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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="limits"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Limits</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> <hr /> <h4 id="2category_theory">2-Category theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/2-category+theory">2-category theory</a></strong></p> <p><strong>Definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-category">2-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/strict+2-category">strict 2-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bicategory">bicategory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+bicategory">enriched bicategory</a></p> </li> </ul> <p><strong>Transfors between 2-categories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-functor">2-functor</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/pseudofunctor">pseudofunctor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/lax+functor">lax functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivalence+of+2-categories">equivalence of 2-categories</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-natural+transformation">2-natural transformation</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/lax+natural+transformation">lax natural transformation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/icon">icon</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/modification">modification</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yoneda+lemma+for+bicategories">Yoneda lemma for bicategories</a></p> </li> </ul> <p><strong>Morphisms in 2-categories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fully+faithful+morphism">fully faithful morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/faithful+morphism">faithful morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conservative+morphism">conservative morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pseudomonic+morphism">pseudomonic morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/discrete+morphism">discrete morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/eso+morphism">eso morphism</a></p> </li> </ul> <p><strong>Structures in 2-categories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mate">mate</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monad">monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cartesian+object">cartesian object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fibration+in+a+2-category">fibration in a 2-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/codiscrete+cofibration">codiscrete cofibration</a></p> </li> </ul> <p><strong>Limits in 2-categories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-limit">2-limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-pullback">2-pullback</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/comma+object">comma object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/inserter">inserter</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/inverter">inverter</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equifier">equifier</a></p> </li> </ul> <p><strong>Structures on 2-categories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-monad">2-monad</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/lax-idempotent+2-monad">lax-idempotent 2-monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pseudomonad">pseudomonad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pseudoalgebra+for+a+2-monad">pseudoalgebra for a 2-monad</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+2-category">monoidal 2-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/cartesian+bicategory">cartesian bicategory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gray+tensor+product">Gray tensor product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/proarrow+equipment">proarrow equipment</a></p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <ul> <li><a href='#Terminology'>Strictness and terminology</a></li> </ul> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#2limits_over_diagrams_of_special_shape'>2-limits over diagrams of special shape</a></li> <li><a href='#finite_2limits'>Finite 2-Limits</a></li> <li><a href='#(2,1)limit'><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(2,1)</annotation></semantics></math>-limits</a></li> <li><a href='#lax'>Lax limits</a></li> <li><a href='#2ColimitsInCat'>2-Colimits in <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></a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>A <strong>2-limit</strong> is the type of <a class="existingWikiWord" href="/nlab/show/limit">limit</a> that is appropriate in a (weak) <a class="existingWikiWord" href="/nlab/show/2-category">2-category</a>. (Since general 2-categories are often called <em><a class="existingWikiWord" href="/nlab/show/bicategories">bicategories</a></em>, 2-limits are often called <em><a class="existingWikiWord" href="/nlab/show/bilimits">bilimits</a></em>.)</p> <p>There are three notable changes when passing from ordinary 1-limits to 2-limits:</p> <ol> <li> <p>In order to satisfy the <a class="existingWikiWord" href="/nlab/show/principle+of+equivalence">principle of equivalence</a>, the “cones” in a 2-limit are required to commute only up to <a class="existingWikiWord" href="/nlab/show/2-morphism">2-isomorphism</a>.</p> </li> <li> <p>The <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a> of the limit is expressed by an <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence of categories</a> rather than a <a class="existingWikiWord" href="/nlab/show/bijection">bijection</a> of <a class="existingWikiWord" href="/nlab/show/sets">sets</a>. This means that</p> <ol> <li> <p>every other cone over the diagram that commutes up to isomorphism factors through the limit, up to isomorphism, and</p> </li> <li> <p>every transformation <em>between</em> cones also factors through a 2-cell in the limit. We will give some examples below.</p> </li> </ol> </li> <li> <p>Since 2-categories are <a class="existingWikiWord" href="/nlab/show/enriched+category">enriched</a> over <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a> (this is precise in the <a class="existingWikiWord" href="/nlab/show/strict+2-category">strict</a> case, and <a class="existingWikiWord" href="/nlab/show/bicategory">weakly</a> true otherwise), <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>-<a class="existingWikiWord" href="/nlab/show/weighted+limit">weighted limit</a>s become important. This means that both the diagrams we take limits of and the shape of “cones” that limits represent can involve <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-cells as well as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math>-cells.