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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="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> <h4 id="yoneda_lemma">Yoneda lemma</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a></strong></p> <p><strong>Ingredients</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category">category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+presheaves">category of presheaves</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/representable+presheaf">representable presheaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yoneda+embedding">Yoneda embedding</a></p> </li> </ul> <p><strong>Incarnations</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+Yoneda+lemma">enriched Yoneda lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/co-Yoneda+lemma">co-Yoneda lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yoneda+reduction">Yoneda reduction</a></p> </li> </ul> <p><strong>Properties</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/free+cocompletion">free cocompletion</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yoneda+extension">Yoneda extension</a></p> </li> </ul> <p><strong>Universal aspects</strong></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/universal+construction">universal construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+element">universal element</a></p> </li> </ul> <p><strong>Classification</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a>, <a class="existingWikiWord" href="/nlab/show/classifying+stack">classifying stack</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/moduli+space">moduli space</a>, <a class="existingWikiWord" href="/nlab/show/moduli+stack">moduli stack</a>, <a class="existingWikiWord" href="/nlab/show/derived+moduli+space">derived moduli space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/classifying+topos">classifying topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/subobject+classifier">subobject classifier</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+principal+bundle">universal principal bundle</a>, <a class="existingWikiWord" href="/nlab/show/universal+principal+%E2%88%9E-bundle">universal principal ∞-bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/classifying+morphism">classifying morphism</a></p> </li> </ul> <p><strong>Induced theorems</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a></li> </ul> <p>…</p> <p><strong>In higher category theory</strong></p> <p><a class="existingWikiWord" href="/nlab/show/Yoneda+lemma+for+higher+categories">Yoneda lemma for higher categories</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Yoneda+lemma+for+%28%E2%88%9E%2C1%29-categories">Yoneda lemma for (∞,1)-categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yoneda+lemma+for+bicategories">Yoneda lemma for bicategories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yoneda+lemma+for+tricategories">Yoneda lemma for tricategories</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/Yoneda+lemma+-+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='#technical_details'>Technical details</a></li> <ul> <li><a href='#proofs'>Proofs</a></li> </ul> <li><a href='#free_cocompletion_of_large_categories'>Free cocompletion of large categories</a></li> <li><a href='#free_cocompletion_as_a_pseudomonad'>Free cocompletion as a pseudomonad</a></li> <li><a href='#in_higher_category_theory'>In higher category theory</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>Passing from a <a class="existingWikiWord" href="/nlab/show/category">category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> to its <a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a> category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PSh</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">PSh(C) := [C^{op},Set]</annotation></semantics></math> may be regarded as the operation of “freely adjoining <a class="existingWikiWord" href="/nlab/show/colimits">colimits</a> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>”.</p> <p>A slightly more precise version of this statement is that the <a class="existingWikiWord" href="/nlab/show/Yoneda+embedding">Yoneda embedding</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>:</mo><mi>C</mi><mo>↪</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Y : C \hookrightarrow PSh(C) </annotation></semantics></math></div> <p>is the <strong><a class="existingWikiWord" href="/nlab/show/free+construction">free</a> <a class="existingWikiWord" href="/nlab/show/cocomplete+category">cocompletion</a></strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a> of the <a class="existingWikiWord" href="/nlab/show/Yoneda+embedding">Yoneda embedding</a> is expressed in terms of the <a class="existingWikiWord" href="/nlab/show/Yoneda+extension">Yoneda extension</a> of any <a class="existingWikiWord" href="/nlab/show/functor">functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">F : C \to D</annotation></semantics></math> to 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> with colimits.