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value="Search" style="display:inline-block; 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"> <p class="show_diff"> Showing changes from revision #17 to #18: <ins class="diffins">Added</ins> | <del class="diffdel">Removed</del> | <del class="diffmod">Chan</del><ins class="diffmod">ged</ins> </p> <div class='rightHandSide'> <div class='toc clickDown' tabindex='0'> <h3 id='context'>Context</h3> <h4 id='limits_and_colimits'>Limits and colimits</h4> <div class='hide'> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/limit'>limits and colimits</a></strong></p> <h2 id='1categorical'>1-Categorical</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/limit'>limit and colimit</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/limits+and+colimits+by+example'>limits and colimits by example</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/commutativity+of+limits+and+colimits'>commutativity of limits and colimits</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/small+limit'>small limit</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/filtered+colimit'>filtered colimit</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/directed+colimit'>directed colimit</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/sequential+limit'>sequential colimit</a></li> </ul> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/sifted+colimit'>sifted colimit</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/connected+limit'>connected limit</a>, <a class='existingWikiWord' href='/nlab/show/diff/wide+pullback'>wide pullback</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/preserved+limit'>preserved limit</a>, <a class='existingWikiWord' href='/nlab/show/diff/reflected+limit'>reflected limit</a>, <a class='existingWikiWord' href='/nlab/show/diff/created+limit'>created limit</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cartesian+product'>product</a>, <a class='existingWikiWord' href='/nlab/show/diff/pullback'>fiber product</a>, <a class='existingWikiWord' href='/nlab/show/diff/base+change'>base change</a>, <a class='existingWikiWord' href='/nlab/show/diff/coproduct'>coproduct</a>, <a class='existingWikiWord' href='/nlab/show/diff/pullback'>pullback</a>, <a class='existingWikiWord' href='/nlab/show/diff/pushout'>pushout</a>, <a class='existingWikiWord' href='/nlab/show/diff/cobase+change'>cobase change</a>, <a class='existingWikiWord' href='/nlab/show/diff/equalizer'>equalizer</a>, <a class='existingWikiWord' href='/nlab/show/diff/coequalizer'>coequalizer</a>, <a class='existingWikiWord' href='/nlab/show/diff/join'>join</a>, <a class='existingWikiWord' href='/nlab/show/diff/meet'>meet</a>, <a class='existingWikiWord' href='/nlab/show/diff/terminal+object'>terminal object</a>, <a class='existingWikiWord' href='/nlab/show/diff/initial+object'>initial object</a>, <a class='existingWikiWord' href='/nlab/show/diff/direct+product'>direct product</a>, <a class='existingWikiWord' href='/nlab/show/diff/direct+sum'>direct sum</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/finite+limit'>finite limit</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/exact+functor'>exact functor</a></li> </ul> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Kan+extension'>Kan extension</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Yoneda+extension'>Yoneda extension</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/weighted+limit'>weighted limit</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/end'>end and coend</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fibered+limit'>fibered limit</a></p> </li> </ul> <h2 id='2categorical'>2-Categorical</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/2-limit'>2-limit</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/inserter'>inserter</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/isoinserter'>isoinserter</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/equifier'>equifier</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/inverter'>inverter</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/PIE-limit'>PIE-limit</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/2-pullback'>2-pullback</a>, <a class='existingWikiWord' href='/nlab/show/diff/comma+object'>comma object</a></p> </li> </ul> <h2 id='1categorical_2'>(∞,1)-Categorical</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28%E2%88%9E%2C1%29-limit'>(∞,1)-limit</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-pullback'>(∞,1)-pullback</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/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/diff/homotopy+Kan+extension'>homotopy Kan extension</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+limit'>homotopy limit</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+product'>homotopy product</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+equalizer'>homotopy equalizer</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fiber+sequence'>homotopy fiber</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/mapping+cone'>mapping cone</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+pullback'>homotopy pullback</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+totalization'>homotopy totalization</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+coend'>homotopy end</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+limit'>homotopy colimit</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+coproduct'>homotopy coproduct</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+coequalizer'>homotopy coequalizer</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cofiber+sequence'>homotopy cofiber</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/mapping+cocone'>mapping cocone</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+pushout'>homotopy pushout</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+realization'>homotopy realization</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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> <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 class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_1' xmlns='http://www.w3.org/1998/Math/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/diff/diagram'>diagram</a> and <math class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math> is its <a class='existingWikiWord' href='/nlab/show/diff/limit'>limit</a> in <math class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_3' xmlns='http://www.w3.org/1998/Math/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/diff/functor'>functor</a> <math class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_4' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_5' xmlns='http://www.w3.org/1998/Math/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/diff/composition'>composite</a> diagram <math class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_6' xmlns='http://www.w3.org/1998/Math/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/diff/cone'>cone</a>. Furthermore, we can use a trick involving the <a class='existingWikiWord' href='/nlab/show/diff/Yoneda+embedding'>Yoneda embedding</a> to get a meaningful condition even if <math class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>J</mi></mrow><annotation encoding='application/x-tex'>J</annotation></semantics></math> has no limit in <math class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_8' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_9' xmlns='http://www.w3.org/1998/Math/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/diff/diagram'>diagram</a> and let <math class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_10' xmlns='http://www.w3.org/1998/Math/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/diff/functor'>functor</a>.