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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 #14 to #15: <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='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='#see_also'>See also</a></li><li><a href='#references'>References</a></li></ul></div> <h2 id='definition'>Definition</h2> <p>A <em>sifted colimit</em> is a <a class='existingWikiWord' href='/nlab/show/diff/colimit'>colimit</a> of a <a class='existingWikiWord' href='/nlab/show/diff/diagram'>diagram</a> <math class='maruku-mathml' display='inline' id='mathml_9959b52033b1428cb96f8c87a422e27b379cab11_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>D \to C</annotation></semantics></math> where <math class='maruku-mathml' display='inline' id='mathml_9959b52033b1428cb96f8c87a422e27b379cab11_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi></mrow><annotation encoding='application/x-tex'>D</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/sifted+category'>sifted category</a> (in analogy with a <a class='existingWikiWord' href='/nlab/show/diff/filtered+colimit'>filtered colimit</a>, involving diagrams of shape a <a class='existingWikiWord' href='/nlab/show/diff/filtered+category'>filtered category</a>). Such colimits <a class='existingWikiWord' href='/nlab/show/diff/commutativity+of+limits+and+colimits'>commute</a> with finite <a class='existingWikiWord' href='/nlab/show/diff/cartesian+product'>products</a> in <a class='existingWikiWord' href='/nlab/show/diff/Set'>Set</a>, by definition.</p> <h2 id='examples'>Examples</h2> <div class='num_example'> <h6 id='example'>Example</h6> <p>A motivating example is a <a class='existingWikiWord' href='/nlab/show/diff/reflexive+coequalizer'>reflexive coequalizer</a>. In fact, sifted colimits can “almost” be characterized as combinations of <a class='existingWikiWord' href='/nlab/show/diff/filtered+colimit'>filtered colimits</a> and reflexive coequalizers. (<a href='#AdamekRosickyVitale10'>Adamek-Rosicky-Vitale 10</a>)</p> </div> <div class='num_example' id='CategoriesWithFiniteProductsAreCosifted'> <h6 id='example_2'>Example</h6> <p><strong>(<a class='existingWikiWord' href='/nlab/show/diff/categories+with+finite+products+are+cosifted'>categories with finite products are cosifted</a>)</strong></p> <p>Let <math class='maruku-mathml' display='inline' id='mathml_9959b52033b1428cb96f8c87a422e27b379cab11_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'>\mathcal{C}</annotation></semantics></math> be a <a class='existingWikiWord' href='/nlab/show/diff/small+category'>small category</a> which has <a class='existingWikiWord' href='/nlab/show/diff/finite+product'>finite products</a>. Then <math class='maruku-mathml' display='inline' id='mathml_9959b52033b1428cb96f8c87a422e27b379cab11_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'>\mathcal{C}</annotation></semantics></math> is a <em><a class='existingWikiWord' href='/nlab/show/diff/sifted+category'>cosifted category</a></em>, equivalently its <a class='existingWikiWord' href='/nlab/show/diff/opposite+category'>opposite category</a> <math class='maruku-mathml' display='inline' id='mathml_9959b52033b1428cb96f8c87a422e27b379cab11_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>𝒞</mi> <mi>op</mi></msup></mrow><annotation encoding='application/x-tex'>\mathcal{C}^{op}</annotation></semantics></math> is a <em><a class='existingWikiWord' href='/nlab/show/diff/sifted+category'>sifted category</a></em>, equivalently <a class='existingWikiWord' href='/nlab/show/diff/colimit'>colimits</a> over <math class='maruku-mathml' display='inline' id='mathml_9959b52033b1428cb96f8c87a422e27b379cab11_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>𝒞</mi> <mi>op</mi></msup></mrow><annotation encoding='application/x-tex'>\mathcal{C}^{op}</annotation></semantics></math> with values in <a class='existingWikiWord' href='/nlab/show/diff/Set'>Set</a> are <em><a class='existingWikiWord' href='/nlab/show/diff/sifted+colimit'>sifted colimits</a></em>, equivalently <a class='existingWikiWord' href='/nlab/show/diff/colimit'>colimits</a> over <math class='maruku-mathml' display='inline' id='mathml_9959b52033b1428cb96f8c87a422e27b379cab11_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>𝒞</mi> <mi>op</mi></msup></mrow><annotation encoding='application/x-tex'>\mathcal{C}^{op}</annotation></semantics></math> with