</p> </li> </ol> <h2 id="definition">Definition</h2> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> 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> be <a class="existingWikiWord" href="/nlab/show/2-categories">2-categories</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi><mo lspace="verythinmathspace">:</mo><mi>D</mi><mo>→</mo><mi>Cat</mi></mrow><annotation encoding="application/x-tex">J\colon D\to Cat</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo lspace="verythinmathspace">:</mo><mi>D</mi><mo>→</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">F\colon D\to K</annotation></semantics></math> be <a class="existingWikiWord" href="/nlab/show/2-functors">2-functors</a>. A <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math>-weighted (2-)limit of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math></strong> is an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo>∈</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">L\in K</annotation></semantics></math> equipped with a <a class="existingWikiWord" href="/nlab/show/pseudonatural+equivalence">pseudonatural equivalence</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>L</mi><mo stretchy="false">)</mo><mo>≃</mo><mo stretchy="false">[</mo><mi>D</mi><mo>,</mo><mi>Cat</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mi>J</mi><mo>,</mo><mi>K</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>F</mi><mo>−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex"> K(X,L) \simeq [D,Cat](J,K(X,F-)). </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>D</mi><mo>,</mo><mi>Cat</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[D,Cat]</annotation></semantics></math> denotes the 2-category of <a class="existingWikiWord" href="/nlab/show/2-functors">2-functors</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>→</mo><mi>Cat</mi></mrow><annotation encoding="application/x-tex">D\to Cat</annotation></semantics></math>, <a class="existingWikiWord" href="/nlab/show/pseudonatural+transformations">pseudonatural transformations</a> between them, and <a class="existingWikiWord" href="/nlab/show/modifications">modifications</a> between those.</p> <p>A 2-limit in the <a class="existingWikiWord" href="/nlab/show/opposite+2-category">opposite 2-category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>K</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">K^{op}</annotation></semantics></math> is called a <strong>2-colimit</strong> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>. Everything below applies dually to 2-colimits, the higher analogues of <a class="existingWikiWord" href="/nlab/show/colimits">colimits</a>. (But somebody might want to make a separate page that gives appropriate examples of these.)</p> <h3 id="Terminology">Strictness and terminology</h3> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> 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> are <a class="existingWikiWord" href="/nlab/show/strict+2-categories">strict 2-categories</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/strict+2-functors">strict 2-functors</a>, and if we replace this pseudonatural equivalence by a (strictly 2-natural) isomorphism <em>and</em> the 2-category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>D</mi><mo>,</mo><mi>Cat</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[D,Cat]</annotation></semantics></math> by the 2-category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>D</mi><mo>,</mo><mi>Cat</mi><msub><mo stretchy="false">]</mo> <mi>strict</mi></msub></mrow><annotation encoding="application/x-tex">[D,Cat]_{strict}</annotation></semantics></math> of strict 2-functors and strict 2-natural transformations, then we obtain the definition of a <strong><a class="existingWikiWord" href="/nlab/show/strict+2-limit">strict 2-limit</a></strong>. This is precisely a Cat-weighted limit in the sense of ordinary <a class="existingWikiWord" href="/nlab/show/enriched+category">enriched category</a> theory. See <a class="existingWikiWord" href="/nlab/show/strict+2-limit">strict 2-limit</a> for details.</p> <p>On the other hand, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>, <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>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> are strict as above, and we replace the equivalence by an isomorphism but keep the weak meaning of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>D</mi><mo>,</mo><mi>Cat</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[D,Cat]</annotation></semantics></math>, then we obtain the notion of a <strong>strict pseudolimit</strong>. Strict pseudolimits are, in particular, 2-limits, whereas strict 2-limits are not always (although some, such as <a class="existingWikiWord" href="/nlab/show/PIE-limits">PIE-limits</a> and <a class="existingWikiWord" href="/nlab/show/flexible+limits">flexible limits</a>, are). In a strict 2-category, these types of strict limits are often technically useful in constructing the “up-to-isomorphism” 2-limits we consider here.</p> <p>When we know we are working in a (weak) 2-category, the only type of limit that makes sense is a (non-strict) 2-limit. Therefore, we usually call these simply “limits.” To emphasize the distinction with the strict 2-limits in a strict 2-category, the “up-to-isomorphism” 2-limits were historically often called <em>bilimits</em> (by analogy with <a class="existingWikiWord" href="/nlab/show/bicategory">bicategory</a> for “weak 2-category”). However, this terminology is somewhat unfortunate, not only because it doesn’t generalize well to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>, but because it leads to words like “biproduct,” which also has the <a class="existingWikiWord" href="/nlab/show/biproduct">completely unrelated meaning</a> of an object that is both a product and a coproduct (which is common in <a class="existingWikiWord" href="/nlab/show/additive+category">additive categories</a>).</p> <p>Unfortunately, we probably shouldn’t use “weak limit” to emphasize the “up-to-isomorphism” nature of these limits, because that also has the <a class="existingWikiWord" href="/nlab/show/weak+limit">completely unrelated meaning</a> of an object in a 1-category satisfying the existence, but not the uniqueness property of an ordinary limit.</p> <h2 id="examples">Examples</h2> <h3 id="2limits_over_diagrams_of_special_shape">2-limits over diagrams of special shape</h3> <p>Any ordinary type of limit can be “2-ified” by boosting its ordinary universal property up to a 2-categorical one. In the following examples we work in a 2-category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>.</p> <ul> <li> <p>A <strong><a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a></strong> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> is an object 1 such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(X,1)</annotation></semantics></math> is equivalent to the <a class="existingWikiWord" href="/nlab/show/terminal+category">terminal category</a> for any object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. This means that for any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> there is a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">X\to 1</annotation></semantics></math> and for any two morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi><mo>:</mo><mi>X</mi><mo>→</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">f,g:X\to 1</annotation></semantics></math> there is a unique morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>→</mo><mi>g</mi></mrow><annotation encoding="application/x-tex">f\to g</annotation></semantics></math>, and this morphism is an isomorphism.