</p> <p>Recall (see <a class='existingWikiWord' href='/nlab/show/diff/limit'>limit</a>) that a <a class='existingWikiWord' href='/nlab/show/diff/cone'>cone</a> over <math class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>J</mi></mrow><annotation encoding='application/x-tex'>J</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_12' xmlns='http://www.w3.org/1998/Math/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/diff/object'>object</a> <math class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math> of <math class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_14' xmlns='http://www.w3.org/1998/Math/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/diff/natural+transformation'>natural transformation</a> to <math class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>J</mi></mrow><annotation encoding='application/x-tex'>J</annotation></semantics></math> from the composite <math class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_16' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_17' xmlns='http://www.w3.org/1998/Math/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/diff/terminal+category'>terminal category</a>. Then a <a class='existingWikiWord' href='/nlab/show/diff/terminal+object'>terminal object</a> in the category of these cones (if it exists) is a <a class='existingWikiWord' href='/nlab/show/diff/limit'>limit</a> of <math class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>J</mi></mrow><annotation encoding='application/x-tex'>J</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math>. Thus, a limit consists of an object <math class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math> and a natural transformation <math class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_21' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math> <strong>preserves</strong> the limit <math class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_23' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_24' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_25' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi></mrow><annotation encoding='application/x-tex'>D</annotation></semantics></math>. (Here, <math class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_27' xmlns='http://www.w3.org/1998/Math/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/diff/whiskering'>whiskering</a>.)</p> <p>Dually, <math class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_28' xmlns='http://www.w3.org/1998/Math/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/diff/colimit'>colimit</a> of <math class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_29' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>J</mi></mrow><annotation encoding='application/x-tex'>J</annotation></semantics></math> if <math class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_30' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_31' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_32' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>I</mi></mrow><annotation encoding='application/x-tex'>I</annotation></semantics></math> be the <a class='existingWikiWord' href='/nlab/show/diff/empty+category'>empty category</a>, so that a limit of the unique functor <math class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_33' xmlns='http://www.w3.org/1998/Math/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/diff/terminal+object'>terminal object</a> <math class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_34' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>1</mn></mrow><annotation encoding='application/x-tex'>1</annotation></semantics></math>. Then <math class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_35' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_36' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_37' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi></mrow><annotation encoding='application/x-tex'>D</annotation></semantics></math>.</p> </li> <li> <p>Let <math class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_38' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>I</mi></mrow><annotation encoding='application/x-tex'>I</annotation></semantics></math> be the <a class='existingWikiWord' href='/nlab/show/diff/discrete+category'>discrete category</a> <math class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_39' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mn>2</mn></mstyle></mrow><annotation encoding='application/x-tex'>\mathbf{2}</annotation></semantics></math>, so that <math class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_40' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>J</mi></mrow><annotation encoding='application/x-tex'>J</annotation></semantics></math> picks out two objects <math class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_41' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>a</mi></mrow><annotation encoding='application/x-tex'>a</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_42' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>b</mi></mrow><annotation encoding='application/x-tex'>b</annotation></semantics></math> of <math class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_43' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> and the limit of <math class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_44' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>J</mi></mrow><annotation encoding='application/x-tex'>J</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/cartesian+product'>product</a> <math class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_45' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_46' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>a</mi></mrow><annotation encoding='application/x-tex'>a</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_47' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_48' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_49' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_50' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math> preserves this product if and only if <math class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_51' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_52' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_53' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_54' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_55' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_56' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_57' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>I</mi></mrow><annotation encoding='application/x-tex'>I</annotation></semantics></math>), we simply say that <math class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_58' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math> preserves that sort of limit (e.g. <math class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_59' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math> preserves <a class='existingWikiWord' href='/nlab/show/diff/cartesian+product'>products</a>, <math class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_60' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math> preserves <a class='existingWikiWord' href='/nlab/show/diff/equalizer'>equalizers</a>, etc.).