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float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="category_theory">Category theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></strong></p> <h2 id="sidebar_concepts">Concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category">category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cat">Cat</a></p> </li> </ul> <h2 id="sidebar_universal_constructions">Universal constructions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+construction">universal construction</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/representable+functor">representable functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+functor">adjoint functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/limit">limit</a>/<a class="existingWikiWord" href="/nlab/show/colimit">colimit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/weighted+limit">weighted limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/end">end</a>/<a class="existingWikiWord" href="/nlab/show/coend">coend</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kan+extension">Kan extension</a></p> </li> </ul> </li> </ul> <h2 id="sidebar_theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+construction">Grothendieck construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+functor+theorem">adjoint functor theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monadicity+theorem">monadicity theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+lifting+theorem">adjoint lifting theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gabriel-Ulmer+duality">Gabriel-Ulmer duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freyd-Mitchell+embedding+theorem">Freyd-Mitchell embedding theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relation+between+type+theory+and+category+theory">relation between type theory and category theory</a></p> </li> </ul> <h2 id="sidebar_extensions">Extensions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf+and+topos+theory">sheaf and topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a></p> </li> </ul> <h2 id="sidebar_applications">Applications</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/applications+of+%28higher%29+category+theory">applications of (higher) category theory</a></li> </ul> <div> <p> <a href="/nlab/edit/category+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="limits_and_colimits">Limits and colimits</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/limit">limits and colimits</a></strong></p> <h2 id="1categorical">1-Categorical</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/limit">limit and colimit</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/limits+and+colimits+by+example">limits and colimits by example</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/commutativity+of+limits+and+colimits">commutativity of limits and colimits</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/small+limit">small limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/filtered+colimit">filtered colimit</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/directed+colimit">directed colimit</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/sequential+colimit">sequential colimit</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sifted+colimit">sifted colimit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connected+limit">connected limit</a>, <a class="existingWikiWord" href="/nlab/show/wide+pullback">wide pullback</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/preserved+limit">preserved limit</a>, <a class="existingWikiWord" href="/nlab/show/reflected+limit">reflected limit</a>, <a class="existingWikiWord" href="/nlab/show/created+limit">created limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/product">product</a>, <a class="existingWikiWord" href="/nlab/show/fiber+product">fiber product</a>, <a class="existingWikiWord" href="/nlab/show/base+change">base change</a>, <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a>, <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a>, <a class="existingWikiWord" href="/nlab/show/pushout">pushout</a>, <a class="existingWikiWord" href="/nlab/show/cobase+change">cobase change</a>, <a class="existingWikiWord" href="/nlab/show/equalizer">equalizer</a>, <a class="existingWikiWord" href="/nlab/show/coequalizer">coequalizer</a>, <a class="existingWikiWord" href="/nlab/show/join">join</a>, <a class="existingWikiWord" href="/nlab/show/meet">meet</a>, <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a>, <a class="existingWikiWord" href="/nlab/show/initial+object">initial object</a>, <a class="existingWikiWord" href="/nlab/show/direct+product">direct product</a>, <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/finite+limit">finite limit</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/exact+functor">exact functor</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kan+extension">Kan extension</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Yoneda+extension">Yoneda extension</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/weighted+limit">weighted limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/end">end and coend</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fibered+limit">fibered limit</a></p> </li> </ul> <h2 