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width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/333/#Item_14" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" 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"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="categorification">Categorification</h4> <div class="hide"><div> <p><strong>categorification</strong></p> <h2 id="background">Background</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+theory">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="contents">Contents</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/vertical+categorification">vertical categorification</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/horizontal+categorification">horizontal categorification</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-categorification">(∞,1)-categorification</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/decategorification">decategorification</a>, <a class="existingWikiWord" href="/nlab/show/Grothendieck+group">Grothendieck group</a></p> </li> </ul> <h2 id="examples">Examples</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/categorification+in+representation+theory">categorification in representation theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Khovanov+homology">Khovanov homology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kazhdan-Lusztig+theory">Kazhdan-Lusztig theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/categorification+via+groupoid+schemes">categorification via groupoid schemes</a></p> </li> <li> <p>The <a class="existingWikiWord" href="/nlab/show/geometric+Langlands">geometric Langlands</a> program</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+Satake">geometric Satake</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/categorification+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#Idea'>Idea</a></li> <li><a href='#Definition'>Definitions</a></li> <ul> <li><a href='#DefinitionForCategories'>For categories</a></li> <li><a href='#DefinitionForHigherCategories'>For higher categories</a></li> </ul> <li><a href='#ExtraStructure'>Extra structure</a></li> <li><a href='#Examples'>Examples</a></li> <li><a href='#parable'>Parable</a></li> </ul> </div> <h2 id="Idea">Idea</h2> <p>In <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a>, by “decategorification” one means (see <a href="#Definition">below</a>) the process which turns a <a class="existingWikiWord" href="/nlab/show/category">category</a> into a <a class="existingWikiWord" href="/nlab/show/set">set</a>, namely into its set of <a class="existingWikiWord" href="/nlab/show/isomorphism+classes">isomorphism classes</a>.</p> <p>Typically one is interested in the case where the category is equipped with extra <a class="existingWikiWord" href="/nlab/show/higher+structure">higher</a> <a class="existingWikiWord" href="/nlab/show/structure">structure</a> (see <a href="#ExtraStructure">further below</a>), whence its set of isomorphism classes will carry the corresponding ordinary <a class="existingWikiWord" href="/nlab/show/structure">structure</a>. For example: the decategorification of a <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> is canonically a <a class="existingWikiWord" href="/nlab/show/monoid">monoid</a>, the decategorification of a <a class="existingWikiWord" href="/nlab/show/rig+category">rig category</a> is canonically a <a class="existingWikiWord" href="/nlab/show/rig">rig</a>, the decategorification of a <a class="existingWikiWord" href="/nlab/show/2-group">2-group</a> is canonically a plain <a class="existingWikiWord" href="/nlab/show/group">group</a>, etc.</p> <p>(Combined with <a class="existingWikiWord" href="/nlab/show/group+completion">group completion</a>, decategorification of <a class="existingWikiWord" href="/nlab/show/monoidal+categories">monoidal categories</a> is often known as a form of <a class="existingWikiWord" href="/nlab/show/K-theory">K-theory</a> in degree 0, see for instance at <em><a class="existingWikiWord" href="/nlab/show/K-theory+of+a+permutative+category">K-theory of a permutative category</a></em>.)</p> <p>In this sense decategorification is a “<a class="existingWikiWord" href="/nlab/show/left+inverse">left inverse</a>” to (<a class="existingWikiWord" href="/nlab/show/vertical+categorification">vertical</a>) “<a class="existingWikiWord" href="/nlab/show/categorification">categorification</a>” (see there for more), namely to the process of asking for <a class="existingWikiWord" href="/nlab/show/category+theory">category theoretic</a> <a class="existingWikiWord" href="/nlab/show/higher+structures">higher structures</a> analogous to given <a class="existingWikiWord" href="/nlab/show/set+theory">set theoretic</a> <a class="existingWikiWord" href="/nlab/show/structures">structures</a>.</p> <p>Crucially, though, decategorification is a systematic process (in fact a <a class="existingWikiWord" href="/nlab/show/2-functor">2-functor</a>, see <a href="#Definition">below</a>) while <a class="existingWikiWord" href="/nlab/show/categorification">categorification</a>, being a (local) <a class="existingWikiWord" href="/nlab/show/section">section</a> of this functor involves making choices: There are in general several categorical structures which have the same decategorification. For instance, the <a class="existingWikiWord" href="/nlab/show/rig+category">rig</a> <a class="existingWikiWord" href="/nlab/show/monoidal+categories">monoidal categories</a> <a class="existingWikiWord" href="/nlab/show/FinSet">FinSet</a> (with its <a class="existingWikiWord" href="/nlab/show/cartesian+product">cartesian product</a>) and <a class="existingWikiWord" href="/nlab/show/FinDimVect">FinDimVect</a> (with its <a class="existingWikiWord" href="/nlab/show/tensor+product+of+vector+spaces">tensor product of vector spaces</a>) both decategorify to the <a class="existingWikiWord" href="/nlab/show/rig">rig</a> <a class="existingWikiWord" href="/nlab/show/monoid">monoid</a> of <a class="existingWikiWord" href="/nlab/show/natural+numbers">natural numbers</a> (see further examples <a href="#Examples">below</a>).