</p> <p>A functor that preserves all small limits in <math class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_61' xmlns='http://www.w3.org/1998/Math/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/diff/continuous+functor'>continuous functor</a></strong>. Usually this term is only used when <math class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_62' xmlns='http://www.w3.org/1998/Math/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/diff/complete+category'>complete category</a>.</p> <h2 id='Examples'>Examples</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/limits+commute+with+limits'>limits preserve limits</a></p> </li> <ins class='diffins'><li> <p><a class='existingWikiWord' href='/nlab/show/diff/hom-functor+preserves+limits'>hom-functors preserve limits</a></p> </li></ins><ins class='diffins'> </ins><li> <p><a class='existingWikiWord' href='/nlab/show/diff/adjoints+preserve+%28co-%29limits'>adjoints preserve (co-)limits</a></p> </li> <li> <p><del class='diffmod'><a class='existingWikiWord' href='/nlab/show/diff/hom-functor+preserves+limits'>hom-functor preserves limits</a></del><ins class='diffmod'>the </ins><ins class='diffins'><a class='existingWikiWord' href='/nlab/show/diff/Yoneda+embedding'>Yoneda embedding</a></ins><ins class='diffins'> preserves limits (see </ins><ins class='diffins'><a href='Yoneda+embedding#PreservesLimits'>there</a></ins><ins class='diffins'>)</ins></p> </li><del class='diffdel'> </del><del class='diffdel'><li> <p>the <a class='existingWikiWord' href='/nlab/show/diff/Yoneda+embedding'>Yoneda embedding</a> preserves limits</p> </li></del> </ul> <h2 id='preservation_of_weighted_limits'>Preservation of weighted limits</h2> <p>Analogously, an <a class='existingWikiWord' href='/nlab/show/diff/enriched+functor'>enriched functor</a> between <a class='existingWikiWord' href='/nlab/show/diff/enriched+category'>enriched categories</a> may preserve <a class='existingWikiWord' href='/nlab/show/diff/weighted+limit'>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 class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_63' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_64' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_65' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/finitely+complete+category'>finitely complete category</a>, then this is fine; such a functor is called <strong><a class='existingWikiWord' href='/nlab/show/diff/exact+functor'>left exact</a></strong>. But what if <math class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_66' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_67' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_68' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi></mrow><annotation encoding='application/x-tex'>D</annotation></semantics></math> are <a class='existingWikiWord' href='/nlab/show/diff/locally+small+category'>locally small</a>, then we can use the <a class='existingWikiWord' href='/nlab/show/diff/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/diff/category+of+presheaves'>presheaf categories</a> <math class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_69' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_70' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_71' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_72' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_73' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi></mrow><annotation encoding='application/x-tex'>V</annotation></semantics></math>-enriched categories, use <math class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_74' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_75' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_76' xmlns='http://www.w3.org/1998/Math/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/diff/Kan+extension'>Kan extension</a> of the composite <math class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_77' xmlns='http://www.w3.org/1998/Math/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/diff/Yoneda+embedding'>Yoneda embedding</a> <math class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_78' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_79' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_80' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_81' xmlns='http://www.w3.org/1998/Math/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/diff/bimodule'>bimodule</a> nature of <a class='existingWikiWord' href='/nlab/show/diff/profunctor'>profunctors</a>). A diagram <math class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_82' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_83' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_84' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_85' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_86' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math> <strong>preserves</strong> the hypothetical limit of <math class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_87' xmlns='http://www.w3.org/1998/Math/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/diff/reflected+limit'>reflects</a> all limits, if <math class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_88' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>J</mi></mrow><annotation encoding='application/x-tex'>J</annotation></semantics></math> has a limit in <math class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_89' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_90' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_91' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_c54cb191f217255ec5a98a0271a804d433682020_92' xmlns='http://www.w3.org/1998/Math/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/diff/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/diff/cocontinuous+functor'>co</a>)<a class='existingWikiWord' href='/nlab/show/diff/continuous+functor'>continuous functor</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/reflected+limit'>reflected limit</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/created+limit'>created limit</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/lifted+limit'>lifted limit</a></p> </li> </ul> <h2 id='references'>References</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Saunders+Mac+Lane'>Saunders MacLane</a>, §V.4 of: <em><a class='existingWikiWord' href='/nlab/show/diff/Categories+for+the+Working+Mathematician'>Categories for the Working Mathematician</a></em>, Graduate Texts in Mathematics <strong>5</strong> Springer (1971, second ed. 1997) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Francis+Borceux'>Francis Borceux</a>, §2.4 in: <em><a class='existingWikiWord' href='/nlab/show/diff/Handbook+of+Categorical+Algebra'>Handbook of Categorical Algebra</a></em> Vol. 1: <em>Basic Category Theory</em> [[doi:10.1017/CBO9780511525858](https://doi.org/10.1017/CBO9780511525858)]</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Emily+Riehl'>Emily Riehl</a>, §3.3 in: <em><a class='existingWikiWord' href='/nlab/show/diff/Category+Theory+in+Context'>Category Theory in Context</a></em>, Dover Publications (2017) [[pdf](http://www.math.jhu.edu/~eriehl/context.pdf), <a href='http://www.math.jhu.edu/~eriehl/context/'>book website</a>]</p> </li> </ul> <p> </p> <p> </p> <p> </p> </div> <div class="revisedby"> <p> Last revised on May 10, 2024 at 10:24:13. 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