id="2categorical">2-Categorical</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-limit">2-limit</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/inserter">inserter</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/isoinserter">isoinserter</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equifier">equifier</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/inverter">inverter</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/PIE-limit">PIE-limit</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-pullback">2-pullback</a>, <a class="existingWikiWord" href="/nlab/show/comma+object">comma object</a></p> </li> </ul> <h2 id="1categorical_2">(∞,1)-Categorical</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-limit">(∞,1)-limit</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-pullback">(∞,1)-pullback</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fiber+sequence">fiber sequence</a></li> </ul> </li> </ul> </li> </ul> <h3 id="modelcategorical">Model-categorical</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+Kan+extension">homotopy Kan extension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+limit">homotopy limit</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+product">homotopy product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+equalizer">homotopy equalizer</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+totalization">homotopy totalization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+end">homotopy end</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+colimit">homotopy colimit</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coproduct">homotopy coproduct</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coequalizer">homotopy coequalizer</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+cofiber">homotopy cofiber</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+pushout">homotopy pushout</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+realization">homotopy realization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coend">homotopy coend</a></p> </li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/infinity-limits+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <li><a href='#examples'>Examples</a></li> <li><a href='#properties'>Properties</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 concept of <em>colimit</em> is that <a class="existingWikiWord" href="/nlab/show/duality">dual</a> to a <a class="existingWikiWord" href="/nlab/show/limit">limit</a>:</p> <p>a colimit of a <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a> in a <a class="existingWikiWord" href="/nlab/show/category">category</a> is, if it exists, the <a class="existingWikiWord" href="/nlab/show/representable+functor">co-classifying space</a> for morphisms <em>out</em> of that diagram.</p> <p>The intuitive general idea of a colimit is that it defines an object obtained by sewing together the objects of the diagram, according to the instructions given by the morphisms of the diagram.</p> <p>We have</p> <ul> <li> <p>the notion of <em>colimit</em> generalizes the notion of <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a>;</p> </li> <li> <p>the notion of <em><a class="existingWikiWord" href="/nlab/show/weighted+colimit">weighted colimit</a></em> generalizes the notion of <em>weighted (direct) sum</em>.</p> </li> </ul> <p>Sometimes colimits (or some colimits) are called <em><a class="existingWikiWord" href="/nlab/show/inductive+limits">inductive limits</a></em> or <em><a class="existingWikiWord" href="/nlab/show/direct+limits">direct limits</a></em>; see the discussion of terminology at <a class="existingWikiWord" href="/nlab/show/limit">limit</a>.</p> <p>A <a class="existingWikiWord" href="/nlab/show/weighted+colimit">weighted colimit</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/weighted+limit">weighted limit</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">C^{op}</annotation></semantics></math>.</p> <h2 id="definition">Definition</h2> <p>A <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a> in 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> is the same as a <a class="existingWikiWord" href="/nlab/show/limit">limit</a> in the <a class="existingWikiWord" href="/nlab/show/opposite+category">opposite category</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">C^{op}</annotation></semantics></math>.</p> <p>More in detail, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><msup><mi>D</mi> <mi>op</mi></msup><mo>→</mo><msup><mi>C</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">F : D^{op} \to C^{op}</annotation></semantics></math> a functor, its <a class="existingWikiWord" href="/nlab/show/limit">limit</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>lim</mi><mi>F</mi></mrow><annotation encoding="application/x-tex">\lim F</annotation></semantics></math> is the colimit of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>F</mi> <mi>op</mi></msup><mo>:</mo><mi>D</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">F^{op} : D \to C</annotation></semantics></math>.</p> <h2 id="examples">Examples</h2> <p>Here are some important examples of colimits:</p> <ul> <li>A colimit of the <a class="existingWikiWord" href="/nlab/show/diagram">empty diagram</a> is an <a class="existingWikiWord" href="/nlab/show/initial+object">initial object</a>.</li> <li>A colimit of a diagram consisting of two (or more) objects and no nontrivial morphisms is their <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a>.</li> <li>A colimit of a <a class="existingWikiWord" href="/nlab/show/span">span</a> is a <a class="existingWikiWord" href="/nlab/show/pushout">pushout</a>.