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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> <h4 id="idempotents">Idempotents</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/idempotent">idempotent</a></strong>,</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/split+idempotent">split idempotent</a>, <a class="existingWikiWord" href="/nlab/show/retract">retract</a></li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/completion">completion</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/idempotent+complete+category">idempotent complete category</a>, <a class="existingWikiWord" href="/nlab/show/Cauchy+complete+category">Cauchy complete category</a>, <a class="existingWikiWord" href="/nlab/show/idempotent+complete+%28%E2%88%9E%2C1%29-category">idempotent complete (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Karoubi+envelope">Karoubi envelope</a>, <a class="existingWikiWord" href="/nlab/show/Cauchy+completion">Cauchy completion</a></p> </li> </ul> <h3 id="sidebar_examples">Examples</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/projector">projector</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/idempotent+monad">idempotent monad</a>, <a class="existingWikiWord" href="/nlab/show/idempotent+adjunction">idempotent adjunction</a></p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#definition'>Definition</a></li> <li><a href='#properties'>Properties</a></li> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#to_the_point'>To the point</a></li> <li><a href='#of_simplices'>Of simplices</a></li> <li><a href='#in_arrow_categories'>In arrow categories</a></li> <li><a href='#RetractsOfDiagrams'>Retracts of diagrams</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="definition">Definition</h2> <p>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>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> in a <a class="existingWikiWord" href="/nlab/show/category">category</a> is called a <strong>retract</strong> of an object <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> if there are <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">i\colon A\to B</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo lspace="verythinmathspace">:</mo><mi>B</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">r \colon B\to A</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>∘</mo><mi>i</mi><mo>=</mo><msub><mi>id</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">r \circ i = id_A</annotation></semantics></math>. In this case <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math> is called a <strong>retraction</strong> of <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> onto <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>.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>id</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>A</mi><munderover><mo>→</mo><mi>section</mi><mi>i</mi></munderover><mi>B</mi><munderover><mo>→</mo><mi>retraction</mi><mi>r</mi></munderover><mi>A</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> id \;\colon\; A \underoverset{section}{i}{\to} B \underoverset{retraction}{r}{\to} A \,. </annotation></semantics></math></div> <p>Here <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> may also be called a <em><a class="existingWikiWord" href="/nlab/show/section">section</a></em> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math>. (In particular if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math> is thought of as exhibiting a <a class="existingWikiWord" href="/nlab/show/bundle">bundle</a>; the terminology originates from <a class="existingWikiWord" href="/nlab/show/topology">topology</a>.)</p> <p>Hence a <strong>retraction</strong> of a <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">i \;\colon\; A \to B</annotation></semantics></math> is a <em>left-<a class="existingWikiWord" href="/nlab/show/inverse">inverse</a></em>.</p> <p>In this situation, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/split+epimorphism">split epimorphism</a> and <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> is a <a class="existingWikiWord" href="/nlab/show/split+monomorphism">split monomorphism</a>. The entire situation is said to be a <em>splitting of the <a class="existingWikiWord" href="/nlab/show/idempotent">idempotent</a></em></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>B</mi><mover><mo>⟶</mo><mi>r</mi></mover><mi>A</mi><mover><mo>⟶</mo><mi>i</mi></mover><mi>B</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> B \stackrel{r}{\longrightarrow} A \stackrel{i}{\longrightarrow} B \,. </annotation></semantics></math></div> <p>Accordingly, a <strong><a class="existingWikiWord" href="/nlab/show/split+monomorphism">split monomorphism</a></strong> is a morphism that <em>has</em> a retraction; a <strong><a class="existingWikiWord" href="/nlab/show/split+epimorphism">split epimorphism</a></strong> is a morphism that <em>is</em> a retraction.