>V</mi><mo>↦</mo><msub><mi mathvariant="normal">colim</mi> <mrow><mi>V</mi><mo>↪</mo><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>W</mi></mrow></msub><mi>G</mi><mo stretchy="false">(</mo><mi>W</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="normal">sheafify</mi><mo stretchy="false">(</mo><mi>V</mi><mo>↦</mo><msub><mi mathvariant="normal">colim</mi> <mrow><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">↓</mo><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></mrow></msub><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> f^{-1}G = \mathrm{sheafify}(V\mapsto\mathrm{colim}_{V\hookrightarrow f^{-1}W} G(W)) = \mathrm{sheafify}(V\mapsto\mathrm{colim}_{(V\downarrow f^{-1})} G) </annotation></semantics></math></div> <p>which is a sheaf, and one can analyze this construction to show that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">f^{-1}</annotation></semantics></math> is a left adjoint to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">f_*</annotation></semantics></math>. This usage of left Kan extension persists in the more general case of Grothendieck topologies.</p> <h3 id="ExamplesKanExtensionsInPhysics">Kan extension in physics</h3> <p>We list here some occurrences of Kan extensions in <a class="existingWikiWord" href="/nlab/show/physics">physics</a>.</p> <p>Notice that since, by the above discussion, Kan extensions are ubiquitous in <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a> and are essentially equivalent to other standard <a class="existingWikiWord" href="/nlab/show/universal+constructions">universal constructions</a> such as notably <a class="existingWikiWord" href="/nlab/show/colimit">co</a>/<a class="existingWikiWord" href="/nlab/show/limits">limits</a>, to the extent that there is a relation between <a class="existingWikiWord" href="/nlab/show/higher+category+theory+and+physics">category theory and physics</a> at all, it necessarily also involves Kan extensions, in some guise. But here is a list of some example where they appear rather explicitly.</p> <ul> <li> <p>In <a class="existingWikiWord" href="/nlab/show/extended+quantum+field+theory">extended quantum field theory</a> on open and closed manifolds, usually the theory “in the bulk” (on closed manifolds) is induced by “extending” that “<a class="existingWikiWord" href="/nlab/show/boundary+field+theory">on the boundary</a>”, and in good cases this extension is explicitly a (<a class="existingWikiWord" href="/nlab/show/homotopy+Kan+extension">homotopy</a>)-Kan extension. This is the case notably for <a class="existingWikiWord" href="/nlab/show/2d+TQFT">2d TQFT</a> in the form of <a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a> (<a href="TCFT#Costello04">Costello 04</a>), see at <em><a href="TCFT#Classification">TCFT – Classification</a></em> for details.</p> </li> <li> <p>When <a class="existingWikiWord" href="/nlab/show/path+integral">path integral</a> <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a> is formalized in terms of <a class="existingWikiWord" href="/nlab/show/fiber+integration+in+generalized+cohomology">fiber integration in generalized cohomology</a> (as surveyed at <em><a class="existingWikiWord" href="/nlab/show/motivic+quantization">motivic quantization</a></em>) then the push-forward step, hence the path integral itself, is given by left <a class="existingWikiWord" href="/nlab/show/homotopy+Kan+extension">homotopy Kan extension</a> of <a class="existingWikiWord" href="/nlab/show/parameterized+spectrum">parameterized spectra</a>. For explicit details see (<a class="existingWikiWord" href="/schreiber/show/master+thesis+Nuiten">Nuiten 13, section 4.1</a>), also (<a class="existingWikiWord" href="/schreiber/show/Homotopy-type+semantics+for+quantization">Schreiber 14, section 6.2</a>). By example 6.3, a special case of this is the integration formulas via Kan extension in (<a class="existingWikiWord" href="/nlab/show/Ambidexterity+in+K%28n%29-Local+Stable+Homotopy+Theory">Hopkins-Lurie 14, section 4</a>).