</p> </li> <li> <p>A <strong><a class="existingWikiWord" href="/nlab/show/product">product</a></strong> of two objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>,</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A,B</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> is an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>×</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A\times B</annotation></semantics></math> together with a natural equivalence of categories <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo>×</mo><mi>B</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>K</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>×</mo><mi>K</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(X,A\times B) \simeq K(X,A)\times K(X,B)</annotation></semantics></math>. This means that we have projections <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><mi>A</mi><mo>×</mo><mi>B</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">p:A\times B\to A</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mo>:</mo><mi>A</mi><mo>×</mo><mi>B</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">q:A\times B\to B</annotation></semantics></math> such that (1) for any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">f:X\to A</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">g:X\to B</annotation></semantics></math>, there exists an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>A</mi><mo>×</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">h:X\to A\times B</annotation></semantics></math> and isomorphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mi>h</mi><mo>≅</mo><mi>f</mi></mrow><annotation encoding="application/x-tex">p h\cong f</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mi>h</mi><mo>≅</mo><mi>g</mi></mrow><annotation encoding="application/x-tex">q h\cong g</annotation></semantics></math>, and (2) for any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>,</mo><mi>k</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>A</mi><mo>×</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">h,k:X\to A\times B</annotation></semantics></math> and 2-cells <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>:</mo><mi>p</mi><mi>h</mi><mo>→</mo><mi>p</mi><mi>k</mi></mrow><annotation encoding="application/x-tex">\alpha:p h \to p k</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>β</mi><mo>:</mo><mi>q</mi><mi>h</mi><mo>→</mo><mi>q</mi><mi>k</mi></mrow><annotation encoding="application/x-tex">\beta: q h \to q k</annotation></semantics></math>, there exists a unique <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>γ</mi><mo>:</mo><mi>h</mi><mo>→</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">\gamma:h \to k</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mi>γ</mi><mo>=</mo><mi>α</mi></mrow><annotation encoding="application/x-tex">p \gamma = \alpha</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mi>γ</mi><mo>=</mo><mi>β</mi></mrow><annotation encoding="application/x-tex">q \gamma = \beta</annotation></semantics></math>.</p> </li> <li> <p>A <strong><a class="existingWikiWord" href="/nlab/show/pullback">pullback</a></strong> of a <a class="existingWikiWord" href="/nlab/show/co-span">co-span</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mover><mo>→</mo><mi>f</mi></mover><mi>C</mi><mover><mo>←</mo><mi>g</mi></mover><mi>B</mi></mrow><annotation encoding="application/x-tex">A \overset{f}{\to} C \overset{g}{\leftarrow} B</annotation></semantics></math> consists of an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><msub><mo>×</mo> <mi>C</mi></msub><mi>B</mi></mrow><annotation encoding="application/x-tex">A\times_C B</annotation></semantics></math> and projections <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><mi>A</mi><msub><mo>×</mo> <mi>C</mi></msub><mi>B</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">p:A\times_C B\to A</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mo>:</mo><mi>A</mi><msub><mo>×</mo> <mi>C</mi></msub><mi>B</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">q:A\times_C B\to B</annotation></semantics></math> together with an isomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>:</mo><mi>f</mi><mi>p</mi><mo>≅</mo><mi>g</mi><mi>q</mi></mrow><annotation encoding="application/x-tex">\phi:f p \cong g q</annotation></semantics></math>, such that (1) for any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">m:X\to A</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">n:X\to B</annotation></semantics></math> with an isomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi><mo>:</mo><mi>f</mi><mi>m</mi><mo>≅</mo><mi>g</mi><mi>n</mi></mrow><annotation encoding="application/x-tex">\psi:f m \cong g n</annotation></semantics></math>, there exists an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>A</mi><msub><mo>×</mo> <mi>C</mi></msub><mi>B</mi></mrow><annotation encoding="application/x-tex">h:X\to A\times_C B</annotation></semantics></math> and isomorphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>:</mo><mi>p</mi><mi>h</mi><mo>≅</mo><mi>m</mi></mrow><annotation encoding="application/x-tex">\alpha:p h \cong m</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>β</mi><mo>:</mo><mi>q</mi><mi>h</mi><mo>≅</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">\beta:q h \cong n</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mi>β</mi><mo>.</mo><mi>ϕ</mi><mi>h</mi><mo>.</mo><mi>f</mi><msup><mi>α</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>=</mo><mi>ψ</mi></mrow><annotation encoding="application/x-tex">g\beta . \phi h . f \alpha^{-1} = \psi</annotation></semantics></math>, and (2) given any two morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>,</mo><mi>k</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>A</mi><msub><mo>×</mo> <mi>C</mi></msub><mi>B</mi></mrow><annotation encoding="application/x-tex">h,k:X\to A\times_C B</annotation></semantics></math> and 2-cells <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi><mo>:</mo><mi>p</mi><mi>h</mi><mo>→</mo><mi>p</mi><mi>k</mi></mrow><annotation encoding="application/x-tex">\mu:p h \to p k</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ν</mi><mo>:</mo><mi>q</mi><mi>h</mi><mo>→</mo><mi>q</mi><mi>k</mi></mrow><annotation encoding="application/x-tex">\nu:q h \to q k</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mi>μ</mi><mo>=</mo><mi>g</mi><mi>ν</mi></mrow><annotation encoding="application/x-tex">f \mu = g \nu</annotation></semantics></math> (modulo the given isomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mi>p</mi><mo>≅</mo><mi>g</mi><mi>q</mi></mrow><annotation encoding="application/x-tex">f p \cong g q</annotation></semantics></math>), i.e., <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mi>k</mi><mo>.