</p> <h2 id="properties">Properties</h2> <div class="num_lemma" id="LeftInverseWithLeftInverseIsLeftInverse"> <h6 id="lemma">Lemma</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/left+inverse">left inverse</a> with <a class="existingWikiWord" href="/nlab/show/left+inverse">left inverse</a> is <a class="existingWikiWord" href="/nlab/show/inverse">inverse</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/category">category</a>, and let <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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> be <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</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">\mathcal{C}</annotation></semantics></math>, such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/left+inverse">left inverse</a> to <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>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>∘</mo><mi>f</mi><mo>=</mo><mi>id</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> g \circ f = id \,. </annotation></semantics></math></div> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> itself has a left inverse <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>∘</mo><mi>g</mi><mo>=</mo><mi>id</mi></mrow><annotation encoding="application/x-tex"> h \circ g = id </annotation></semantics></math></div> <p>then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>=</mo><mi>f</mi></mrow><annotation encoding="application/x-tex">h = f</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>=</mo><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">g = f^{-1}</annotation></semantics></math> is an actual (two-sided) <a class="existingWikiWord" href="/nlab/show/inverse+morphism">inverse morphism</a> to <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>.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>Since <a class="existingWikiWord" href="/nlab/show/inverse+morphisms">inverse morphisms</a> are unique if they exists, it is sufficient to show that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∘</mo><mi>g</mi><mo>=</mo><mi>id</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> f \circ g = id \,. </annotation></semantics></math></div> <p>Compute as follows:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mi>f</mi><mo>∘</mo><mi>g</mi></mtd> <mtd><mo>=</mo><munder><munder><mrow><mi>h</mi><mo>∘</mo><mi>g</mi></mrow><mo>⏟</mo></munder><mrow><mo>=</mo><mi>id</mi></mrow></munder><mo>∘</mo><mi>f</mi><mo>∘</mo><mi>g</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>h</mi><mo>∘</mo><munder><munder><mrow><mi>g</mi><mo>∘</mo><mi>f</mi></mrow><mo>⏟</mo></munder><mrow><mo>=</mo><mi>id</mi></mrow></munder><mo>∘</mo><mi>g</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>h</mi><mo>∘</mo><mi>g</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>id</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} f \circ g & = \underset{ = id}{\underbrace{h \circ g}} \circ f \circ g \\ & = h \circ \underset{= id}{\underbrace{g \circ f}} \circ g \\ & = h \circ g \\ & = id \end{aligned} </annotation></semantics></math></div></div> <div class="num_remark" id="RetractsPreservedByFunctor"> <h6 id="remark">Remark</h6> <p>Retracts are clearly preserved by any <a class="existingWikiWord" href="/nlab/show/functor">functor</a>.</p> </div> <div class="num_remark" id="SplitEpisAndMonos"> <h6 id="remark_2">Remark</h6> <p>A <a class="existingWikiWord" href="/nlab/show/split+epimorphism">split epimorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>;</mo><mi>B</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">r; B \to A</annotation></semantics></math> is the strongest of various notions of <a class="existingWikiWord" href="/nlab/show/epimorphism">epimorphism</a> (e.g., it is a <a class="existingWikiWord" href="/nlab/show/regular+epimorphism">regular epimorphism</a>, in fact an <span class="newWikiWord">absolute<a href="/nlab/new/abolute+limit">?</a></span> <a class="existingWikiWord" href="/nlab/show/coequalizer">coequalizer</a>, being the coequalizer of a pair <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><msub><mn>1</mn> <mi>B</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(e, 1_B)</annotation></semantics></math> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi><mo>=</mo><mi>i</mi><mo>∘</mo><mi>r</mi><mo>:</mo><mi>B</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">e = i \circ r: B \to B</annotation></semantics></math> is idempotent). Dually, a <a class="existingWikiWord" href="/nlab/show/split+monomorphism">split monomorphism</a> is the strongest of various notions of monomorphism.</p> </div> <div class="num_prop" id="RetractOfObjectWithLLP"> <h6 id="proposition">Proposition</h6> <p>If an object <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> has the <a class="existingWikiWord" href="/nlab/show/left+lifting+property">left lifting property</a> against a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X \to Y</annotation></semantics></math>, then so does every of its retracts <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 \to B</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><mo>(</mo><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mo>∃</mo></mpadded></msup><mo>↗</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>A</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>:</mo><mo>=</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mo>∃</mo></mpadded></msup><mo>↗</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>A</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>B</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>A</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> \left( \array{ && X \\ & {}^{\mathllap{\exists}}\nearrow& \downarrow \\ A &\to& Y } \right) \;\;\;\; := \;\;\;\; \left( \array{ && && && X \\ &&& {}^{\mathllap{\exists}}\nearrow& && \downarrow \\ A &\to& B &\to& A &\to& Y } \right) </annotation></semantics></math></div></div> <div class="num_prop" id="RetractOfRepresentable"> <h6 id="proposition_2">Proposition</h6> <p>Let <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> be a <a class="existingWikiWord" href="/nlab/show/category">category</a> with <a class="existingWikiWord" href="/nlab/show/split+idempotent">split idempotent</a>s and write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PSh</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><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">PSh(C) = [C^{op}, Set]</annotation></semantics></math> for its <a class="existingWikiWord" href="/nlab/show/presheaf+category">presheaf category</a>. Then a retract of a <a class="existingWikiWord" href="/nlab/show/representable+functor">representable functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>=</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F = PSh(C)</annotation></semantics></math> is itself representable.