</p> </li> </ul> <h2 id="remark_on_terminology_pushforward_vs_pullback">Remark on terminology: pushforward vs. pullback</h2> <p>Generally, for <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>C</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">p : C \to C'</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/functor">functor</a>, the induced “precomposition” functor on <a class="existingWikiWord" href="/nlab/show/functor+category">functor categories</a></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><mo>,</mo><mi>D</mi><mo stretchy="false">]</mo><mover><mo>→</mo><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>∘</mo><mi>p</mi></mrow></mover><mo stretchy="false">[</mo><mi>C</mi><mo>,</mo><mi>D</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> [C', D] \stackrel{- \circ p}{\to} [C,D] </annotation></semantics></math></div> <p>is spoken of as <em>pulling back</em> a functor on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">C'</annotation></semantics></math> to a functor on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>, as this operation goes in the direction opposite to that of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> itself. For this reason, we have above denoted this functor by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>p</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">p^*</annotation></semantics></math>. Likewise, one might call the (left or right) Kan extensions along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> a <em>push forward</em> of functors from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> to functors on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">C'</annotation></semantics></math>.</p> <p>This notation also coincides with that for <a class="existingWikiWord" href="/nlab/show/geometric+morphisms">geometric morphisms</a> in one case: any functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo lspace="verythinmathspace">:</mo><mi>C</mi><mo>→</mo><mi>C</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">p\colon C\to C'</annotation></semantics></math> between small categories induces a geometric morphism <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>Set</mi><mo stretchy="false">]</mo><mo>→</mo><mo stretchy="false">[</mo><mi>C</mi><mo>′</mo><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[C,Set] \to [C',Set]</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/presheaf+topos">presheaf toposes</a>, whose <a class="existingWikiWord" href="/nlab/show/inverse+image">inverse image</a> is the above <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>p</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">p^*</annotation></semantics></math> and whose <a class="existingWikiWord" href="/nlab/show/direct+image">direct image</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">p_*</annotation></semantics></math> is the right Kan extension functor. Note that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>p</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">p^*</annotation></semantics></math> preserves (finite) limits, as required of an inverse image functor, since it has a left adjoint, namely <em>left</em> Kan extension.</p> <p>On the other hand, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> is additionally a <a class="existingWikiWord" href="/nlab/show/flat+functor">flat functor</a>, then the above precomposition functor is also the <em>direct</em> image of a geometric morphism, whose inverse image is given by <em>left</em> Kan extension (which preserves finite limits when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> is flat). More generally, if <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> and <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><msup><mo stretchy="false">)</mo> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">(C')^{op}</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/sites">sites</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>p</mi> <mi>op</mi></msup><mo lspace="verythinmathspace">:</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>→</mo><mo stretchy="false">(</mo><mi>C</mi><mo>′</mo><msup><mo stretchy="false">)</mo> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">p^{op}\colon C^{op}\to (C')^{op}</annotation></semantics></math> is flat and preserves covering families (i.e. it is a <a class="existingWikiWord" href="/nlab/show/morphism+of+sites">morphism of sites</a>), then precomposition is the direct image of a geometric morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sh</mi><mo stretchy="false">(</mo><msup><mi>C</mi> <mi>op</mi></msup><mo stretchy="false">)</mo><mo>→</mo><mi>Sh</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>C</mi><mo>′</mo><msup><mo stretchy="false">)</mo> <mi>op</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh(C^{op})\to Sh((C')^{op})</annotation></semantics></math> between sheaf toposes.