</mo><mi>f</mi><mi>μ</mi><mo>=</mo><mi>g</mi><mi>ν</mi><mo>.</mo><mi>ϕ</mi><mi>h</mi></mrow><annotation encoding="application/x-tex">\phi k . f\mu = g\nu . \phi h</annotation></semantics></math>, there exists a unique 2-cell <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>γ</mi><mo>:</mo><mi>h</mi><mo>→</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">\gamma:h\to k</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mi>γ</mi><mo>=</mo><mi>μ</mi></mrow><annotation encoding="application/x-tex">p \gamma = \mu</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mi>γ</mi><mo>=</mo><mi>ν</mi></mrow><annotation encoding="application/x-tex">q \gamma = \nu</annotation></semantics></math>. This is sometimes called the <em>pseudo-pullback</em> but that term more properly refers to a particular <a class="existingWikiWord" href="/nlab/show/strict+2-limit">strict 2-limit</a>.</p> </li> <li> <p>An <strong><a class="existingWikiWord" href="/nlab/show/equalizer">equalizer</a></strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">f,g:A\to B</annotation></semantics></math> consists of an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> and a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi><mo>:</mo><mi>E</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">e:E\to A</annotation></semantics></math> together with an isomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mi>e</mi><mo>≅</mo><mi>g</mi><mi>e</mi></mrow><annotation encoding="application/x-tex">f e \cong g e</annotation></semantics></math>, which is universal in a sense the reader should now be able to write down. This is sometimes called the <em>pseudo-equalizer</em> but that term more properly refers to a particular <a class="existingWikiWord" href="/nlab/show/strict+2-limit">strict 2-limit</a>. Note that frequently, such as in the construction of all limits from basic ones, equalizers need to be replaced by <a class="existingWikiWord" href="/nlab/show/descent+object">descent object</a>s.</p> </li> </ul> <p>There are also various important types of 2-limits that involve 2-cells in the diagram shape or in the weight, and hence are not just “boosted-up” versions of 1-limits.</p> <ul> <li> <p>The <strong><a class="existingWikiWord" href="/nlab/show/comma+object">comma object</a></strong> of a cospan <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mover><mo>→</mo><mi>f</mi></mover><mi>C</mi><mover><mo>←</mo><mi>g</mi></mover><mi>B</mi></mrow><annotation encoding="application/x-tex">A \overset{f}{\to} C \overset{g}{\leftarrow} B</annotation></semantics></math> is a universal object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">/</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(f/g)</annotation></semantics></math> and projections <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">/</mo><mi>g</mi><mo stretchy="false">)</mo><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">p:(f/g)\to A</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mo>:</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">/</mo><mi>g</mi><mo stretchy="false">)</mo><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">q:(f/g)\to B</annotation></semantics></math> together with a transformation (not an isomorphism) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mi>p</mi><mo>→</mo><mi>g</mi><mi>q</mi></mrow><annotation encoding="application/x-tex">f p \to g q</annotation></semantics></math>. In <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a>, <a class="existingWikiWord" href="/nlab/show/comma+objects">comma objects</a> are <a class="existingWikiWord" href="/nlab/show/comma+category">comma categories</a>. Comma objects are sometimes called <em>lax pullbacks</em> but that term more properly refers to the lax version of a pullback; see “lax limits” below.</p> </li> <li> <p>The <strong><a class="existingWikiWord" href="/nlab/show/inserter">inserter</a></strong> of a pair of parallel arrows <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi><mo>:</mo><mi>A</mi><mspace width="thickmathspace"></mspace><mo>⇉</mo><mspace width="thickmathspace"></mspace><mi>B</mi></mrow><annotation encoding="application/x-tex">f,g:A \;\rightrightarrows\; B</annotation></semantics></math> is a universal object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> equipped with a map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>:</mo><mi>I</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">i:I\to A</annotation></semantics></math> and a 2-cell <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mi>i</mi><mo>→</mo><mi>g</mi><mi>i</mi></mrow><annotation encoding="application/x-tex">f i \to g i</annotation></semantics></math>.</p> </li> <li> <p>The <strong><a class="existingWikiWord" href="/nlab/show/equifier">equifier</a></strong> of a pair of parallel 2-cells <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>,</mo><mi>β</mi><mo>:</mo><mi>f</mi><mo>→</mo><mi>g</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">\alpha,\beta: f\to g: A\to B</annotation></semantics></math> is a universal object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> equipped with a map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi><mo>:</mo><mi>E</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">e:E\to A</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mi>e</mi><mo>=</mo><mi>β</mi><mi>e</mi></mrow><annotation encoding="application/x-tex">\alpha e = \beta e</annotation></semantics></math>.</p> </li> <li> <p>The <strong><a class="existingWikiWord" href="/nlab/show/inverter">inverter</a></strong> of a 2-cell <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>:</mo><mi>f</mi><mo>→</mo><mi>g</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">\alpha:f\to g:A\to B</annotation></semantics></math> is a universal object <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> with a map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi><mo>:</mo><mi>V</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">v:V\to A</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mi>v</mi></mrow><annotation encoding="application/x-tex">\alpha v</annotation></semantics></math> is invertible.</p> </li> <li> <p>The <strong><a class="existingWikiWord" href="/nlab/show/power">power</a></strong> of an object <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">A</annotation></semantics></math> by a category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is a universal object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>A</mi> <mi>C</mi></msup></mrow><annotation encoding="application/x-tex">A^C</annotation></semantics></math> equipped with a functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>→</mo><mi>K</mi><mo stretchy="false">(</mo><msup><mi>A</mi> <mi>C</mi></msup><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C\to K(A^C,A)</annotation></semantics></math>. Of particular importance is the case when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/walking+arrow">walking arrow</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mn>2</mn></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{2}</annotation></semantics></math>.