</p> </div> <p>This appears as (<a href="#Borceux">Borceux, lemma 6.5.6</a>)</p> <h2 id="examples">Examples</h2> <h3 id="to_the_point">To the point</h3> <ul> <li>In a category with <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><mo>*</mo></mrow><annotation encoding="application/x-tex">*</annotation></semantics></math> every morphism of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">* \to X</annotation></semantics></math> is a section, and the unique morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">X \to *</annotation></semantics></math> is the corresponding retraction.</li> </ul> <h3 id="of_simplices">Of simplices</h3> <p>The inclusion of standard topological <a class="existingWikiWord" href="/nlab/show/horn">horn</a>s into the topological <a class="existingWikiWord" href="/nlab/show/simplex">simplex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Λ</mi> <mi>k</mi> <mi>n</mi></msubsup><mo>↪</mo><msup><mi>Δ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\Lambda^n_k \hookrightarrow \Delta^n</annotation></semantics></math> is a retract in <a class="existingWikiWord" href="/nlab/show/Top">Top</a>.</p> <h3 id="in_arrow_categories">In arrow categories</h3> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo>=</mo><mo stretchy="false">{</mo><mn>0</mn><mo>→</mo><mn>1</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\Delta[1] = \{0 \to 1\}</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/interval+category">interval category</a>. For every category <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> the <a class="existingWikiWord" href="/nlab/show/functor+category">functor category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo>,</mo><mi>C</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\Delta[1], C]</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/arrow+category">arrow category</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>.</p> <p>In the theory of <a class="existingWikiWord" href="/nlab/show/weak+factorization+systems">weak factorization systems</a> and <a class="existingWikiWord" href="/nlab/show/model+categories">model categories</a>, an important role is played by retracts in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow></msup></mrow><annotation encoding="application/x-tex">C^{\Delta[1]}</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/arrow+category">arrow category</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>. Explicitly spelled out in terms of the original category <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 morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f:X\to Y</annotation></semantics></math> is a retract of a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>:</mo><mi>Z</mi><mo>→</mo><mi>W</mi></mrow><annotation encoding="application/x-tex">g:Z\to W</annotation></semantics></math> if we have commutative squares</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>id</mi> <mi>X</mi></msub><mo lspace="verythinmathspace">:</mo></mtd> <mtd><mi>X</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>Z</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mi>f</mi><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mi>g</mi><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo><mi>f</mi></mtd></mtr> <mtr><mtd><msub><mi>id</mi> <mi>Y</mi></msub><mo lspace="verythinmathspace">:</mo></mtd> <mtd><mi>Y</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>W</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ id_X \colon & X & \to & Z & \to & X \\ & f \downarrow & & g \downarrow & & \downarrow f \\ id_Y \colon & Y & \to & W & \to & Y } </annotation></semantics></math></div> <p>such that the top and bottom rows compose to identities.</p> <div class="num_prop" id="RetractsOfMorphismWithLiftingProperty"> <h6 id="proposition_3">Proposition</h6> <p>Classes of morphisms in a category <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 are given by a left or right <a class="existingWikiWord" href="/nlab/show/lifting+property">lifting property</a> are preserved under retracts in the <a class="existingWikiWord" href="/nlab/show/arrow+category">arrow category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo>,</mo><mi>C</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\Delta[1],C]</annotation></semantics></math>. In particular the defining classes of a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> are closed under retracts.</p> </div> <p>This is fairly immediate, a proof is made explicit <a href="ClosurePropertiesOfInjectiveAndProjectiveMorphisms">here</a>.