</p> <p>There is also an alternative construction of the free cocomplete category, given by the <a class="existingWikiWord" href="/nlab/show/free+strict+cocompletion">free strict cocompletion</a>.</p> <h2 id="technical_details">Technical details</h2> <p>The rough statement is that the <a class="existingWikiWord" href="/nlab/show/Yoneda+embedding">Yoneda embedding</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>y</mi> <mi>S</mi></msub><mo>:</mo><mi>S</mi><mo>→</mo><msup><mi>Set</mi> <mrow><msup><mi>S</mi> <mi>op</mi></msup></mrow></msup></mrow><annotation encoding="application/x-tex">y_S: S \to Set^{S^{op}}</annotation></semantics></math></div> <p>of a <a class="existingWikiWord" href="/nlab/show/small+category">small category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> into the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Set</mi> <mrow><msup><mi>S</mi> <mi>op</mi></msup></mrow></msup></mrow><annotation encoding="application/x-tex">Set^{S^{op}}</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/presheaf">presheaves</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> is universal among functors from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> into <a class="existingWikiWord" href="/nlab/show/cocomplete+category">cocomplete categories</a>. Technically, this should be understood in an appropriate <a class="existingWikiWord" href="/nlab/show/2-category">2-categorical</a> sense: given a <a class="existingWikiWord" href="/nlab/show/functor">functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mi>S</mi><mo>→</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">F: S \to D</annotation></semantics></math> where <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/small+colimit">small</a>-)cocomplete, there exists a unique (up to <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>) <a class="existingWikiWord" href="/nlab/show/cocontinuous+functor">cocontinuous</a> <a class="existingWikiWord" href="/nlab/show/extension">extension</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mi>F</mi><mo stretchy="false">^</mo></mover><mo>:</mo><msup><mi>Set</mi> <mrow><msup><mi>S</mi> <mi>op</mi></msup></mrow></msup><mo>→</mo><mi>D</mi><mo>,</mo></mrow><annotation encoding="application/x-tex">\hat{F}: Set^{S^{op}} \to D,</annotation></semantics></math></div> <p>called the <a class="existingWikiWord" href="/nlab/show/Yoneda+extension">Yoneda extension</a>, meaning that <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><msub><mi>y</mi> <mi>S</mi></msub><mo>≅</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">\hat{F} y_S \cong F</annotation></semantics></math> and <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> <a class="existingWikiWord" href="/nlab/show/preserved+limit">preserves</a> <a class="existingWikiWord" href="/nlab/show/small+colimits">small colimits</a>.</p> <p>Put slightly differently: let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cocomp</mi></mrow><annotation encoding="application/x-tex">Cocomp</annotation></semantics></math> denote the 2-category of cocomplete categories, cocontinuous functors, and natural transformations between them. Then for cocomplete <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>, the Yoneda embedding <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mi>S</mi><mo>:</mo><mi>S</mi><mo>→</mo><msup><mi>Set</mi> <mrow><msup><mi>S</mi> <mi>op</mi></msup></mrow></msup></mrow><annotation encoding="application/x-tex">y S: S \to Set^{S^{op}}</annotation></semantics></math> induces by restriction 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>res</mi> <mrow><mi>y</mi><mi>S</mi></mrow></msub><mi>D</mi><mo>:</mo><mi>Cocomp</mi><mo stretchy="false">(</mo><msup><mi>Set</mi> <mrow><msup><mi>S</mi> <mi>op</mi></msup></mrow></msup><mo>,</mo><mi>D</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Cat</mi><mo stretchy="false">(</mo><mi>S</mi><mo>,</mo><mi>D</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">res_{y S} D: Cocomp(Set^{S^{op}}, D) \to Cat(S, D)</annotation></semantics></math></div> <p>that is an <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence</a> for each <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>, one that is <a