</p> </li> </ul> <h3 id="finite_2limits">Finite 2-Limits</h3> <p>A 2-limit is called <strong>finite</strong> if its diagram shape and its weight are both “finitely presentable” in a suitable sense (defined in terms of <a class="existingWikiWord" href="/nlab/show/computads">computads</a>; see <a href="#StreetLimitsIndexed">Street’s article</a> <em>Limits indexed by category-valued 2-functors</em> ). Pullbacks, comma objects, inserters, equifiers, and so on are all finite limits, as are powers by any finitely presented category. All finite limits can be constructed from pullbacks, a terminal object, and powers with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mn>2</mn></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{2}</annotation></semantics></math>.</p> <h3 id="(2,1)limit"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(2,1)</annotation></semantics></math>-limits</h3> <p>If the ambient <a class="existingWikiWord" href="/nlab/show/2-category">2-category</a> is in fact a <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-category">(2,1)-category</a> in that all <a class="existingWikiWord" href="/nlab/show/2-morphism">2-morphism</a>s are invertible then there is a rich set of tools available for handling the 2-limits in this context. We may say <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(2,1)</annotation></semantics></math>-limits</strong> and <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(2,1)</annotation></semantics></math>-colimits</strong> in this case.</p> <p>These are then a special case of the more general <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-limit">(∞,1)-limit</a>s and <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-colimit">(∞,1)-colimit</a>s in a <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a>. A <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-category">(2,1)-category</a> is a special case of an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a>.</p> <p>Traditionally, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-limit">(∞,1)-limit</a>s are best known in terms of the presentation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categories by <a class="existingWikiWord" href="/nlab/show/categories+with+weak+equivalences">categories with weak equivalences</a> in general and <a class="existingWikiWord" href="/nlab/show/model+categories">model categories</a> in particular. (2,1)-limits can often also be viewed in this way. The corresponding presentation of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-limits / <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(2,1)</annotation></semantics></math>-limits is called <strong><a class="existingWikiWord" href="/nlab/show/homotopy+limit">homotopy limit</a>s</strong> and <strong><a class="existingWikiWord" href="/nlab/show/homotopy+colimit">homotopy colimit</a>s</strong>.</p> <p>For instance 2-limits in the <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-category">(2,1)-category</a> <a class="existingWikiWord" href="/nlab/show/Grpd">Grpd</a> of <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a>s, <a class="existingWikiWord" href="/nlab/show/functor">functor</a>s and (necessarily) <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a>s. Are equivalently computed as <a class="existingWikiWord" href="/nlab/show/homotopy+limit">homotopy limit</a>s in the <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+sets">model structure on simplicial sets</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>sSet</mi> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">sSet_{Quillen}</annotation></semantics></math> of diagrams of <a class="existingWikiWord" href="/nlab/show/1-truncated">1-truncated</a> <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a>es. (The equivalence of homotopy limits with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-limits is discussed at <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-limit">(∞,1)-limit</a>).</p> <p>Or for instance, more generally, the 2-limits in any <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-sheaf">(2,1)-sheaf</a>(=<a class="existingWikiWord" href="/nlab/show/stack">stack</a>) <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-topos">(2,1)-topos</a> may be computed as <a class="existingWikiWord" href="/nlab/show/homotopy+limit">homotopy limit</a>s in a <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">model structure on simplicial presheaves</a> over the given <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-site">(2,1)-site</a> of diagrams of <a class="existingWikiWord" href="/nlab/show/1-truncated">1-truncated</a> <a class="existingWikiWord" href="/nlab/show/simplicial+presheaves">simplicial presheaves</a>. This includes as examples <a class="existingWikiWord" href="/nlab/show/big+topos">big (2,1)-toposes</a> such as those over the large sites <a class="existingWikiWord" href="/nlab/show/Top">Top</a> or <a class="existingWikiWord" href="/nlab/show/SmoothMfd">SmoothMfd</a> where computations with <a class="existingWikiWord" href="/nlab/show/topological+groupoid">topological groupoid</a>s/<a class="existingWikiWord" href="/nlab/show/topological+stack">topological stack</a>s, <a class="existingWikiWord" href="/nlab/show/Lie+groupoid">Lie groupoid</a>s/<a class="existingWikiWord" href="/nlab/show/differentiable+stack">differentiable stack</a>s etc. take place.</p> <h3 id="lax">Lax limits</h3> <p>A <strong>lax limit</strong> can be defined like a 2-limit, except that the triangles in the definition of a cone are required only to commute up to a specified <em>transformation</em>, not necessarily an isomorphism. In other words, in place of the 2-category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>D</mi><mo>,</mo><mi>Cat</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[D,Cat]</annotation></semantics></math> we use the 2-category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>D</mi><mo>,</mo><mi>Cat</mi><msub><mo stretchy="false">]</mo> <mi>l</mi></msub></mrow><annotation encoding="application/x-tex">[D,Cat]_l</annotation></semantics></math> whose morphisms are <a class="existingWikiWord" href="/nlab/show/lax+natural+transformations">lax natural transformations</a>; thus the lax limit <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> of a diagram <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> weighted by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math> is equipped with a universal lax natural transformation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi><mo>→</mo><mi>K</mi><mo stretchy="false">(</mo><mi>L</mi><mo>,</mo><mi>F</mi><mo>−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">J\to K(L,F-)</annotation></semantics></math>.