</p> <p>This implies:</p> <div class="num_prop" id="RetractOfIso"> <h6 id="proposition_4">Proposition</h6> <p>In every category <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> the class of <a class="existingWikiWord" href="/nlab/show/isomorphisms">isomorphisms</a> is preserved under retracts in the <a class="existingWikiWord" href="/nlab/show/arrow+category">arrow category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo>,</mo><mi>C</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\Delta[1], C]</annotation></semantics></math></p> </div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>This is also checked directly: for</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><mi>id</mi><mo lspace="verythinmathspace">:</mo></mtd> <mtd><msub><mi>a</mi> <mn>1</mn></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>a</mi> <mn>2</mn></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>a</mi> <mn>1</mn></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>id</mi><mo lspace="verythinmathspace">:</mo></mtd> <mtd><msub><mi>b</mi> <mn>1</mn></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>b</mi> <mn>2</mn></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>b</mi> <mn>1</mn></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ id \colon & a_1 &\to& a_2 &\to& a_1 \\ & \downarrow && \downarrow && \downarrow \\ id \colon & b_1 &\to& b_2 &\to& b_1 } </annotation></semantics></math></div> <p>a retract diagram and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>a</mi> <mn>2</mn></msub><mo>→</mo><msub><mi>b</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">a_2 \to b_2</annotation></semantics></math> an isomorphism, the inverse to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>a</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>b</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">a_1 \to b_1</annotation></semantics></math> is given by the composite</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></mtd> <mtd><msub><mi>a</mi> <mn>2</mn></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>a</mi> <mn>1</mn></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↑</mo></mtd> <mtd></mtd> <mtd></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mi>b</mi> <mn>1</mn></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>b</mi> <mn>2</mn></msub></mtd> <mtd></mtd> <mtd></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ & & & a_2 &\to& a_1 \\ & && \uparrow && \\ & b_1 &\to& b_2 && } \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>b</mi> <mn>2</mn></msub><mo>→</mo><msub><mi>a</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">b_2 \to a_2</annotation></semantics></math> is the inverse of the middle morphism.</p> </div> <h3 id="RetractsOfDiagrams">Retracts of diagrams</h3> <p>For the following, let <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>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math> be categories and write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>J</mi> <mo>◃</mo></msup></mrow><annotation encoding="application/x-tex">J^{\triangleleft}</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/join+of+quasi-categories">join</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> with a single <a class="existingWikiWord" href="/nlab/show/initial+object">initial object</a>, so that <a class="existingWikiWord" href="/nlab/show/functor">functor</a>s <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>J</mi> <mo>◃</mo></msup><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">J^{\triangleleft} \to C</annotation></semantics></math> are precisely <a class="existingWikiWord" href="/nlab/show/cone">cone</a>s over functors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">J \to C</annotation></semantics></math>. Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>:</mo><mi>J</mi><mo>→</mo><msup><mi>J</mi> <mo>◃</mo></msup></mrow><annotation encoding="application/x-tex"> i : J \to J^{\triangleleft} </annotation></semantics></math></div> <p>for the canonical inclusion and hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>i</mi> <mo>*</mo></msup><mi>F</mi></mrow><annotation encoding="application/x-tex">i^* F</annotation></semantics></math> for the underlying diagram of a cone <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><msup><mi>J</mi> <mo>◃</mo></msup><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">F : J^{\triangleleft} \to C</annotation></semantics></math>. Finally, write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>J</mi> <mo>◃</mo></msup><mo>,</mo><mi>C</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[J^{\triangleleft}, C]</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/functor+category">functor category</a>.</p> <div class="num_prop" id="RetractsOfLimits"> <h6 id="proposition_5">Proposition</h6> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Id</mi><mo>:</mo><msub><mi>F</mi> <mn>1</mn></msub><mo>↪</mo><msub><mi>F</mi> <mn>2</mn></msub><mo>→</mo><msub><mi>F</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">Id: F_1 \hookrightarrow F_2 \to F_1</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/retract">retract</a> in the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>J</mi> <mo>◃</mo></msup><mo>,</mo><mi>C</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[J^{\triangleleft}, C]</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mn>2</mn></msub><mo>:</mo><msup><mi>J</mi> <mo>◃</mo></msup><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">F_2 : J^{\triangleleft} \to C</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/limit">limit</a> <a class="existingWikiWord" href="/nlab/show/cone">cone</a> over the <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>i</mi> <mo>*</mo></msup><msub><mi>F</mi> <mn>2</mn></msub><mo>:</mo><mi>J</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">i^* F_2 : J \to C</annotation></semantics></math>, then also <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">F_1</annotation></semantics></math> is a limit cone over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>i</mi> <mo>*</mo></msup><msub><mi>F</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">i^* F_1</annotation></semantics></math>.