class="existingWikiWord" href="/nlab/show/2-natural+transformation">2-natural</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>. In fact we show that the Yoneda extension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>↦</mo><mover><mi>F</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">F \mapsto \hat{F}</annotation></semantics></math> gives a functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ext</mi> <mrow><mi>y</mi><mi>S</mi></mrow></msub><mi>D</mi><mo>:</mo><mi>Cat</mi><mo stretchy="false">(</mo><mi>S</mi><mo>,</mo><mi>D</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Cocomp</mi><mo stretchy="false">(</mo><msup><mi>Set</mi> <mrow><msup><mi>S</mi> <mi>op</mi></msup></mrow></msup><mo>,</mo><mi>D</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">ext_{y S}D: Cat(S, D) \to Cocomp(Set^{S^{op}}, D)</annotation></semantics></math> that is a left adjoint to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>res</mi> <mrow><mi>y</mi><mi>S</mi></mrow></msub><mi>D</mi></mrow><annotation encoding="application/x-tex">res_{y S}D</annotation></semantics></math>, giving a 2-natural adjoint equivalence.</p> <p>As a first step, we construct the desired extension <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> as a <a class="existingWikiWord" href="/nlab/show/adjoint+functor">left adjoint</a> to the functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>→</mo><msup><mi>Set</mi> <mrow><msup><mi>S</mi> <mi>op</mi></msup></mrow></msup><mo>:</mo><mi>d</mi><mo>↦</mo><msub><mi>hom</mi> <mi>D</mi></msub><mo stretchy="false">(</mo><mi>F</mi><mo>−</mo><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex">D \to Set^{S^{op}}: d \mapsto \hom_D(F-, d).</annotation></semantics></math></div> <p>Being a left adjoint, <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> is cocontinuous.</p> <p>An explicit formula for the left adjoint is given by the <a class="existingWikiWord" href="/nlab/show/weighted+colimit">weighted colimit</a>, <a class="existingWikiWord" href="/nlab/show/end">coend</a>, or <a class="existingWikiWord" href="/nlab/show/tensor+product+of+functors">tensor product of functors</a> formula</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mi>F</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><mi>X</mi><msub><mo>⊗</mo> <mi>S</mi></msub><mi>F</mi><mo>=</mo><msup><mo>∫</mo> <mrow><mi>s</mi><mo>:</mo><mi>S</mi></mrow></msup><mi>X</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>F</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\hat{F}(X) = X \otimes_S F = \int^{s: S} X(s) \cdot F(s)</annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>⋅</mo><mi>d</mi></mrow><annotation encoding="application/x-tex">S \cdot d</annotation></semantics></math> is notation for <a class="existingWikiWord" href="/nlab/show/power">copowering</a> (or tensoring) an object <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> by a set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> (in this case, a coproduct of an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-indexed set of copies 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>). This formula recurs frequently throughout this wiki; see also <a class="existingWikiWord" href="/nlab/show/nerve">nerve</a>, <a class="existingWikiWord" href="/nlab/show/Day+convolution">Day convolution</a>.</p> <p>This “free cocompletion” property generalizes to <a class="existingWikiWord" href="/nlab/show/enriched+category">enriched category</a> theory. If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> is complete, cocomplete, symmetric monoidal closed, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> is a small <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-enriched category, then the enriched presheaf category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>V</mi> <mrow><msup><mi>S</mi> <mi>op</mi></msup></mrow></msup></mrow><annotation encoding="application/x-tex">V^{S^{op}}</annotation></semantics></math> is a free <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>-cocompletion of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>. The explicit meaning is analogous to the case where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>=</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">V = Set</annotation></semantics></math>, where all ordinary category concepts are replaced by their <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-enriched analogues; in particular, the notion of “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-cocontinuous functor” refers to preservation of enriched weighted colimits (not just ordinary conical colimits).