</p> <p>This may look like a different concept, but in fact, for any weight <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math> there is another weight <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Q</mi> <mi>l</mi></msub><mo stretchy="false">(</mo><mi>J</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Q_l(J)</annotation></semantics></math> such that lax <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math>-weighted limits are the same as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Q</mi> <mi>l</mi></msub><mo stretchy="false">(</mo><mi>J</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Q_l(J)</annotation></semantics></math>-weighted 2-limits. Here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Q</mi> <mi>l</mi></msub></mrow><annotation encoding="application/x-tex">Q_l</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/lax+morphism+classifier">lax morphism classifier</a> for 2-functors. Therefore, lax limits are really a special case of 2-limits. Similarly, <strong>oplax limits</strong>, in which we use oplax natural transformations, are also a special case of 2-limits.</p> <p>There is a further simplification of lax limits in the case of “conical” lax limits where the weight <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi><mo>=</mo><mi>Δ</mi><mn>1</mn></mrow><annotation encoding="application/x-tex">J=\Delta 1</annotation></semantics></math> is constant at the <a class="existingWikiWord" href="/nlab/show/terminal+category">terminal category</a>. In this case, it is easy to check that a lax natural transformation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mn>1</mn><mo>→</mo><mi>K</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>F</mi><mo>−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Delta 1 \to K(X,F-)</annotation></semantics></math> is the same as a lax natural transformation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mi>X</mi><mo>→</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">\Delta X \to F</annotation></semantics></math>. Thus, a conical lax limit of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> is a representing object for such lax transformations.</p> <p>Here are some examples.</p> <ul> <li> <p>Lax terminal objects and lax products are the same as ordinary ones, since there are no commutativity conditions on the cones.</p> </li> <li> <p>The <strong>lax limit of an arrow</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">f:A\to B</annotation></semantics></math> is a universal object <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> equipped with projections <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><mi>L</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">p:L\to A</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mo>:</mo><mi>L</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">q:L\to B</annotation></semantics></math> and a 2-cell <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mi>p</mi><mo>→</mo><mi>q</mi></mrow><annotation encoding="application/x-tex">f p \to q</annotation></semantics></math>. Note that this is equivalent to a comma object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">/</mo><msub><mn>1</mn> <mi>B</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(f/1_B)</annotation></semantics></math>.</p> </li> <li> <p>The <strong>lax pullback</strong> of a cospan <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mover><mo>→</mo><mi>f</mi></mover><mi>C</mi><mover><mo>←</mo><mi>g</mi></mover><mi>B</mi></mrow><annotation encoding="application/x-tex">A \overset{f}{\to} C \overset{g}{\leftarrow} B</annotation></semantics></math> is a universal object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> equipped with projections <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><mi>P</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">p:P\to A</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mo>:</mo><mi>P</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">q:P\to B</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>:</mo><mi>P</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">r:P\to C</annotation></semantics></math>, and 2-cells <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mi>p</mi><mo>→</mo><mi>r</mi></mrow><annotation encoding="application/x-tex">f p \to r</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mi>q</mi><mo>→</mo><mi>r</mi></mrow><annotation encoding="application/x-tex">g q \to r</annotation></semantics></math>.</p> </li> </ul> <p>Note that lax pullbacks are <em>not</em> the same as <a class="existingWikiWord" href="/nlab/show/comma+objects">comma objects</a>. In general comma objects are much more useful, but there are 2-categories that admit all lax limits but do not admit comma objects, so using “lax pullback” to mean “comma object” can be misleading.</p> <p>A <strong>lax colimit</strong> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> is, of course, a lax limit in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>K</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">K^{op}</annotation></semantics></math>. Thus, it is a representing object for lax natural transformations <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi><mo>→</mo><mi>K</mi><mo stretchy="false">(</mo><mi>F</mi><mo>−</mo><mo>,</mo><mi>L</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">J \to K(F-,L)</annotation></semantics></math>. There is a subtlety here, however, because in the case <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi><mo>=</mo><mi>Δ</mi><mn>1</mn></mrow><annotation encoding="application/x-tex">J=\Delta 1</annotation></semantics></math>, a lax natural transformation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mn>1</mn><mo>→</mo><mi>K</mi><mo stretchy="false">(</mo><mi>F</mi><mo>−</mo><mo>,</mo><mi>L</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Delta 1 \to K(F-,L)</annotation></semantics></math> is the same as an <em>oplax</em> natural transformation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>→</mo><mi>Δ</mi><mi>L</mi></mrow><annotation encoding="application/x-tex">F \to \Delta L</annotation></semantics></math>. Thus, it is easy to mistakenly say “lax colimit” when one really means “oplax colimit” and vice versa.</p> <div class="un_remark"> <h6 id="remark">Remark</h6> <p>With this in mind, one might be tempted to switch the meanings of “lax colimit” and “oplax colimit”, but there is a good reason not to. Recall that a lax <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math>-weighted limit is the same as an ordinary <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Q</mi> <mi>l</mi></msub><mo stretchy="false">(</mo><mi>J</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Q_l(J)</annotation></semantics></math>-weighted limit. Standard terminology in enriched category theory is that a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math>-weighted colimit in an enriched category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> is the same as a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math>-weighted limit in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>K</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">K^{op}</annotation></semantics></math>, and indeed in that generality there is no other option. Thus, a lax <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math>-weighted colimit in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> should be an ordinary <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Q</mi> <mi>l</mi></msub><mo stretchy="false">(</mo><mi>J</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Q_l(J)</annotation></semantics></math>-weighted colimit in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>, hence a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Q</mi> <mi>l</mi></msub><mo stretchy="false">(</mo><mi>J</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Q_l(J)</annotation></semantics></math>-weighted limit in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>K</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">K^{op}</annotation></semantics></math>, and thus a lax <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math>-weighted limit in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>K</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">K^{op}</annotation></semantics></math>.