</p> </div> <div class="proof"> <h6 id="proof_3">Proof</h6> <p>We give a direct and a more abstract argument.</p> <p><strong>Direct argument</strong>. We can directly check the <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a> of the limit: for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> any other <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><msup><mi>i</mi> <mo>*</mo></msup><msub><mi>F</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">i^* F_1</annotation></semantics></math>, the composite <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>i</mi> <mo>*</mo></msup><mi>G</mi><mo>=</mo><msup><mi>i</mi> <mo>*</mo></msup><msub><mi>F</mi> <mn>1</mn></msub><mo>→</mo><msup><mi>i</mi> <mo>*</mo></msup><msub><mi>F</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">i^* G = i^* F_1 \to i^* F_2</annotation></semantics></math> exhibits <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> also as a cone over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>i</mi> <mo>*</mo></msup><msub><mi>F</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">i^* F_2</annotation></semantics></math>. By the pullback property of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">F_2</annotation></semantics></math> this extends to a morphism of cones <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>→</mo><msub><mi>F</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G \to F_2</annotation></semantics></math>. Postcomposition with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mn>2</mn></msub><mo>→</mo><msub><mi>F</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">F_2 \to F_1</annotation></semantics></math> makes this a morphism of cones <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>→</mo><msub><mi>F</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">G \to F_1</annotation></semantics></math>. By the injectivity of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>F</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">F_1 \to F_2</annotation></semantics></math> and the universality of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">F_2</annotation></semantics></math>, any two such cone morphisms are equals.</p> <p><strong>More abstract argument</strong>. The limiting cone over a diagram <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>:</mo><mi>J</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">D : J \to C</annotation></semantics></math> may be regarded as the right <a class="existingWikiWord" href="/nlab/show/Kan+extension">Kan extension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mo>*</mo></msub><mi>D</mi><mo>:</mo><mo>=</mo><msub><mi>Ran</mi> <mi>i</mi></msub><mi>D</mi></mrow><annotation encoding="application/x-tex">i_* D := Ran_i D</annotation></semantics></math> along <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></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><mi>J</mi></mtd> <mtd><mover><mo>→</mo><mi>D</mi></mover></mtd> <mtd><mi>C</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>i</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><msub><mo>↗</mo> <mrow><msub><mi>i</mi> <mo>*</mo></msub><mi>D</mi></mrow></msub></mtd></mtr> <mtr><mtd><msup><mi>J</mi> <mo>◃</mo></msup></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ J &\stackrel{D}{\to}& C \\ {}^{\mathllap{i}}\downarrow & \nearrow_{i_* D} \\ J^{\triangleleft} } \,. </annotation></semantics></math></div> <p>Therefore a cone <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><msup><mi>J</mi> <mo>◃</mo></msup><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">F : J^{\triangleleft} \to C</annotation></semantics></math> is limiting precisely if the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>i</mi> <mo>*</mo></msup><mo>⊣</mo><msub><mi>i</mi> <mo>*</mo></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(i^* \dashv i_*)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/unit+of+an+adjunction">unit</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>F</mi><mover><mo>→</mo><mrow></mrow></mover><msub><mi>i</mi> <mo>*</mo></msub><msup><mi>i</mi> <mo>*</mo></msup><mi>F</mi></mrow><annotation encoding="application/x-tex"> F \stackrel{}{\to} i_* i^* F </annotation></semantics></math></div> <p>is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>. Since this unit is a <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a> it follows that applied to the retract diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Id</mi><mo>:</mo><msub><mi>F</mi> <mn>1</mn></msub><mo>↪</mo><msub><mi>F</mi> <mn>2</mn></msub><mo>→</mo><msub><mi>F</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex"> Id : F_1 \hookrightarrow F_2 \to F_1 </annotation></semantics></math></div> <p>it yields the retract diagram</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><mi>Id</mi><mo>:</mo></mtd> <mtd><msub><mi>F</mi> <mn>1</mn></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>F</mi> <mn>2</mn></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>F</mi> <mn>1</mn></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>Id</mi><mo>:</mo></mtd> <mtd><msub><mi>i</mi> <mo>*</mo></msub><msup><mi>i</mi> <mo>*</mo></msup><msub><mi>F</mi> <mn>1</mn></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>i</mi> <mo>*</mo></msub><msup><mi>i</mi> <mo>*</mo></msup><msub><mi>F</mi> <mn>2</mn></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>i</mi> <mo>*</mo></msub><msup><mi>i</mi> <mo>*</mo></msup><msub><mi>F</mi> <mn>1</mn></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ Id : & F_1 &\to& F_2 &\to& F_1 \\ & \downarrow && \downarrow && \downarrow \\ Id : & i_* i^* F_1 &\to& i_* i^* F_2 &\to& i_* i^* F_1 } </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo>,</mo><mo stretchy="false">[</mo><msup><mi>J</mi> <mo>◃</mo></msup><mo>,</mo><mi>C</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\Delta[1], [J^{\triangleleft}, C]]</annotation></semantics></math>. Here by assumption the middle morphism is an isomorphism. Since isomorphisms are stable under retract, by prop. <a class="maruku-ref" href="#RetractOfIso"></a>, also the left and right vertical morphism is an isomorphism, hence also <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">F_1</annotation></semantics></math> is a limiting cone.