</p> <p>For example, <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> and <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><msup><mo stretchy="false">)</mo> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">(C')^{op}</annotation></semantics></math> might be the <a class="existingWikiWord" href="/nlab/show/posets">posets</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Open</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Open(X)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Open</mi><mo stretchy="false">(</mo><mi>X</mi><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Open(X')</annotation></semantics></math> of open subsets of <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> (or <a class="existingWikiWord" href="/nlab/show/locales">locales</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">X'</annotation></semantics></math> and inclusions, in which case</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Open</mi><mo stretchy="false">(</mo><mi>X</mi><msup><mo stretchy="false">)</mo> <mi>op</mi></msup><mo>→</mo><mi>Open</mi><mo stretchy="false">(</mo><mi>X</mi><mo>′</mo><msup><mo stretchy="false">)</mo> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex"> Open(X)^{op} \to Open(X')^{op} </annotation></semantics></math></div> <p>come from continuous maps of topological spaces going the other way</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>←</mo><mi>X</mi><mo>′</mo><mo>:</mo><mi>f</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> X \leftarrow X' : f \,, </annotation></semantics></math></div> <p>via the usual inverse image <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>:</mo><mi>O</mi><mo stretchy="false">(</mo><mi>X</mi><msup><mo stretchy="false">)</mo> <mi>op</mi></msup><mo>→</mo><mi>O</mi><mo stretchy="false">(</mo><mi>X</mi><mo>′</mo><msup><mo stretchy="false">)</mo> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">f^{-1} : O(X)^{op} \to O(X')^{op}</annotation></semantics></math> of open subsets.</p> <p>Thus, in such cases, the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>p</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">p^*</annotation></semantics></math>, which looks like a pullback of functors along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>=</mo><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">p = f^{-1}</annotation></semantics></math>, corresponds geometrically to a <em>push-forward</em> of (pre)sheaves along <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>. Therefore, in <a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a> literature (such as <a class="existingWikiWord" href="/nlab/show/Categories+and+Sheaves">Categories and Sheaves</a>) the precomposition functor induced by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> is usually denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">p_*</annotation></semantics></math> and not <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>p</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">p^*</annotation></semantics></math>.</p> <p>It is however noteworthy that also the opposite perspective does occur in geometrically motivated examples. For instance</p> <ul> <li> <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 the <a class="existingWikiWord" href="/nlab/show/discrete+category">discrete category</a> on smooth space and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>=</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">D = U(1)</annotation></semantics></math> is the discrete category on the smooth space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> underlying the Lie group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(1)</annotation></semantics></math>, then smooth functors (i.e. functors <a class="existingWikiWord" href="/nlab/show/internal+category">internal to</a> <a class="existingWikiWord" href="/nlab/show/generalized+smooth+space">smooth spaces</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> can be identified with smooth <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(1)</annotation></semantics></math>-valued functions on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, and the functor on these functor categories induced by a smooth 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>C</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">p : C \to C'</annotation></semantics></math> does correspond to the familiar notion of <em>pullback</em> of functions;</p> </li> <li> <p>and similar in higher degrees: if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>=</mo><msub><mi>P</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C = P_1(X)</annotation></semantics></math> is the smooth path groupoid of a smooth space and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>=</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">D = \mathbf{B} U(1)</annotation></semantics></math> the smooth <a class="existingWikiWord" href="/nlab/show/group">group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(1)</annotation></semantics></math> regarded as a one-object <a class="existingWikiWord" href="/nlab/show/Lie+groupoid">Lie groupoid</a>, then smooth functors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>→</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">C \to D</annotation></semantics></math> correspond to smooth 1-forms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∈</mo><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\in \Omega^1(X)</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, and precomposition with a smooth functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><msub><mi>P</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>P</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p : P_1(X) \to P_1(X')</annotation></semantics></math> corresponds to the familiar notion of <em>pullback</em> of 1-forms.</p> </li> </ul> <p>This means that whether or not Kan extensions correspond geometrically to pushforward or to pullback depends on the way (covariant or contravariant) in which the domain categories <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>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">C'</annotation></semantics></math> are identified with geometric entities.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><strong>Kan extension</strong>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Kan+extension">(∞,1)-Kan extension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/base+change">base change</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/dependent+sum">dependent sum</a>, <a class="existingWikiWord" href="/nlab/show/dependent+product">dependent product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dependent+sum+type">dependent sum type</a>, <a class="existingWikiWord" href="/nlab/show/dependent+product+type">dependent product type</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/internal+%28co-%29limit">internal (co-)limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/codensity+monad">codensity monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/shape+theory">shape theory</a></p> </li> </ul> <h2 id="references">References</h2> <p>The original definition is due to <a class="existingWikiWord" href="/nlab/show/Daniel+M.+Kan">Daniel M. Kan</a>, found in the paper that also defined <a class="existingWikiWord" href="/nlab/show/adjoint+functors">adjoint functors</a> and <a class="existingWikiWord" href="/nlab/show/limits">limits</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Daniel+M.+Kan">Daniel M. Kan</a>, <em>Adjoint functors</em>, Transactions of the American Mathematical Society 87:2 (1958), 294–294 (<a href="https://doi.org/10.1090/s0002-9947-1958-0131451-0">doi:10.1090/s0002-9947-1958-0131451-0</a>).</li> </ul> <p>Textbook sources include</p> <ul> <li id="Borceux"> <p><a class="existingWikiWord" href="/nlab/show/Francis+Borceux">Francis Borceux</a>, section 3.7 of <em><a class="existingWikiWord" href="/nlab/show/Handbook+of+Categorical+Algebra">Handbook of Categorical Algebra</a> I</em></p> </li> <li> <p>Kashiwara and Shapira, section 2.3 in <em><a class="existingWikiWord" href="/nlab/show/Categories+and+Sheaves">Categories and Sheaves</a></em></p> </li> </ul> <p>The book</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Saunders+MacLane">Saunders MacLane</a>, <em><a class="existingWikiWord" href="/nlab/show/Categories+Work">Categories for the working mathematician</a></em></li> </ul> <p>has a famous treatment of Kan extensions with a statement: “The notion of Kan extensions subsumes all the other fundamental concepts in category theory”. Of course, many other fundamental concepts of category theory can also be regarded as subsuming all the others.</p> <p>Lecture notes with an eye towards applications in <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> include</p> <ul> <li id="Riehl"><a class="existingWikiWord" href="/nlab/show/Emily+Riehl">Emily Riehl</a>, Chapter 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>]</li> </ul> <p>For Kan extensions in the context of <a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a> see</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Eduardo+Dubuc">Eduardo Dubuc</a>, <em>Kan extensions in enriched category theory</em>, Lecture Notes in Mathematics, Vol. 145 Springer-Verlag, Berlin-New York 1970 xvi+173 pp.