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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='#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="definition">Definition</h2> <p>A <strong>finite limit</strong> is a <a class="existingWikiWord" href="/nlab/show/limit">limit</a> over a finite <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a> - that is, one whose shape is a <a class="existingWikiWord" href="/nlab/show/finite+category">finite category</a>.</p> <p>More generally, in <a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a>, a finite limit is a <a class="existingWikiWord" href="/nlab/show/limit">limit</a> of a diagram that is a finite <a class="existingWikiWord" href="/nlab/show/%28n%2Cr%29-category">(n,r)-category</a>.</p> <p>A <a class="existingWikiWord" href="/nlab/show/category">category</a> that has all finite limits is called a <em><a class="existingWikiWord" href="/nlab/show/finitely+complete+category">finitely complete category</a></em> or a (finitary) <em><a class="existingWikiWord" href="/nlab/show/essentially+algebraic+theory">essentially algebraic theory</a></em>.</p> <p>A <a class="existingWikiWord" href="/nlab/show/functor">functor</a> that <a class="existingWikiWord" href="/nlab/show/preserved+limit">preserves</a> all finite limits is called <em><a class="existingWikiWord" href="/nlab/show/left+exact+functor">left exact functor</a></em>, a <em>lex functor</em>, a <em>cartesian functor</em>, or a <em>finitely continuous functor</em>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/2-category">2-category</a> of <a class="existingWikiWord" href="/nlab/show/finitely+complete+categories">finitely complete categories</a>, <a class="existingWikiWord" href="/nlab/show/left+exact+functors">left exact functors</a> and <a class="existingWikiWord" href="/nlab/show/natural+transformations">natural transformations</a> is often denoted <em><a class="existingWikiWord" href="/nlab/show/Lex">Lex</a></em>.</p> <h2 id="Examples">Examples</h2> <p> <div class='num_remark'> <h6>Example</h6> <p>A <a class="existingWikiWord" href="/nlab/show/product">product</a> is a finite limit iff it is a <a class="existingWikiWord" href="/nlab/show/finite+product">finite product</a>, hence if its factors are <a class="existingWikiWord" href="/nlab/show/indexed+set">indexed</a> by a <a class="existingWikiWord" href="/nlab/show/finite+set">finite set</a>.</p> <p>In particular, <a class="existingWikiWord" href="/nlab/show/binary+products">binary products</a> are finite limits and the <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a> (being the <a class="existingWikiWord" href="/nlab/show/limit">limit</a> over the <a class="existingWikiWord" href="/nlab/show/empty+diagram">empty diagram</a>) is a finite limit.</p> </div> </p> <p> <div class='num_remark'> <h6>Example</h6> <p>Other common finite limits are <a class="existingWikiWord" href="/nlab/show/pullbacks">pullbacks</a> and <a class="existingWikiWord" href="/nlab/show/equalizers">equalizers</a>.</p> </div> </p> <p> <div class='num_remark'> <h6>Example</h6> <p>The <a class="existingWikiWord" href="/nlab/show/fixed+locus">fixed locus</a> of a <a class="existingWikiWord" href="/nlab/show/group+action">group action</a> is a finite limit if the group in question is a <a class="existingWikiWord" href="/nlab/show/finite+group">finite group</a>.</p> </div> </p> <p> <div class='num_remark'> <h6>Example</h6> <p>The <a class="existingWikiWord" href="/nlab/show/functor">functor</a> of <a class="existingWikiWord" href="/nlab/show/geometric+realization">geometric realization</a> of <a class="existingWikiWord" href="/nlab/show/simplicial+sets">simplicial sets</a> into <a class="existingWikiWord" href="/nlab/show/compactly+generated+topological+spaces">compactly generated topological spaces</a> <a class="existingWikiWord" href="/nlab/show/preserved+limit">preserves</a> finite limits (is a <a class="existingWikiWord" href="/nlab/show/left+exact+functor">left exact functor</a>), see <a href="geometric+realization#GeometricRealizationIsLeftExact">there</a>.</p> <p>More generally, the <a class="existingWikiWord" href="/nlab/show/functor">functor</a> of <a class="existingWikiWord" href="/nlab/show/geometric+realization+of+simplicial+topological+spaces">geometric realization of simplicial topological spaces</a> (with respect to <a class="existingWikiWord" href="/nlab/show/compactly+generated+topological+spaces">compactly generated topological spaces</a>) preserves finite limits (by <a href="geometric+realization+of+simplicial+topological+spaces#GeometricRealizationPreservesFiniteLimits">this Prop.</a>).</p> </div> </p> <h2 id="Properties">Properties</h2> <div class="num_prop" id="FiniteLimitsFromPullbacksAndTerminalObject"> <h6 id="proposition">Proposition</h6> <p>For 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">\mathcal{C}</annotation></semantics></math> the following are equivalent:</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> has all <a class="existingWikiWord" href="/nlab/show/finite+limits">finite limits</a>.