</li> <li>A colimit of two (or more) <a class="existingWikiWord" href="/nlab/show/parallel+morphisms">parallel morphisms</a> is a <a class="existingWikiWord" href="/nlab/show/coequalizer">coequalizer</a>.</li> <li>A colimit of a diagram whose domain is a <a class="existingWikiWord" href="/nlab/show/sifted+category">sifted category</a> is a <a class="existingWikiWord" href="/nlab/show/sifted+colimit">sifted colimit</a>.</li> <li>A colimit of a diagram whose domain is a <a class="existingWikiWord" href="/nlab/show/filtered+category">filtered category</a> is a <a class="existingWikiWord" href="/nlab/show/filtered+colimit">filtered colimit</a>.</li> <li>A colimit of a connected diagram is a <a class="existingWikiWord" href="/nlab/show/connected+colimit">connected colimit</a>.</li> <li>A colimit of a nonempty diagram is a <span class="newWikiWord">nonempty colimit<a href="/nlab/new/nonempty+colimit">?</a></span>.</li> </ul> <p>See also <em><a class="existingWikiWord" href="/nlab/show/limits+and+colimits+by+example">limits and colimits by example</a></em>.</p> <h2 id="properties">Properties</h2> <p>The properties of colimits are of course <a class="existingWikiWord" href="/nlab/show/formal+duality">dual</a> to those of <a class="existingWikiWord" href="/nlab/show/limits">limits</a>. It is still worthwhile to make some of them explicit:</p> <p> <div class='num_prop' id='ColimitsInTermsOfCoequalizers'> <h6>Proposition</h6> <p>All colimits may be expressed via <a class="existingWikiWord" href="/nlab/show/coequalizers">coequalizers</a> of maps between <a class="existingWikiWord" href="/nlab/show/coproducts">coproducts</a>.</p> </div> This was historically first observed by <a href="#Maranda62">Maranda 1962, Thm. 1</a>. See the <a class="existingWikiWord" href="/nlab/show/formal+dual">dual</a> discussion (<a href="limit#ConstructionFromProductsAndEqualizers">here</a>) of <a class="existingWikiWord" href="/nlab/show/limits">limits</a> via <a class="existingWikiWord" href="/nlab/show/products">products</a> and <a class="existingWikiWord" href="/nlab/show/equalizers">equalizers</a>.</p> <p> <div class='num_prop'> <h6>Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/hom-functor+preserves+limits">contravariant Hom sends colimits to limits</a>)</strong> <br /> For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/locally+small">locally small</a> category, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mi>D</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">F : D \to C</annotation></semantics></math> a functor, for <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> an object and writing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</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>c</mi><mo stretchy="false">)</mo><mo>:</mo><mi>D</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">C(F(-), c) : D \to Set</annotation></semantics></math>, we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>colim</mi><mi>F</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>lim</mi><mi>C</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>c</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> C(colim F, c) \simeq lim C(F(-), c) \,. </annotation></semantics></math></div> <p></p> </div> </p> <p>Depending on how one introduces limits this holds by definition or is an easy consequence. In fact, this is just rewriting the fact that the covariant Hom respects <a class="existingWikiWord" href="/nlab/show/limit">limit</a>s (as described there) in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">C^{op}</annotation></semantics></math> in terms 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> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mi>C</mi><mo stretchy="false">(</mo><mi>colim</mi><mi>F</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>≃</mo><msup><mi>C</mi> <mi>op</mi></msup><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>colim</mi><mi>F</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><msup><mi>C</mi> <mi>op</mi></msup><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>lim</mi><msup><mi>F</mi> <mi>op</mi></msup><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mi>lim</mi><msup><mi>C</mi> <mi>op</mi></msup><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><msup><mi>F</mi> <mi>op</mi></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mi>lim</mi><mi>C</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>c</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} C(colim F, c) & \simeq C^{op}(c, colim F) \\ & \simeq C^{op}(c, lim F^{op}) \\ & \simeq lim C^{op}(c, F^{op}(-)) \\ & \simeq lim C(F(-), c) \end{aligned} </annotation></semantics></math></div> <p>Notice that this actually says that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>:</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>×</mo><mi>C</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">C(-,-) : C^{op} \times C \to Set</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/continuous+functor">continuous functor</a> in both variables: in the first it sends limits in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">C^{op}</annotation></semantics></math> and hence equivalently colimits in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> to limits in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Set</mi></mrow><annotation encoding="application/x-tex">Set</annotation></semantics></math>.