values in <a class='existingWikiWord' href='/nlab/show/diff/Set'>Set</a> <em><a class='existingWikiWord' href='/nlab/show/diff/commutativity+of+limits+and+colimits'>commute</a> with <a class='existingWikiWord' href='/nlab/show/diff/finite+product'>finite products</a></em>, as follows:</p> <p>For <math class='maruku-mathml' display='inline' id='mathml_9959b52033b1428cb96f8c87a422e27b379cab11_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>X</mi></mstyle><mo>,</mo><mstyle mathvariant='bold'><mi>Y</mi></mstyle><mo>∈</mo><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'>\mathbf{X}, \mathbf{Y} \in [\mathcal{C}^{op}, Set]</annotation></semantics></math> to <a class='existingWikiWord' href='/nlab/show/diff/functor'>functors</a> on the <a class='existingWikiWord' href='/nlab/show/diff/opposite+category'>opposite category</a> of <math class='maruku-mathml' display='inline' id='mathml_9959b52033b1428cb96f8c87a422e27b379cab11_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'>\mathcal{C}</annotation></semantics></math> (hence two <a class='existingWikiWord' href='/nlab/show/diff/presheaf'>presheaves</a> on <math class='maruku-mathml' display='inline' id='mathml_9959b52033b1428cb96f8c87a422e27b379cab11_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'>\mathcal{C}</annotation></semantics></math>) we have a <a class='existingWikiWord' href='/nlab/show/diff/natural+isomorphism'>natural isomorphism</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_9959b52033b1428cb96f8c87a422e27b379cab11_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><munder><mi>lim</mi><munder><mo>⟶</mo><mrow><msup><mi>𝒞</mi> <mi>op</mi></msup></mrow></munder></munder><mrow><mo>(</mo><mstyle mathvariant='bold'><mi>X</mi></mstyle><mo>×</mo><mstyle mathvariant='bold'><mi>Y</mi></mstyle><mo>)</mo></mrow><mspace width='thickmathspace' /><mo>≃</mo><mspace width='thickmathspace' /><mrow><mo>(</mo><munder><mi>lim</mi><munder><mo>⟶</mo><mrow><msup><mi>𝒞</mi> <mi>op</mi></msup></mrow></munder></munder><mstyle mathvariant='bold'><mi>X</mi></mstyle><mo>)</mo></mrow><mo>×</mo><mrow><mo>(</mo><munder><mi>lim</mi><munder><mo>⟶</mo><mrow><msup><mi>𝒞</mi> <mi>op</mi></msup></mrow></munder></munder><mstyle mathvariant='bold'><mi>Y</mi></mstyle><mo>)</mo></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \underset{\underset{\mathcal{C}^{op}}{\longrightarrow}}{\lim} \left( \mathbf{X} \times \mathbf{Y} \right) \;\simeq\; \left( \underset{\underset{\mathcal{C}^{op}}{\longrightarrow}}{\lim} \mathbf{X} \right) \times \left( \underset{\underset{\mathcal{C}^{op}}{\longrightarrow}}{\lim} \mathbf{Y} \right) \,. </annotation></semantics></math></div></div> <h2 id='properties'>Properties</h2> <p>See at <em><a class='existingWikiWord' href='/nlab/show/diff/distributivity+of+limits+over+colimits'>distributivity of products and colimits</a></em>.</p> <h2 id='see_also'>See also</h2> <ul> <ins class='diffins'><li> <p><a class='existingWikiWord' href='/nlab/show/diff/sind-object'>sind-object</a>, an object in the <a class='existingWikiWord' href='/nlab/show/diff/free+cocompletion'>free cocompletion</a> under sifted colimits</p> </li></ins><ins class='diffins'> </ins><li> <p><a class='existingWikiWord' href='/nlab/show/diff/sifted+%28infinity%2C1%29-colimit'>sifted $(\infty,1)$-colimit</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/finite+product'>finite product</a></p> </li> </ul> <h2 id='references'>References</h2> <ul> <li id='GabrielUlmer71'> <p><a class='existingWikiWord' href='/nlab/show/diff/Pierre+Gabriel'>Pierre Gabriel</a>, <a class='existingWikiWord' href='/nlab/show/diff/Friedrich+Ulmer'>Friedrich Ulmer</a>, <em>Lokal präsentierbare Kategorien</em>, Springer LNM 221, Springer-Verlag 1971</p> </li> <li id='AdamekRosickyVitale10'> <p><a class='existingWikiWord' href='/nlab/show/diff/Ji%C5%99%C3%AD+Ad%C3%A1mek'>Jiri Adamek</a>, <a class='existingWikiWord' href='/nlab/show/diff/Ji%C5%99%C3%AD+Rosick%C3%BD'>Jiri Rosicky</a>, <a class='existingWikiWord' href='/nlab/show/diff/Enrico+Vitale'>Enrico Vitale</a>, <em>What are sifted colimits?</em>, TAC <strong>23</strong> (2010) pp. 251–260. (<a href='http://www.tac.mta.ca/tac/volumes/23/13/23-13abs.html'>web</a>)</p> </li> </ul> <p> </p> </div> <div class="revisedby"> <p> Last revised on June 7, 2024 at 06:34:18. 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