</p> <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 not small, then its free cocompletion still exists, but it is not the category of all presheaves 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>. Rather, it is the category of <a class="existingWikiWord" href="/nlab/show/small+presheaves">small presheaves</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>, i.e. presheaves that are small colimits of representables.</p> <h3 id="proofs">Proofs</h3> <div class="num_prop" id="FreeCocompletion"> <h6 id="proposition">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/small+category">small category</a>, its <a class="existingWikiWord" href="/nlab/show/Yoneda+embedding">Yoneda embedding</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mover><mo>↪</mo><mi>y</mi></mover><mo stretchy="false">[</mo><msup><mi>𝒞</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathcal{C} \overset{y}{\hookrightarrow} [\mathcal{C}^{op}, Set]</annotation></semantics></math> exhibits the <a class="existingWikiWord" href="/nlab/show/category+of+presheaves">category of presheaves</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>𝒞</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\mathcal{C}^{op}, Set]</annotation></semantics></math> as the <em>free co-completion</em> 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">\mathcal{C}</annotation></semantics></math>, in that it is a <a class="existingWikiWord" href="/nlab/show/universal+morphism">universal morphism</a> (as in <a href="adjoint+functor#UniversalArrow">this Def.</a> but “up to natural isomorphism”) into a <a class="existingWikiWord" href="/nlab/show/cocomplete+category">cocomplete category</a>, in that:</p> <ol> <li> <p>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">\mathcal{D}</annotation></semantics></math> any <a class="existingWikiWord" href="/nlab/show/cocomplete+category">cocomplete category</a>;</p> </li> <li> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>𝒞</mi><mo>⟶</mo><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">F \;\colon\; \mathcal{C} \longrightarrow \mathcal{D}</annotation></semantics></math> any <a class="existingWikiWord" href="/nlab/show/functor">functor</a>;</p> </li> </ol> <p>there is 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><mover><mi>F</mi><mo>˜</mo></mover><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">[</mo><msup><mi>𝒞</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo><mo>⟶</mo><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">\widetilde F \;\colon\; [\mathcal{C}^{op}, Set] \longrightarrow \mathcal{D}</annotation></semantics></math>, unique up to <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a>, such that</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>F</mi><mo>˜</mo></mover></mrow><annotation encoding="application/x-tex">\widetilde F</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/preserved+limit">preserves</a> all <a class="existingWikiWord" href="/nlab/show/colimits">colimits</a>,</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>F</mi><mo>˜</mo></mover></mrow><annotation encoding="application/x-tex">\widetilde F</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/extension">extends</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> through the <a class="existingWikiWord" href="/nlab/show/Yoneda+embedding">Yoneda embedding</a>, in that the following <a class="existingWikiWord" href="/nlab/show/commuting+diagram">diagram commutes</a>, up to <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a>:</p> </li> </ol> <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></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mi>y</mi></msup><mo>↙</mo></mtd> <mtd><mo>⇙</mo></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mi>F</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mpadded width="0"><mrow><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mo stretchy="false">[</mo><msup><mi>𝒞</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow></mpadded></mtd> <mtd></mtd> <mtd><munder><mo>⟶</mo><mover><mi>F</mi><mo>˜</mo></mover></munder></mtd> <mtd></mtd> <mtd><mi>𝒟</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && \mathcal{C} \\ & {}^{y}\swarrow &\swArrow& \searrow^{\mathrlap{F}} \\ \mathrlap{ \!\!\!\!\!\!\!\!\!\!\!\!