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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="preservation_of_limits">Preservation of limits</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definitions'>Definitions</a></li> <li><a href='#Examples'>Examples</a></li> <li><a href='#preservation_of_weighted_limits'>Preservation of weighted limits</a></li> <li><a href='#preservation_of_limits_that_dont_exist'>Preservation of limits that don't exist</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi><mo lspace="verythinmathspace">:</mo><mi>I</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">J\colon I \to C</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a> and <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> is its <a class="existingWikiWord" href="/nlab/show/limit">limit</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>, then we may naïvely say that this limit is <em>preserved</em> by a <a class="existingWikiWord" href="/nlab/show/functor">functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo lspace="verythinmathspace">:</mo><mi>C</mi><mo>→</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">F\colon C \to D</annotation></semantics></math> if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(x)</annotation></semantics></math> is the limit of the <a class="existingWikiWord" href="/nlab/show/composite">composite</a> diagram <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mover><mo>→</mo><mi>J</mi></mover><mi>C</mi><mover><mo>→</mo><mi>F</mi></mover><mi>D</mi></mrow><annotation encoding="application/x-tex">I \overset{J}\to C \overset{F}\to D</annotation></semantics></math>. However, it is not enough to state this at the level of objects; we also need to impose some coherence conditions, preserving the entire universal <a class="existingWikiWord" href="/nlab/show/cone">cone</a>. Furthermore, we can use a trick involving the <a class="existingWikiWord" href="/nlab/show/Yoneda+embedding">Yoneda embedding</a> to get a meaningful condition even if <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> has no limit 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> at all.</p> <h2 id="definitions">Definitions</h2> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi><mo lspace="verythinmathspace">:</mo><mi>I</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">J\colon I \to C</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a> and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo lspace="verythinmathspace">:</mo><mi>C</mi><mo>→</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">F\colon C \to D</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/functor">functor</a>.</p> <p>Recall (see <a class="existingWikiWord" href="/nlab/show/limit">limit</a>) that a <a class="existingWikiWord" href="/nlab/show/cone">cone</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</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> may be defined as an <a class="existingWikiWord" href="/nlab/show/object">object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> together with a <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a> to <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> from the composite <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mo lspace="0em" rspace="thinmathspace">const</mo> <mi>x</mi> <mi>I</mi></msubsup><mo lspace="verythinmathspace">:</mo><mi>I</mi><mover><mo>→</mo><mo>!</mo></mover><mstyle mathvariant="bold"><mn>1</mn></mstyle><mover><mo>→</mo><mrow><mo stretchy="false">{</mo><mi>x</mi><mo stretchy="false">}</mo></mrow></mover><mi>C</mi></mrow><annotation encoding="application/x-tex">\const^I_x\colon I \overset{!}\to \mathbf{1} \overset{\{x\}}\to C</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mn>1</mn></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{1}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/terminal+category">terminal category</a>. Then a <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a> in the category of these cones (if it exists) is a <a class="existingWikiWord" href="/nlab/show/limit">limit</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</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>. Thus, a limit consists of an 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> and a natural transformation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo lspace="verythinmathspace">:</mo><msubsup><mo lspace="0em" rspace="thinmathspace">const</mo> <mi>x</mi> <mi>I</mi></msubsup><mo>→</mo><mi>J</mi></mrow><annotation encoding="application/x-tex">\eta\colon \const^I_x \to J</annotation></semantics></math>.</p> <p>The functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> <strong>preserves</strong> the limit <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>η</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x,\eta)</annotation></semantics></math> if <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>x</mi><mo stretchy="false">)</mo><mo>,</mo><mi>F</mi><mo>⋅</mo><mi>η</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(F(x),F\cdot\eta)</annotation></semantics></math> is a limit of the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>∘</mo><mi>J</mi></mrow><annotation encoding="application/x-tex">F \circ J</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>. (Here, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>⋅</mo><mi>η</mi><mo lspace="verythinmathspace">:</mo><msubsup><mo lspace="0em" rspace="thinmathspace">const</mo> <mrow><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow> <mi>I</mi></msubsup><mo>→</mo><mi>F</mi><mo>∘</mo><mi>J</mi></mrow><annotation encoding="application/x-tex">F\cdot\eta\colon \const^I_{F(x)} \to F \circ J</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/whiskering">whiskering</a>.)