</li> </ul> <p>and chapter 4 of</p> <ul> <li id="Kelly"><a class="existingWikiWord" href="/nlab/show/Max+Kelly">Max Kelly</a>, <em>Basic Concepts of Enriched Category Theory</em>, <p>Cambridge University Press, Lecture Notes in Mathematics 64, 1982, Republished in: Reprints in Theory and Applications of Categories, No. 10 (2005) pp. 1-136 (<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> </li> </ul> <p>The <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+theory">(∞,1)-category theory</a> notion is discussed in section 4.3 of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, <em><a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">Higher Topos Theory</a></em></li> </ul> <p>For uses of Kan extension in the study of <a class="existingWikiWord" href="/nlab/show/algebra+over+a+Lawvere+theory">algebras over an algebraic theory</a> see</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Paul-Andr%C3%A9+Melli%C3%A8s">Paul-André Melliès</a> and <a class="existingWikiWord" href="/nlab/show/Nicholas+Tabareau">Nicholas Tabareau</a>, <em>Free models of T-algebraic theories computed as Kan extensions</em> (<a href="http://hal.archives-ouvertes.fr/hal-00339331/fr/">web</a>)</li> </ul> <p>Preservation of certain limits by left Kan extended functors is discussed in</p> <ul> <li id="BorceuxDay"> <p><a class="existingWikiWord" href="/nlab/show/Francis+Borceux">Francis Borceux</a>, and <a class="existingWikiWord" href="/nlab/show/Brian+Day">Brian Day</a>, <em>On product-preserving Kan extension</em>, Bulletin of the Australian Mathematical Society, Vol 17 (1977), 247-255 (<a href="http://journals.cambridge.org/download.php?file=%2FBAZ%2FBAZ17_02%2FS0004972700010455a.pdf&code=e8c9ebb215a55a0891f4e05f6aebbcf2">pdf</a>)</p> </li> <li id="KarazerisProtsonis"> <p>Panagis Karazeris, Grigoris Protsonis, <em>Left Kan extensions preserving finite products</em>, (<a href="http://www.math.upatras.gr/~pkarazer/publications/topsift.pdf">pdf</a>)</p> </li> </ul> <p>The general notion of extensions of <a class="existingWikiWord" href="/nlab/show/1-morphisms">1-morphisms</a> in <a class="existingWikiWord" href="/nlab/show/2-categories">2-categories</a> is discussed in</p> <ul> <li>John W. Gray, <em>Quasi-Kan extensions for 2-categories</em>, Bull. Amer. Math. Soc. 80:1 (1974) 142-147 <a href="https://pdfs.semanticscholar.org/346b/9ae758d63f12e8e5c635c85cb30f6c568fd9.pdf">pdf</a></li> <li id="Lack09"><a class="existingWikiWord" href="/nlab/show/Steve+Lack">Steve Lack</a>, <em>A 2-categories companion</em>, in <a class="existingWikiWord" href="/nlab/show/John+Baez">John Baez</a>, <a class="existingWikiWord" href="/nlab/show/Peter+May">Peter May</a>, <em><a class="existingWikiWord" href="/nlab/show/Towards+Higher+Categories">Towards Higher Categories</a></em>, Springer, (2009) (<a href="http://arxiv.org/abs/math/0702535">arXiv:math/0702535</a>)</li> <li><a class="existingWikiWord" href="/nlab/show/Ross+Street">Ross Street</a>, <em>Pointwise extensions and sketches in bicategories</em>, <a href="https://arxiv.org/abs/1409.6427">arXiv:1409.6427</a></li> <li id="Roald13">Seerp Roald Koudenburg, <em>Algebraic weighted colimits</em>, <a href="http://arxiv.org/abs/1304.4079">arXiv</a></li> </ul> <p>For the notion of (2-dimensional) (pointwise) bi-Kan extensions of pseudofunctors, see</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Fernando+Lucatelli+Nunes">Fernando Lucatelli Nunes</a>, <em>On biadjoint triangles</em>, TAC <a href="http://tac.mta.ca/tac/volumes/31/9/31-09abs.html">31-9</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fernando+Lucatelli+Nunes">Fernando Lucatelli Nunes</a>, <em>Pseudo-Kan extensions and descent theory</em>, TAC <a href="http://www.tac.mta.ca/tac/volumes/33/15/33-15abs.html">33-15</a></p> </li> </ul> <p>and its applications to the theory of (2-dimensional) <a class="existingWikiWord" href="/nlab/show/flat+functors">flat functors</a> can be seen in</p> <ul> <li>M.E. Descotte, E.J. Dubuc, M. Szyld, <em>On the notion of flat 2-functors</em>, <a href="https://www.sciencedirect.com/science/article/abs/pii/S0001870818301968">Adv. Math</a>, arXiv:<a href="https://arxiv.org/abs/1610.09429">1610.09429</a></li> </ul> <p>For a treatment of left Kan extensions as ‘partial colimits’, see</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Paolo+Perrone">Paolo Perrone</a>, <a class="existingWikiWord" href="/nlab/show/Walter+Tholen">Walter Tholen</a>, <em>Kan extensions are <a class="existingWikiWord" href="/nlab/show/partial+evaluation">partial</a> colimits</em>, Applied Categorical Structures, 2022. (<a href="https://arxiv.org/abs/2101.04531">arXiv:2101.04531</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on November 13, 2024 at 18:40:15. 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