\! [\mathcal{C}^{op}, Set] } && \underset{ \widetilde F }{\longrightarrow} && \mathcal{D} } </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof">Proof</h6> <p>The last condition says that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>F</mi><mo>˜</mo></mover></mrow><annotation encoding="application/x-tex">\widetilde F</annotation></semantics></math> is fixed on <a class="existingWikiWord" href="/nlab/show/representable+presheaves">representable presheaves</a>, up to isomorphism, by</p> <div class="maruku-equation" id="eq:YonedaRestriction"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mi>F</mi><mo>˜</mo></mover><mo stretchy="false">(</mo><mi>y</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"> \widetilde F( y(c) ) \simeq F(c) </annotation></semantics></math></div> <p>and in fact <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">naturally</a> so:</p> <div class="maruku-equation" id="eq:FunctorialYonedaRestriction"><span class="maruku-eq-number">(2)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>c</mi> <mn>1</mn></msub></mtd> <mtd></mtd> <mtd><mover><mi>F</mi><mo>˜</mo></mover><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mo>≃</mo></mtd> <mtd><mi>F</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>f</mi></mpadded></msup><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd> <mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mpadded></msup><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo maxsize="1.2em" minsize="1.2em">↓</mo> <mpadded width="0"><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>c</mi> <mn>2</mn></msub></mtd> <mtd></mtd> <mtd><mover><mi>F</mi><mo>˜</mo></mover><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mo>≃</mo></mtd> <mtd><mi>F</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ c_1&& \widetilde F( y(c_1) ) &\simeq& F(c_1) \\ {}^{\mathllap{f}}\big\downarrow && {}^{\mathllap{ F(y(f)) }}\big\downarrow && \big\downarrow^{\mathrlap{ F(f) }} \\ c_2 && \widetilde F (y(c_2)) &\simeq& F(c_2) } </annotation></semantics></math></div> <p>But the <a class="existingWikiWord" href="/nlab/show/co-Yoneda+lemma">co-Yoneda lemma</a> expresses every <a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo>∈</mo><mo stretchy="false">[</mo><msup><mi>𝒞</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathbf{X} \in [\mathcal{C}^{op}, Set]</annotation></semantics></math> as a <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a> of <a class="existingWikiWord" href="/nlab/show/representable+presheaves">representable presheaves</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>X</mi></mstyle><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><msup><mo>∫</mo> <mrow><mi>c</mi><mo>∈</mo><mi>𝒞</mi></mrow></msup><mi>y</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>⋅</mo><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{X} \;\simeq\; \int^{c \in \mathcal{C}} y(c) \cdot \mathbf{X}(c) \,. </annotation></semantics></math></div> <p>Since <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">\tilde F</annotation></semantics></math> is required to preserve any colimit and hence these particular colimits, <a class="maruku-eqref" href="#eq:YonedaRestriction">(1)</a> implies that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>F</mi><mo>˜</mo></mover></mrow><annotation encoding="application/x-tex">\widetilde F</annotation></semantics></math> is fixed to act, up to isomorphism, as</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><mover><mi>F</mi><mo>˜</mo></mover><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mover><mi>F</mi><mo>˜</mo></mover><mrow><mo>(</mo><msup><mo>∫</mo> <mrow><mi>c</mi><mo>∈</mo><mi>𝒞</mi></mrow></msup><mi>y</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>⋅</mo><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>)</mo></mrow></mtd> <mtd><mo>≔</mo><msup><mo>∫</mo> <mrow><mi>c</mi><mo>∈</mo><mi>𝒞</mi></mrow></msup><mi>F</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>⋅</mo><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>∈</mo><mi>𝒟</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \widetilde F(\mathbf{X}) & = \widetilde F \left( \int^{c \in \mathcal{C}} y(c) \cdot \mathbf{X}(c) \right) & \coloneqq \int^{c \in \mathcal{C}} F(c) \cdot \mathbf{X}(c) \;\;\;\;\in \mathcal{D} \end{aligned} </annotation></semantics></math></div> <p>(where the colimit (a <a class="existingWikiWord" href="/nlab/show/coend">coend</a>) on the right is computed in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">\mathcal{D}</annotation></semantics></math>!).