</p> </div> <p>Here are some examples of lax and oplax colimits:</p> <ul> <li> <p>A <a class="existingWikiWord" href="/nlab/show/Kleisli+object">Kleisli object</a> is a lax colimit of a <a class="existingWikiWord" href="/nlab/show/monad">monad</a>, regarded as a diagram in a 2-category.</p> </li> <li> <p>The <a class="existingWikiWord" href="/nlab/show/collage">collage</a> of a <a class="existingWikiWord" href="/nlab/show/profunctor">profunctor</a> is its lax colimit, regarded as a diagram in the 2-category <a class="existingWikiWord" href="/nlab/show/Prof">Prof</a>.</p> </li> <li> <p>When <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is a category, the <a class="existingWikiWord" href="/nlab/show/Grothendieck+construction">Grothendieck construction</a> of a functor <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\to Cat</annotation></semantics></math> is the same as its <em>oplax</em> colimit; see <a href="http://ncatlab.org/nlab/show/Grothendieck+construction#AsALaxColimit">here</a>.</p> </li> </ul> <h3 id="2ColimitsInCat">2-Colimits in <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></h3> <p>As shown <a href="http://ncatlab.org/nlab/show/Grothendieck+construction#AsALaxColimit">here</a>, 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 an ordinary category and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo lspace="verythinmathspace">:</mo><mi>C</mi><mo>→</mo><mi>Cat</mi></mrow><annotation encoding="application/x-tex">F \colon C \to Cat</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/pseudofunctor">pseudofunctor</a>, then the <a class="existingWikiWord" href="/nlab/show/oplax+colimit">oplax colimit</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> is given by the <a class="existingWikiWord" href="/nlab/show/Grothendieck+construction">Grothendieck construction</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∫</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">\int F</annotation></semantics></math> — and its <a class="existingWikiWord" href="/nlab/show/pseudo-colimit">pseudo-colimit</a> is given by <a class="existingWikiWord" href="/nlab/show/localization">formally inverting</a> the <a class="existingWikiWord" href="/nlab/show/cartesian+morphism">opcartesian</a> morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∫</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">\int F</annotation></semantics></math>. This yields a construction of certain pseudo 2-colimits in <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>.</p> <p>Moreover, a similar result holds more generally when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/bicategory">bicategory</a>. In this case, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∫</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">\int F</annotation></semantics></math> is also a bicategory: a 2-cell from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>m</mi><mo lspace="verythinmathspace">:</mo><mi>c</mi><mo>→</mo><mi>d</mi><mo>,</mo><mi>f</mi><mo lspace="verythinmathspace">:</mo><msub><mi>m</mi> <mo>*</mo></msub><mi>x</mi><mo>→</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(m \colon c \to d, f \colon m_*x \to y)</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo lspace="verythinmathspace">:</mo><mi>c</mi><mo>→</mo><mi>d</mi><mo>,</mo><mi>g</mi><mo lspace="verythinmathspace">:</mo><msub><mi>n</mi> <mo>*</mo></msub><mi>x</mi><mo>→</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n \colon c \to d, g \colon n_*x \to y)</annotation></semantics></math> is given by a 2-cell <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi><mo lspace="verythinmathspace">:</mo><mi>m</mi><mo>⇒</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">\mu \colon m \Rightarrow n</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>μ</mi> <mo>*</mo></msub><mi>x</mi></mrow><annotation encoding="application/x-tex">\mu_* x</annotation></semantics></math> is a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>→</mo><mi>g</mi></mrow><annotation encoding="application/x-tex">f \to g</annotation></semantics></math> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math>.</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">\pi_*</annotation></semantics></math> denote the functor that sends a bicategory <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> to the category whose objects are those of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> and whose hom-sets are the <a class="existingWikiWord" href="/nlab/show/connected+category">connected components</a> of the hom-categories of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>; let also <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">d_*</annotation></semantics></math> denote the functor that sends a category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> to the corresponding locally discrete bicategory. Then there is 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><mo stretchy="false">[</mo><mi>K</mi><mo>,</mo><msub><mi>d</mi> <mo>*</mo></msub><mi>X</mi><mo stretchy="false">]</mo><mo>≃</mo><mo stretchy="false">[</mo><msub><mi>π</mi> <mo>*</mo></msub><mi>K</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> [K, d_* X] \simeq [\pi_* K, X] </annotation></semantics></math></div> <p>It is straightforward to check that the first of the above facts extends to the bicategorical case:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Lax</mi><mo stretchy="false">(</mo><mi>F</mi><mo>,</mo><mi>Δ</mi><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><mo stretchy="false">[</mo><mstyle displaystyle="false"><mo>∫</mo></mstyle><mi>F</mi><mo>,</mo><msub><mi>d</mi> <mo>*</mo></msub><mi>X</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> Lax(F, \Delta X) \simeq [{\textstyle \int} F, d_* X] </annotation></semantics></math></div> <p>as does the fact that a lax transformation on the left is pseudo if and only if the corresponding functor on the right inverts the opcartesian morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∫</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">\int