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class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></strong></p> <h2 id="sidebar_concepts">Concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category">category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cat">Cat</a></p> </li> </ul> <h2 id="sidebar_universal_constructions">Universal constructions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+construction">universal construction</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/representable+functor">representable functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+functor">adjoint functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/limit">limit</a>/<a class="existingWikiWord" href="/nlab/show/colimit">colimit</a></p> </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</a>/<a class="existingWikiWord" href="/nlab/show/coend">coend</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kan+extension">Kan extension</a></p> </li> </ul> </li> </ul> <h2 id="sidebar_theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+construction">Grothendieck construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+functor+theorem">adjoint functor theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monadicity+theorem">monadicity theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+lifting+theorem">adjoint lifting theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gabriel-Ulmer+duality">Gabriel-Ulmer duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freyd-Mitchell+embedding+theorem">Freyd-Mitchell embedding theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relation+between+type+theory+and+category+theory">relation between type theory and category theory</a></p> </li> </ul> <h2 id="sidebar_extensions">Extensions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf+and+topos+theory">sheaf and topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a></p> </li> </ul> <h2 id="sidebar_applications">Applications</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/applications+of+%28higher%29+category+theory">applications of (higher) category theory</a></li> </ul> <div> <p> <a href="/nlab/edit/category+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#definition'>Definition</a></li> <ul> <li><a href='#structured_sinks'>Structured sinks</a></li> </ul> <li><a href='#examples'>Examples</a></li> <li><a href='#related_pages'>Related pages</a></li> </ul> </div> <h2 id="definition">Definition</h2> <p>Given 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>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> of a <a class="existingWikiWord" href="/nlab/show/category">category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>, a <strong>sink</strong> or <strong>wide cospan</strong> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> 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> is a <a class="existingWikiWord" href="/nlab/show/family">family</a> of <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> 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> whose <a class="existingWikiWord" href="/nlab/show/targets">targets</a> (codomains) are all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math>, or equivalently, a family of objects in the <a class="existingWikiWord" href="/nlab/show/over+category">over category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">/</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">C / Y</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>X</mi> <mn>1</mn></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mo>↘</mo> <mrow><msub><mi>f</mi> <mn>1</mn></msub></mrow></msup></mtd></mtr> <mtr><mtd><msub><mi>X</mi> <mn>2</mn></msub></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>f</mi> <mn>2</mn></msub></mrow></mover></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mo>↗</mo> <mrow><msub><mi>f</mi> <mn>3</mn></msub></mrow></msub></mtd></mtr> <mtr><mtd><msub><mi>X</mi> <mn>3</mn></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array { X_1 \\ & \searrow^{f_1} \\ X_2 & \overset{f_2}\to & Y \\ & \nearrow_{f_3} \\ X_3 } </annotation></semantics></math></div> <p>We do not, in general, require that this family be <a class="existingWikiWord" href="/nlab/show/small+category">small</a>; if it is so we would call it a “small sink”.</p> <p>The <a class="existingWikiWord" href="/nlab/show/duality">dual</a> concept is a family of morphisms 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> whose <a class="existingWikiWord" href="/nlab/show/sources">sources</a> (domains) are all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math>, or equivalently, a family of objects in the <a class="existingWikiWord" href="/nlab/show/under+category">under category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo stretchy="false">/</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">Y / C</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><msub><mi>X</mi> <mn>1</mn></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mrow><msub><mi>f</mi> <mn>1</mn></msub></mrow></msup><mo>↗</mo></mtd></mtr> <mtr><mtd><mi>Y</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>f</mi> <mn>2</mn></msub></mrow></mover></mtd> <mtd><msub><mi>X</mi> <mn>2</mn></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mrow><msub><mi>f</mi> <mn>3</mn></msub></mrow></msub><mo>↘</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msub><mi>X</mi> <mn>3</mn></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array { & & X_1 \\ & {}^{f_1}\nearrow \\ Y & \overset{f_2}\to & X_2 \\ & {}_{f_3}\searrow \\ & & X_3 } </annotation></semantics></math></div> <p>Confusingly, this dual concept is called a <strong>source</strong> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> 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>, even though the term ‘source’ has another meaning, one which we just used in the definition! One can of course say ‘domain’ instead of ‘source’ for this other meaning, but that leads to <a class="existingWikiWord" href="/nlab/show/domain">other confusions</a>. Or one can say ‘cosink’ or <strong>wide span</strong> for a source in the sense dual to a sink or wide cospan, since a source from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> 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> is the same as a sink to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> in the <a class="existingWikiWord" href="/nlab/show/opposite+category">opposite category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mi mathvariant="normal">op</mi></msup></mrow><annotation encoding="application/x-tex">C^{\mathrm{op}}</annotation></semantics></math>.</p> <h3 id="structured_sinks">Structured sinks</h3> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo lspace="verythinmathspace">:</mo><mi>C</mi><mo>→</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">U\colon C\to D</annotation></semantics></math> is a functor, then a <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math>-structured sink</strong> is a collection of objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>i</mi></msub><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">X_i\in C</annotation></semantics></math> together with a sink 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> of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>U</mi><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>→</mo><mi>Y</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{U(X_i) \to Y\}</annotation></semantics></math>. This notion figures in the definition of a <a class="existingWikiWord" href="/nlab/show/final+lift">final lift</a>.</p> <h2 id="examples">Examples</h2> <ul> <li> <p>Any <a class="existingWikiWord" href="/nlab/show/cocone">cocone</a> under a <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a> is a sink; indeed a cocone is precisely a sink indexed by the objects of the domain of the diagram together with a commutativity condition for the arrows in the diagram.</p> </li> <li> <p>A source with two morphisms is a <a class="existingWikiWord" href="/nlab/show/span">span</a>, a sink with two morphisms is a <a class="existingWikiWord" href="/nlab/show/cospan">cospan</a>.</p> </li> <li> <p>A <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal source</a> is a <a class="existingWikiWord" href="/nlab/show/dependent+product">dependent product</a>, while an <a class="existingWikiWord" href="/nlab/show/initial+object">initial sink</a> is a <a class="existingWikiWord" href="/nlab/show/dependent+sum">dependent sum</a>.</p> </li> </ul> <h2 id="related_pages">Related pages</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/final+lift">final lift</a>, <a class="existingWikiWord" href="/nlab/show/topological+concrete+category">topological concrete category</a>, <a class="existingWikiWord" href="/nlab/show/solid+functor">solid functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/factorization+structure+for+sinks">factorization structure for sinks</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/wide+pullback">wide pullback</a>, <a class="existingWikiWord" href="/nlab/show/wide+pushout">wide pushout</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dependent+product+type">dependent product type</a>, <a class="existingWikiWord" href="/nlab/show/dependent+sum+type">dependent sum type</a></p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on January 1, 2024 at 02:34:22. 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