</p> </div> <h2 id="free_cocompletion_of_large_categories">Free cocompletion of large categories</h2> <p>In general, given a locally small (but possibly large) category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math>, we can define a locally small category of <em>small presheaves</em> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math> as the full subcategory of the functor category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Fun</mi><mo stretchy="false">(</mo><msup><mi>𝒜</mi> <mi mathvariant="normal">op</mi></msup><mo>,</mo><mo lspace="0em" rspace="thinmathspace">Set</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Fun(\mathcal{A}^{\mathrm{op}},\operatorname{Set})</annotation></semantics></math> spanned by those functors that are colimits of small diagrams of representables. Equivalently, a <em>small presheaf</em> is a functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><msup><mi>𝒜</mi> <mi mathvariant="normal">op</mi></msup><mo>→</mo><mo lspace="0em" rspace="thinmathspace">Set</mo></mrow><annotation encoding="application/x-tex">F:\mathcal{A}^{\mathrm{op}} \to \operatorname{Set}</annotation></semantics></math> such that there exists a functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>γ</mi><mo>:</mo><mi>𝒞</mi><mo>→</mo><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\gamma:\mathcal{C} \to \mathcal{A}</annotation></semantics></math> with <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> small and a presheaf <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>′</mo><mo>:</mo><msup><mi>𝒞</mi> <mi mathvariant="normal">op</mi></msup><mo>→</mo><mo lspace="0em" rspace="thinmathspace">Set</mo></mrow><annotation encoding="application/x-tex">F': \mathcal{C}^{\mathrm{op}}\to \operatorname{Set}</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><mi>F</mi><mo>=</mo><msub><mo lspace="0em" rspace="thinmathspace">Lan</mo> <mi>γ</mi></msub><mi>F</mi><mo>′</mo><mo>.</mo></mrow><annotation encoding="application/x-tex">F=\operatorname{Lan}_{\gamma} F'.</annotation></semantics></math></div> <p>This process of free cocompletion defines a functor <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> from the 2-category</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>CAT</mi><mo>=</mo><mo stretchy="false">[</mo><mi>locally</mi><mspace width="thickmathspace"></mspace><mi>small</mi><mspace width="thickmathspace"></mspace><mi>categories</mi><mo>,</mo><mspace width="thickmathspace"></mspace><mi>functors</mi><mo>,</mo><mspace width="thickmathspace"></mspace><mi>natural</mi><mspace width="thickmathspace"></mspace><mi>transformations</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> CAT = [locally \; small \; categories, \; functors, \; natural \; transformations ] </annotation></semantics></math></div> <p>to the 2-category</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Cocomp</mi><mo>=</mo><mo stretchy="false">[</mo><mi>locally</mi><mspace width="thickmathspace"></mspace><mi>small</mi><mspace width="thickmathspace"></mspace><mi>cocomplete</mi><mspace width="thickmathspace"></mspace><mi>categories</mi><mo>,</mo><mspace width="thickmathspace"></mspace><mi>cocontinuous</mi><mspace width="thickmathspace"></mspace><mi>functors</mi><mo>,</mo><mi>natural</mi><mspace width="thickmathspace"></mspace><mi>transformations</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> Cocomp = [locally \; small \; cocomplete \; categories, \; cocontinuous \; functors, natural \; transformations ] </annotation></semantics></math></div> <p>where, just to be clear, ‘cocomplete’ means having all <em>small</em> colimits, as usual. Moreover, this functor <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> is part of a <a class="existingWikiWord" href="/nlab/show/pseudoadjunction">pseudoadjunction</a> where the right adjoint <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>:</mo><mi>Cocomp</mi><mo>→</mo><mi>CAT</mi></mrow><annotation encoding="application/x-tex">R: Cocomp \to CAT</annotation></semantics></math> assigns each locally small coccomplete category its underlying category, and so on.</p> <p>In the case where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒜</mi> <mi mathvariant="normal">op</mi></msup></mrow><annotation encoding="application/x-tex">\mathcal{A}^{\mathrm{op}}</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/locally+presentable">locally presentable</a>, the category of small presheaves on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math> is equivalent to the category of <a class="existingWikiWord" href="/nlab/show/accessible+functors">accessible functors</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒜</mi> <mi mathvariant="normal">op</mi></msup><mo>→</mo><mo lspace="0em" rspace="thinmathspace">Set</mo></mrow><annotation encoding="application/x-tex">\mathcal{A}^{\mathrm{op}}\to \operatorname{Set}</annotation></semantics></math>. In this situation, the category of small functors has many nice properties and is often said to be <em>almost a topos</em>. In particular, under this assumption, the category of small presheaves is complete, cocomplete, and Cartesian-closed.