</p> </div> <p>This argument generalizes form limits to <a class="existingWikiWord" href="/nlab/show/homotopy+limit">homotopy limit</a>s.</p> <p>For that, let now <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> be a <a class="existingWikiWord" href="/nlab/show/category+with+weak+equivalences">category with weak equivalences</a> and write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>:</mo><msup><mi>Diagram</mi> <mi>op</mi></msup><mo>→</mo><mi>Cat</mi></mrow><annotation encoding="application/x-tex">Ho(C) : Diagram^{op} \to Cat</annotation></semantics></math> for the corresponding <a class="existingWikiWord" href="/nlab/show/derivator">derivator</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>J</mi><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><mo stretchy="false">[</mo><mi>J</mi><mo>,</mo><mi>C</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><msup><mi>W</mi> <mi>J</mi></msup><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">Ho(C)(J) := [J,C](W^J)^{-1}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</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>-diagrams 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>, with respect to the degreewise weak equivalences 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> <div class="num_cor" id="RetractOfHomotopyLimits"> <h6 id="corollary">Corollary</h6> <p>Let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Id</mi><mo>:</mo><msub><mi>F</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>F</mi> <mn>2</mn></msub><mo>→</mo><msub><mi>F</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex"> Id : F_1 \to F_2 \to F_1 </annotation></semantics></math></div> <p>be a retract in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>J</mi> <mo>◃</mo></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(C)(J^{\triangleleft})</annotation></semantics></math>. If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">F_2</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/homotopy+limit">homotopy limit</a> cone over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>i</mi> <mo>*</mo></msup><msub><mi>F</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">i^* F_2</annotation></semantics></math>, then also <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">F_1</annotation></semantics></math> is a homotopy limit cone over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>i</mi> <mo>*</mo></msup><msub><mi>F</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">i^* F_1</annotation></semantics></math>.</p> </div> <div class="proof"> <h6 id="proof_4">Proof</h6> <p>By the discussion at <a class="existingWikiWord" href="/nlab/show/derivator">derivator</a> we have that</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mo>*</mo></msub><mo>:</mo><mi>Ho</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>J</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Ho</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>J</mi> <mo>◃</mo></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">i_* : Ho(C)(J) \to Ho(C)(J^{\triangleleft})</annotation></semantics></math> forms <a class="existingWikiWord" href="/nlab/show/homotopy+limit">homotopy limit</a> cones;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>→</mo><msub><mi>i</mi> <mo>*</mo></msub><msup><mi>i</mi> <mo>*</mo></msup><mi>F</mi></mrow><annotation encoding="application/x-tex">F \to i_* i^* F</annotation></semantics></math> is an isomorphism precisely 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> is a homotopy limit cone.</p> </li> </ol> <p>With this the claim follows as in prop. <a class="maruku-ref" href="#RetractsOfLimits"></a>.</p> </div> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/retractive+space">retractive space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/deformation+retract">deformation retract</a>, <a class="existingWikiWord" href="/nlab/show/neighbourhood+retract">neighbourhood retract</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/absolute+retract">absolute retract</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dominant+functor">dominant functor</a></p> </li> </ul> <h2 id="references">References</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Karol+Borsuk">Karol Borsuk</a>, <em>Theory of retracts</em></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Sibe+Marde%C5%A1i%C4%87">Sibe Mardešić</a>, <em>Absolute Neighborhood Retracts and Shape Theory</em> (<a href="https://www.maths.ed.ac.uk/~v1ranick/papers/mardesic.pdf">pdf</a>)</p> </li> <li id="Borceux"> <p><a class="existingWikiWord" href="/nlab/show/Francis+Borceux">Francis Borceux</a>, Def. 1.7.3 and Sec. 6.5 of: <em><a class="existingWikiWord" href="/nlab/show/Handbook+of+Categorical+Algebra">Handbook of Categorical Algebra</a> I</em></p> </li> </ul> <p>See also</p> <ul> <li>Wikipedia, <em><a href="https://en.wikipedia.org/wiki/Retract">Retract</a></em></li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on October 13, 2023 at 15:09:24. 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