</p> <p> <div class='num_prop'> <h6>Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/adjoints+preserve+%28co-%29limits">left adjoint functors preserve colimits</a>)</strong> <br /> Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>C</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">L : C \to C'</annotation></semantics></math> be a functor that is <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> to some functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>:</mo><mi>C</mi><mo>′</mo><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">R : C' \to C</annotation></semantics></math>. Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/small+category">small category</a> such that <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> admits limits of shape <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 <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> commutes with <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>-shaped colimits in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> in that</p> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mi>D</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">F : D \to C</annotation></semantics></math> some diagram, we have</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 stretchy="false">(</mo><mi>colim</mi><mi>F</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>colim</mi><mo stretchy="false">(</mo><mi>L</mi><mo>∘</mo><mi>F</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> L(colim F) \simeq colim (L \circ F) \,. </annotation></semantics></math></div> <p></p> </div> </p> <div class="proof"> <h6 id="proof">Proof</h6> <p>Using the adjunction isomorphism and the above fact that commutes with limits in both arguments, one obtains 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></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mi>C</mi><mo>′</mo><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">(</mo><mi>colim</mi><mi>F</mi><mo stretchy="false">)</mo><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>≃</mo><mi>C</mi><mo stretchy="false">(</mo><mi>colim</mi><mi>F</mi><mo>,</mo><mi>R</mi><mo stretchy="false">(</mo><mi>c</mi><mo>′</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mi>lim</mi><mi>C</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>R</mi><mo stretchy="false">(</mo><mi>c</mi><mo>′</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mi>lim</mi><mi>C</mi><mo>′</mo><mo stretchy="false">(</mo><mi>L</mi><mo>∘</mo><mi>F</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></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mi>C</mi><mo>′</mo><mo stretchy="false">(</mo><mi>colim</mi><mo stretchy="false">(</mo><mi>L</mi><mo>∘</mo><mi>F</mi><mo stretchy="false">)</mo><mo>,</mo><mi>c</mi><mo>′</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} C'(L (colim F), c) & \simeq C(colim F, R(c')) \\ & \simeq lim C(F(-), R(c')) \\ & \simeq lim C'(L \circ F(-), c') \\ & \simeq C'(colim (L \circ F), c') \,. \end{aligned} \,. </annotation></semantics></math></div> <p>Since this holds naturally for every <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>, the <a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma, corollary II</a> on uniqueness of representing objects implies that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo stretchy="false">(</mo><mi>lim</mi><mi>F</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>lim</mi><mo stretchy="false">(</mo><mi>R</mi><mo>∘</mo><mi>F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">R (lim F) \simeq lim (R \circ F)</annotation></semantics></math>.</p> </div> <h2 id="related_concepts">Related concepts</h2> <ul> <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/direct+sum">direct sum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+colimit">homotopy colimit</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-colimit">(∞,1)-colimit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/lax+colimit">lax colimit</a></p> </li> </ul> <h2 id="references">References</h2> <p><a class="existingWikiWord" href="/nlab/show/limit">Limits</a> and colimits were defined in <a class="existingWikiWord" href="/nlab/show/Daniel+M.+Kan">Daniel M. Kan</a> in Chapter II of the paper that also defined <a class="existingWikiWord" href="/nlab/show/adjoint+functors">adjoint functors</a> and <a class="existingWikiWord" href="/nlab/show/Kan+extensions">Kan extensions</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>The observation that colimits may be constructed from <a class="existingWikiWord" href="/nlab/show/coequalizers">coequalizers</a> and set-indexed <a class="existingWikiWord" href="/nlab/show/coproducts">coproducts</a>:</p> <ul> <li id="Maranda62"><a class="existingWikiWord" href="/nlab/show/Jean-Marie+Maranda">Jean-Marie Maranda</a>, Thm. 1 in: <em>Some remarks on limits in categories</em>, Canadian Mathematical Bulletin <strong>5</strong> 2 (1962) 133-146 [<a href="https://doi.org/10.4153/CMB-1962-015-0">doi:10.4153/CMB-1962-015-0</a>]</li> </ul> <p>Beware that these early articles refer to colimits as <em><a class="existingWikiWord" href="/nlab/show/direct+limits">direct limits</a></em>.</p> <p>Textbook account:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Saunders+MacLane">Saunders MacLane</a>, §III.3 of: <em><a class="existingWikiWord" href="/nlab/show/Categories+for+the+Working+Mathematician">Categories for the Working Mathematician</a></em>, Graduate Texts in Mathematics <strong>5</strong> Springer (second ed. 1997) [<a href="https://link.springer.com/book/10.1007/978-1-4757-4721-8">doi:10.1007/978-1-4757-4721-8</a>]</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on November 4, 2023 at 10:56:26. 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