</p> <p>See <a href="#DayLack">Day-Lack</a> for more details on all these matters. They handle the more general case of enriched categories.</p> <h2 id="free_cocompletion_as_a_pseudomonad">Free cocompletion as a pseudomonad</h2> <p><a class="existingWikiWord" href="/nlab/show/David+Corfield">David Corfield</a>: So is this ‘free cocompletion’ part of an adjunction between the category of categories and the category of cocomplete categories (modulo size worries?). Or should we think of it as part of a <a class="existingWikiWord" href="/nlab/show/pseudoadjunction">pseudoadjunction</a> between <em>2-categories</em>? (I would start a page on that, but how are naming conventions going in this area?)</p> <p><a class="existingWikiWord" href="/nlab/show/John+Baez">John Baez</a>: Equations between functors tends to hold only up to natural isomorphism. So, your first guess should not be that there’s an <em>adjunction</em> between the <em>categories</em> <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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cocomp</mi></mrow><annotation encoding="application/x-tex">Cocomp</annotation></semantics></math>, but rather, a <em>pseudoadjunction</em> between the <em>2-categories</em> <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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cocomp</mi></mrow><annotation encoding="application/x-tex">Cocomp</annotation></semantics></math>.</p> <p>This means there’s a forgetful 2-functor:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>:</mo><mi>Cocomp</mi><mo>→</mo><mi>CAT</mi></mrow><annotation encoding="application/x-tex"> R: Cocomp \to CAT </annotation></semantics></math></div> <p>together with a ‘free cocompletion’ 2-functor:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo>:</mo><mi>CAT</mi><mo>→</mo><mi>Cocomp</mi></mrow><annotation encoding="application/x-tex"> L: CAT \to Cocomp </annotation></semantics></math></div> <p>And, it mean there’s an equivalence of categories</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>hom</mi> <mi>Cocomp</mi></msub><mo stretchy="false">(</mo><mi>F</mi><mi>C</mi><mo>,</mo><mi>D</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>hom</mi> <mi>CAT</mi></msub><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mi>U</mi><mi>D</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> hom_{Cocomp} (F C, D) \simeq hom_{CAT} (C, U D) </annotation></semantics></math></div> <p>for every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>∈</mo><mi>CAT</mi></mrow><annotation encoding="application/x-tex">C \in CAT</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>∈</mo><mi>Cocomp</mi></mrow><annotation encoding="application/x-tex">D \in Cocomp</annotation></semantics></math>. And finally, it means that this equivalence is <a class="existingWikiWord" href="/nlab/show/pseudonatural+transformation">pseudonatural</a> as a function 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>.</p> <p>If we have a pseudonatural equivalence of categories</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>hom</mi> <mi>Cocomp</mi></msub><mo stretchy="false">(</mo><mi>F</mi><mi>C</mi><mo>,</mo><mi>D</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>hom</mi> <mi>CAT</mi></msub><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mi>U</mi><mi>D</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> hom_{Cocomp} (F C, D) \simeq hom_{CAT} (C, U D) </annotation></semantics></math></div> <p>instead of a natural isomorphism of sets, then we say we have a ‘pseudoadjunction’ instead of an adjunction. A pseudoadjunction is the right generalization of adjunction when we go to 2-categories; if we were feeling in a modern mood we might just say ‘adjunction’ and expect people to know we meant ‘pseudo’.</p> <p>The details have been worked out by <a href="#DayLack">Day-Lack</a> in even more generality: they consider enriched categories. Note that the free cocompletion of a <em>small</em> category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is the category of all presheaves 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>, but for a more general <em>locally small</em> category it’s just the category of small presheaves.</p> <p>In <a href="http://www.lacim.uqam.ca/~gambino/species.pdf">their work on species</a>, Fiore, Gambino, Hyland and Winskel confronted the size issues in another way. In one draft of this paper they had a very artful and sophisticated device for dealing with this size problem. In the latest draft they seem to have sidestepped it entirely: you’ll see they discuss the ‘free symmetric monoidal category on a category’ pseudomonad, but never the ‘free cocomplete category on a category’ pseudomonad, even though they <em>do</em> use the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>C</mi><mo>^</mo></mover></mrow><annotation encoding="application/x-tex">\widehat{C}</annotation></semantics></math> construction all over the place. Somehow they’ve managed to avoid the need to consider this construction as a pseudomonad!