</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> has all <a class="existingWikiWord" href="/nlab/show/equalizers">equalizers</a> and <a class="existingWikiWord" href="/nlab/show/binary+products">binary products</a>.</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> has all <a class="existingWikiWord" href="/nlab/show/pullbacks">pullbacks</a> and a <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a>.</p> </li> </ol> <p>In fact, either class of limits may be expressed in terms of another, so that for a <a class="existingWikiWord" href="/nlab/show/functor">functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>𝒞</mi><mo>→</mo><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">F \;\colon\; \mathcal{C} \to \mathcal{D}</annotation></semantics></math> the following are equivalent:</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/preserved+limit">preserves</a> <a class="existingWikiWord" href="/nlab/show/finite+limits">finite limits</a>.</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/preserved+limit">preserves</a> <a class="existingWikiWord" href="/nlab/show/equalizers">equalizers</a> and <a class="existingWikiWord" href="/nlab/show/binary+products">binary products</a>.</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/preserved+limit">preserves</a> <a class="existingWikiWord" href="/nlab/show/pullbacks">pullbacks</a> and the <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a>.</p> </li> </ol> </div> <p>(The first statement may be found, e.g., in <a href="#Borceux94">Borceux 1994, Prop. 2.8.2</a>. From the proof there the second statement immediately follows.)</p> <p> <div class='num_remark'> <h6>Remark</h6> <p>The equivalence of the first two items in Prop. <a class="maruku-ref" href="#FiniteLimitsFromPullbacksAndTerminalObject"></a> is the finite analog of how a category with equalizers and all small products has all <a class="existingWikiWord" href="/nlab/show/small+limits">small limits</a>.<br /></p> </div> </p> <p> <div class='num_remark' id='LFiniteLimits'> <h6>Remark</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/saturation+of+a+class+of+limits">saturation</a> to <a class="existingWikiWord" href="/nlab/show/L-finite+limits">L-finite limits</a>)</strong> <br /> Prop. <a class="maruku-ref" href="#FiniteLimitsFromPullbacksAndTerminalObject"></a> implies that finite limits are contained in the <a class="existingWikiWord" href="/nlab/show/saturation+of+a+class+of+limits">saturation</a> of the class containing only <a class="existingWikiWord" href="/nlab/show/finite+products">finite products</a> and <a class="existingWikiWord" href="/nlab/show/equalizers">equalizers</a>, and also that of the class containing only <a class="existingWikiWord" href="/nlab/show/pullbacks">pullbacks</a> and <a class="existingWikiWord" href="/nlab/show/terminal+objects">terminal objects</a>.</p> <p>But the existence of finite limits in fact implies that of the larger class of <a class="existingWikiWord" href="/nlab/show/L-finite+limits">L-finite limits</a>, and this is the largest class of limits implied by the existence of finite limits (<a href="#Pare90">Paré 1990, Prop 7</a>, see <a href="L-finite+category#CharacterizationsOfLFiniteLimits">there</a>).</p> <p>Therefore Prop. <a class="maruku-ref" href="#FiniteLimitsFromPullbacksAndTerminalObject"></a> implies that <a class="existingWikiWord" href="/nlab/show/L-finite+limits">L-finite limits</a> constitute also the full <a class="existingWikiWord" href="/nlab/show/saturation+of+a+class+of+limits">saturation class</a> of both equalizers+finite products as well as pullbacks+terminal objects.</p> </div> </p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/finite+%28%E2%88%9E%2C1%29-limit">finite (∞,1)-limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/finite+set">finite set</a>, <a class="existingWikiWord" href="/nlab/show/finite+group">finite group</a>, <a class="existingWikiWord" href="/nlab/show/finite+category">finite category</a></p> </li> </ul> <h2 id="references">References</h2> <p>Textbook account:</p> <ul> <li id="Borceux94"><a class="existingWikiWord" href="/nlab/show/Francis+Borceux">Francis Borceux</a>, around Def. 2.8.2 in: <em><a class="existingWikiWord" href="/nlab/show/Handbook+of+Categorical+Algebra">Handbook of Categorical Algebra</a></em> Vol. 1: <em>Basic Category Theory</em>, Encyclopedia of Mathematics and its Applications <strong>50</strong>, Cambridge University Press (1994) <p>(<a href="https://doi.org/10.1017/CBO9780511525858">doi:10.1017/CBO9780511525858</a>)</p> </li> </ul> <p>On the saturation to <a class="existingWikiWord" href="/nlab/show/L-finite+limits">L-finite limits</a>:</p> <ul> <li id="Pare90"><a class="existingWikiWord" href="/nlab/show/Robert+Par%C3%A9">Robert Paré</a>, Section 3 of: <em>Simply connected limits</em>. Can. J. Math., Vol. XLH, No. 4, 1990, pp. 731-746 (<a href="https://doi.org/10.4153/CJM-1990-038-6">doi:10.4153/CJM-1990-038-6</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on August 25, 2021 at 07:59:54. 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