</p> <p>Dually, <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>preserves</strong> a <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math> if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>F</mi> <mo lspace="0em" rspace="thinmathspace">op</mo></msup><mo lspace="verythinmathspace">:</mo><msup><mi>C</mi> <mo lspace="0em" rspace="thinmathspace">op</mo></msup><mo>→</mo><msup><mi>D</mi> <mo lspace="0em" rspace="thinmathspace">op</mo></msup></mrow><annotation encoding="application/x-tex">F^\op\colon C^\op \to D^\op</annotation></semantics></math> preserves it as a limit of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>J</mi> <mo lspace="0em" rspace="thinmathspace">op</mo></msup><mo lspace="verythinmathspace">:</mo><msup><mi>I</mi> <mo lspace="0em" rspace="thinmathspace">op</mo></msup><mo>→</mo><msup><mi>C</mi> <mo lspace="0em" rspace="thinmathspace">op</mo></msup></mrow><annotation encoding="application/x-tex">J^\op\colon I^\op \to C^\op</annotation></semantics></math>.</p> <p>For instance:</p> <ul> <li> <p>Let <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> be the <a class="existingWikiWord" href="/nlab/show/empty+category">empty category</a>, so that a limit of the unique functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi><mo lspace="verythinmathspace">:</mo><mi>I</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">J\colon I \to C</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a> <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>. Then <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> preserves this terminal object if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(1)</annotation></semantics></math> is a terminal object of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>.</p> </li> <li> <p>Let <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> be the <a class="existingWikiWord" href="/nlab/show/discrete+category">discrete category</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>, so that <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> picks out two objects <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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> and the limit of <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 a <a class="existingWikiWord" href="/nlab/show/product">product</a> <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> of <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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math>. Note that this product comes equipped with product projections <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi><mo lspace="verythinmathspace">:</mo><mi>a</mi><mo>×</mo><mi>b</mi><mo>→</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">\pi\colon 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>ρ</mi><mo lspace="verythinmathspace">:</mo><mi>a</mi><mo>×</mo><mi>b</mi><mo>→</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">\rho\colon a \times b \to b</annotation></semantics></math>. Then <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> preserves this product if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>a</mi><mo>×</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(a \times b)</annotation></semantics></math> is a product of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(a)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(b)</annotation></semantics></math> and furthermore the product projections are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>π</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(\pi)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(\rho)</annotation></semantics></math>.</p> </li> </ul> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> preserves all limits or colimits of a given type (i.e. over a given category <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>), we simply say that <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> preserves that sort of limit (e.g. <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> preserves <a class="existingWikiWord" href="/nlab/show/products">products</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> preserves <a class="existingWikiWord" href="/nlab/show/equalizers">equalizers</a>, etc.).</p> <p>A functor that preserves all small limits 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> that exist is called a <strong><a class="existingWikiWord" href="/nlab/show/continuous+functor">continuous functor</a></strong>. Usually this term is only used 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> has all small limits, i.e. is a <a class="existingWikiWord" href="/nlab/show/complete+category">complete category</a>.</p> <h2 id="Examples">Examples</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/limits+preserve+limits">limits preserve limits</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hom-functors+preserve+limits">hom-functors preserve limits</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoints+preserve+%28co-%29limits">adjoints preserve (co-)limits</a></p> </li> <li id="ExampleYonedaEmbedding"> <p>the <a class="existingWikiWord" href="/nlab/show/Yoneda+embedding">Yoneda embedding</a> preserves limits (see <a href="Yoneda+embedding#PreservesLimits">there</a>)</p> </li> </ul> <h2 id="preservation_of_weighted_limits">Preservation of weighted limits</h2> <p>Analogously, an <a class="existingWikiWord" href="/nlab/show/enriched+functor">enriched functor</a> between <a class="existingWikiWord" href="/nlab/show/enriched+categories">enriched categories</a> may preserve <a class="existingWikiWord" href="/nlab/show/weighted+limits">weighted limits</a>. Are there any tricky points that we should mention?</p> <h2 id="preservation_of_limits_that_dont_exist">Preservation of limits that don't exist</h2> <p>Sometimes we want to say that a functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo lspace="verythinmathspace">:</mo><mi>C</mi><mo>→</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">F\colon C \to D</annotation></semantics></math> preserves a limit that does not actually exist 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>. For instance, a <strong>finitely continuous functor</strong> is usually defined as one that preserves all finite limits. If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/finitely+complete+category">finitely complete category</a>, then this is fine; such a functor is called <strong><a class="existingWikiWord" href="/nlab/show/left+exact+functor">left exact</a></strong>. But what 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> does not have all finite limits?