F</annotation></semantics></math>. It is almost trivial that the adjunction <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mo>*</mo></msub><mo>⊣</mo><msub><mi>d</mi> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">\pi_* \dashv d_*</annotation></semantics></math> holds when restricted to the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mo stretchy="false">]</mo> <mrow><msup><mi>S</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow></msub></mrow><annotation encoding="application/x-tex">[-, -]_{S^{-1}}</annotation></semantics></math> that takes two categories or bicategories to the full subcategory of functors that invert the class <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> of morphisms. Taking <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> to be the opcartesian morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∫</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">\int F</annotation></semantics></math>, then, we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ps</mi><mo stretchy="false">(</mo><mi>F</mi><mo>,</mo><mi>Δ</mi><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><mo stretchy="false">[</mo><mstyle displaystyle="false"><mo>∫</mo></mstyle><mi>F</mi><mo>,</mo><msub><mi>d</mi> <mo>*</mo></msub><mi>X</mi><msub><mo stretchy="false">]</mo> <mrow><msup><mi>S</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow></msub><mo>≃</mo><mo stretchy="false">[</mo><msub><mi>π</mi> <mo>*</mo></msub><mstyle displaystyle="false"><mo>∫</mo></mstyle><mi>F</mi><mo>,</mo><mi>X</mi><msub><mo stretchy="false">]</mo> <mrow><msup><mi>S</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow></msub><mo>≃</mo><mo stretchy="false">[</mo><mo stretchy="false">(</mo><msub><mi>π</mi> <mo>*</mo></msub><mstyle displaystyle="false"><mo>∫</mo></mstyle><mi>F</mi><mo stretchy="false">)</mo><mo stretchy="false">[</mo><msup><mi>S</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">]</mo><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> Ps(F, \Delta X) \simeq [{\textstyle \int} F, d_* X]_{S^{-1}} \simeq [\pi_* {\textstyle \int} F, X]_{S^{-1}} \simeq [(\pi_* {\textstyle \int} F)[S^{-1}], X] </annotation></semantics></math></div> <p>Hence the pseudo colimit of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> is got by taking its bicategory of elements, applying the ‘local <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\pi_0</annotation></semantics></math>’ functor, and then inverting the (images of the) opcartesian morphisms as usual.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/limit">limit</a></p> </li> <li> <p><strong>2-limit</strong></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/strict+2-limit">strict 2-limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/marked+2-limit">marked 2-limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/double+limit">double limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-limits+and+2-colimits+in+2-categories+of+2-algebras">2-limits and 2-colimits in 2-categories of 2-algebras</a></p> </li> <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/homotopy+limit">homotopy limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/lax+%28%E2%88%9E%2C1%29-colimit">lax (∞,1)-colimit</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quasi-limit">quasi-limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/coherence+for+bicategories+with+finite+limits">coherence for bicategories with finite limits</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/representable+2-category">representable 2-category</a></p> </li> </ul> <h2 id="references">References</h2> <ul> <li id="Street74"> <p><a class="existingWikiWord" href="/nlab/show/Ross+Street">Ross Street</a>, <em>Elementary cosmoi I</em> (§6) in <em>Category Seminar</em>, Lecture Notes in Mathematics <strong>420</strong>, Springer (1974) [<a href="https://doi.org/10.1007/BFb0063103">doi:10.1007/BFb0063103</a>]</p> </li> <li id="StreetLimitsIndexed"> <p><a class="existingWikiWord" href="/nlab/show/Ross+Street">Ross Street</a>, <em>Limits indexed by category-valued 2-functors</em> Journal of Pure and Applied Algebra <strong>8</strong>, Issue 2 (1976) pp 149-181. doi:<a href="https://doi.org/10.1016/0022-4049(76%2990013-X">10.1016/0022-4049(76)90013-X</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Max+Kelly">Max Kelly</a>, <em>Elementary observations on 2-categorical limits</em>, Bulletin of the Australian Mathematical Society (1989), 39: 301-317, doi:<a href="https://doi.org/10.1017/S0004972700002781">10.1017/S0004972700002781</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ross+Street">Ross Street</a>, <em>Fibrations in Bicategories</em>, Cahiers de Topologie et Géométrie Différentielle Catégoriques, Volume <strong>21</strong> (1980) no. 2, pp 111-160. <a href="http://www.numdam.org/item?id=CTGDC_1980__21_2_111_0">Numdam</a> and correction, Cahiers de Topologie et Géométrie Différentielle Catégoriques, Volume <strong>28</strong> (1987) no. 1, pp 53-56 <a href="http://www.numdam.org/item?id=CTGDC_1987__28_1_53_0">Numdam</a></p> </li> </ul> <p>Section 6, page 37 in</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Steve+Lack">Steve Lack</a>, <em>A 2-categories companion</em>. In: Baez J., May J. (eds) Towards Higher Categories. The IMA Volumes in Mathematics and its Applications, vol <strong>152</strong> 2010 Springer, New York, NY. doi:<a href="https://doi.org/10.1007/978-1-4419-1524-5_4">10.1007/978-1-4419-1524-5_4</a>, arXiv:<a href="http://arxiv.org/abs/math.CT/0702535">math.CT/0702535</a>.</p> </li> <li> <p>G. J. Bird, <a class="existingWikiWord" href="/nlab/show/Max+Kelly">Max Kelly</a>, <a class="existingWikiWord" href="/nlab/show/John+Power">John Power</a>, <a class="existingWikiWord" href="/nlab/show/Ross+Street">Ross Street</a>, <em>Flexible limits for 2-categories</em>, Journal of Pure and Applied Algebra <strong>61</strong> Issue 1 (1989) pp 1-27. doi:<a href="http://dx.doi.org/10.1016/0022-4049(89%2990065-0">10.1016/0022-4049(89)90065-0</a> Chapters 3,4,5 in</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Thomas+Fiore">Thomas Fiore</a>, <em>Pseudo Limits, Biadjoints, and Pseudo Algebras: Categorical Foundations of Conformal Field Theory</em>, Mem. Amer. Math. Soc. <strong>182</strong> (2006), no. 860 (<a href="http://arxiv.org/abs/math/0408298">arXiv:math/0408298</a>) (<a href="http://bookstore.ams.org/memo-182-860">AMS page</a>, <a href="https://books.google.com.au/books?id=y_DUCQAAQBAJ">Google Books</a>)</p> </li> </ul> <p>Another example of a 2-limit is the <a class="existingWikiWord" href="/nlab/show/category+of+monoids">category of monoids</a>:</p> <ul> <li>Łukasz Sienkiewicz and <span class="newWikiWord">Marek Zawadowski<a href="/nlab/new/Marek+Zawadowski">?</a></span>, <em>Weights for Monoids and Actions of Monoids</em> <a href="https://arxiv.org/abs/1306.3215">arXiv:1306.3215</a> (2013).</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on August 23, 2024 at 08:38:12. 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