</p> <p>Fiore, Gambino, Hyland and Winskel later exhibited the free cocompletion of a small category as a <a class="existingWikiWord" href="/nlab/show/relative+pseudomonad">relative pseudomonad</a> from the 2-category Cat of small categories into the 2-category CAT of locally-small categories (via the 2-category COC of locally-small cocomplete categories). See <a href="#FGHW">FGHW</a> for more details.</p> <h2 id="in_higher_category_theory">In higher category theory</h2> <p>One can ask for the notion of free cocompletion in the wider context of <a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a>.</p> <ul> <li>for <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> theory there is <a class="existingWikiWord" href="/nlab/show/free+%28%E2%88%9E%2C1%29-cocompletion">free (∞,1)-cocompletion</a>.</li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p>free cocompletions</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/free+strict+cocompletion">free strict cocompletion</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/free+coproduct+completion">free coproduct completion</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/idempotent+completion">idempotent completion</a>, <a class="existingWikiWord" href="/nlab/show/Karoubi+envelope">Karoubi envelope</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cauchy+completion">Cauchy completion</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/free+completion">free completion</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/free+bicompletion">free bicompletion</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conservative+cocompletion">conservative cocompletion</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ind-object">ind-object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sind-object">sind-object</a></p> </li> </ul> <h2 id="references">References</h2> <ul> <li id="DayLack"> <p><a class="existingWikiWord" href="/nlab/show/Brian+Day">Brian Day</a>, <a class="existingWikiWord" href="/nlab/show/Steve+Lack">Steve Lack</a>, <em>Limits of small functors</em> (<a href="https://arxiv.org/abs/math/0610439">web</a>)</p> </li> <li id="FGHW"> <p>Fiore, Gambino, Hyland and Winskel, <em>Relative pseudomonads, Kleisli bicategories, and substitution monoidal structures</em>, (<a href="https://arxiv.org/pdf/1612.03678.pdf">web</a>)</p> </li> <li id="PT"> <p><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 partial colimits</em>, Applied Categorical Structures, 2022. (<a href="https://arxiv.org/abs/2101.04531">arXiv:2101.04531</a>)</p> </li> </ul> <p>On <a class="existingWikiWord" href="/nlab/show/strict+free+cocompletions">strict free cocompletions</a>:</p> <ul> <li>Erwan Beurier, Dominique Pastor, and René Guitart. <em>Presentations of clusters and strict free-cocompletions</em>. <a class="existingWikiWord" href="/nlab/show/Theory+and+Applications+of+Categories">Theory and Applications of Categories</a> 36.17 (2021): 492-513. (<a href="http://www.tac.mta.ca/tac/volumes/36/17/36-17abs.html">link</a>)</li> </ul> <p>This reference might also give helpful clues:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Daniel+Dugger">Daniel Dugger</a>, <em>Sheaves and Homotopy Theory</em> (<a href="http://www.uoregon.edu/~ddugger/cech.html">web</a>, <a href="http://ncatlab.org/nlab/files/cech.pdf">pdf</a>)</li> </ul> <p>A pedagogical explanation of the universal property of the <a class="existingWikiWord" href="/nlab/show/Yoneda+embedding">Yoneda embedding</a> is given starting on page 7. On page 8 there’s an explanation with lots of pictures how a presheaf is an “instruction for how to build a colimit”. Then on p. 9 the universal morphism that we are looking for here is identified as the one that “takes the instructions for building a colimit and actually <em>builds</em> it”.</p> <p>(This text, by the way, contains various other gems. A pity that it is left unfinished.)</p> <p>An early paper on free completions:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/V%C4%9Bra+Trnkov%C3%A1">Věra Trnková</a>. <em>Limits in categories and limit-preserving functors</em>, Commentationes Mathematicae Universitatis Carolinae 7.1 (1966): 1-73.</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on June 7, 2024 at 06:34:57. 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