</p> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> 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/locally+small+category">locally small</a>, then we can use the <a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a> to turn the question into one involving categories that <em>do</em> have the required limits (and in fact have all limits), the <a class="existingWikiWord" href="/nlab/show/presheaf+categories">presheaf categories</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[C^op,Set]</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>D</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[D^op,Set]</annotation></semantics></math>. (For colimits, use <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>C</mi><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[C,Set]</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>D</mi><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[D,Set]</annotation></semantics></math>; for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-enriched categories, use <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>V</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[C^op,V]</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>D</mi> <mi>op</mi></msup><mo>,</mo><mi>V</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[D^op,V]</annotation></semantics></math>, which will work if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> is complete.)</p> <p>The left <a class="existingWikiWord" href="/nlab/show/Kan+extension">Kan extension</a> of the composite <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mover><mo>→</mo><mi>F</mi></mover><mi>D</mi><mover><mo>↪</mo><mi>Yon</mi></mover><mo stretchy="false">[</mo><msup><mi>D</mi> <mo lspace="0em" rspace="thinmathspace">op</mo></msup><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">C \overset{F}\to D \overset{Yon}\hookrightarrow [D^\op,Set]</annotation></semantics></math> along the <a class="existingWikiWord" href="/nlab/show/Yoneda+embedding">Yoneda embedding</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mover><mo>↪</mo><mi>Yon</mi></mover><mo stretchy="false">[</mo><msup><mi>C</mi> <mo lspace="0em" rspace="thinmathspace">op</mo></msup><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">C \overset{Yon}\hookrightarrow [C^\op,Set]</annotation></semantics></math> (which always exists) is a functor from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[C^op,Set]</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>D</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[D^op,Set]</annotation></semantics></math>, which may be written as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>⊗</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">- \otimes F</annotation></semantics></math> (alluding to the <a class="existingWikiWord" href="/nlab/show/bimodule">bimodule</a> nature of <a class="existingWikiWord" href="/nlab/show/profunctors">profunctors</a>). A diagram <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi><mo lspace="verythinmathspace">:</mo><mi>I</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">J\colon I \to C</annotation></semantics></math> becomes a diagram <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mover><mo>→</mo><mi>J</mi></mover><mi>C</mi><mover><mo>↪</mo><mi>Yon</mi></mover><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">I \overset{J}\to C \overset{Yon}\hookrightarrow [C^op,Set]</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[C^op,Set]</annotation></semantics></math>, where it has a limit. If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>⊗</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">- \otimes F</annotation></semantics></math> preserves this limit, then we say that <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>preserves</strong> the hypothetical limit of <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>.</p> <p>Since the Yoneda embedding preserves and <a class="existingWikiWord" href="/nlab/show/reflected+limit">reflects</a> all limits, if <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> has a limit 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>, then this condition is equivalent to the condition that <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> preserve it in the ordinary sense, but in general it is stronger than requiring that <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> preserve the limit only if it exists 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>.</p> <p>Finishing the motivating example, a <strong><a class="existingWikiWord" href="/nlab/show/flat+functor">flat functor</a></strong> may be defined as one that preserves all finite limits, whether or not they exist.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p>(<a class="existingWikiWord" href="/nlab/show/cocontinuous+functor">co</a>)<a class="existingWikiWord" href="/nlab/show/continuous+functor">continuous functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/reflected+limit">reflected limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/created+limit">created limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/lifted+limit">lifted limit</a></p> </li> </ul> <h2 id="references">References</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Saunders+MacLane">Saunders MacLane</a>, §V.4 of: <em><a class="existingWikiWord" href="/nlab/show/Categories+for+the+Working+Mathematician">Categories for the Working Mathematician</a></em>, Graduate Texts in Mathematics <strong>5</strong> Springer (1971, second ed. 1997) [<a href="https://link.springer.com/book/10.1007/978-1-4757-4721-8">doi:10.1007/978-1-4757-4721-8</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Francis+Borceux">Francis Borceux</a>, §2.4 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> [<a href="https://doi.org/10.1017/CBO9780511525858">doi:10.1017/CBO9780511525858</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Emily+Riehl">Emily Riehl</a>, §3.3 in: <em><a class="existingWikiWord" href="/nlab/show/Category+Theory+in+Context">Category Theory in Context</a></em>, Dover Publications (2017) [<a href="http://www.math.jhu.edu/~eriehl/context.pdf">pdf</a>, <a href="http://www.math.jhu.edu/~eriehl/context/">book website</a>]</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on May 10, 2024 at 10:24:13. 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