</p> <p>More generally, in <a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a> there are higher sequences of “higher decategorification” functors which incrementally discard non-invertible <a class="existingWikiWord" href="/nlab/show/higher+morphisms">higher morphisms</a> and <a class="existingWikiWord" href="/nlab/show/quotient">quotient</a> by remaining invertible <a class="existingWikiWord" href="/nlab/show/higher+morphisms">higher morphisms</a> up to some degree (see <a href="#DefinitionForHigherCategories">below</a>).</p> <p>In particular, in the <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> of <a class="existingWikiWord" href="/nlab/show/groupoids">groupoids</a>, <a class="existingWikiWord" href="/nlab/show/higher+groupoids">higher groupoids</a> and <a class="existingWikiWord" href="/nlab/show/infinity-groupoid"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mn>∞</mn> </mrow> <annotation encoding="application/x-tex">\infty</annotation> </semantics> </math>-groupoids</a>, namely in <a class="existingWikiWord" href="/nlab/show/%28infinity%2C0%29-category"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mo stretchy="false">(</mo> <mn>∞</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> <annotation encoding="application/x-tex">(\infty,0)</annotation> </semantics> </math>-category</a>-theory, decategorification is nothing but <a class="existingWikiWord" href="/nlab/show/truncated+object+in+an+%28infinity%2C1%29-category">truncation</a> and the tower of decategorifications/<a class="existingWikiWord" href="/nlab/show/n-truncation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>n</mi> </mrow> <annotation encoding="application/x-tex">n</annotation> </semantics> </math>-truncations</a> is known as the <em><a class="existingWikiWord" href="/nlab/show/Postnikov+tower">Postnikov tower</a></em>.</p> <h2 id="Definition">Definitions</h2> <h3 id="DefinitionForCategories">For categories</h3> <p>Given an (<a class="existingWikiWord" href="/nlab/show/essentially+small+category">essentially</a> <a class="existingWikiWord" href="/nlab/show/small+category">small</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">\mathcal{C}</annotation></semantics></math>, its decategorification is the <a class="existingWikiWord" href="/nlab/show/set">set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(C)</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/isomorphism+classes">isomorphism classes</a> of <a class="existingWikiWord" href="/nlab/show/objects">objects</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">\mathcal{C}</annotation></semantics></math>.</p> <p>This construction extends to a <a class="existingWikiWord" href="/nlab/show/2-functor">2-functor</a></p> <div class="maruku-equation" id="eq:PlainDecategorification"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mtable displaystyle="false" rowspacing="0.5ex" columnalign="right center left"><mtr><mtd><mpadded width="0" lspace="-100%width"><mrow><mi>K</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace></mrow></mpadded><mi>Cat</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Set</mi></mtd></mtr> <mtr><mtd><mi>𝒞</mi></mtd> <mtd><mo>↦</mo></mtd> <mtd><mi>Obj</mi><mo stretchy="false">(</mo><mi>𝒞</mi><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">/</mo><mi>iso</mi></mrow></msub></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex"> \begin{array}{rcl} \mathllap{K \;\colon\; } Cat &\longrightarrow& Set \\ \mathcal{C} &\mapsto& Obj(\mathcal{C})_{/iso} \end{array} </annotation></semantics></math></div> <p>from the (<a class="existingWikiWord" href="/nlab/show/2-category">2-</a>)<a class="existingWikiWord" href="/nlab/show/category">category</a> <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a> of (<a class="existingWikiWord" href="/nlab/show/essentially+small+categories">essentially</a> <a class="existingWikiWord" href="/nlab/show/small+category">small</a>) <a class="existingWikiWord" href="/nlab/show/categories">categories</a> to the category (or <a class="existingWikiWord" href="/nlab/show/locally+discrete+2-category">locally discrete 2-category</a>) <a class="existingWikiWord" href="/nlab/show/Set">Set</a> of <a class="existingWikiWord" href="/nlab/show/sets">sets</a>.</p> <p>Notice that we may think of <a class="existingWikiWord" href="/nlab/show/sets">sets</a> as <a class="existingWikiWord" href="/nlab/show/0-categories">0-categories</a>, so that <a class="maruku-eqref" href="#eq:PlainDecategorification">(1)</a> may equivalently be thought of as being of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo lspace="verythinmathspace">:</mo><mn>1</mn><mi>Cat</mi><mo>⟶</mo><mn>0</mn><mi>Cat</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> K \colon 1Cat \longrightarrow 0Cat \,, </annotation></semantics></math></div> <p>which makes manifest that and how decategorification indeed <em>decreases categorical degree</em>.</p> <p>For generalization of decategorification to <a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a> (<a href="#DefinitionForHigherCategories">below</a>) it is useful to make explicit that the decategorification 2-functor <a class="maruku-eqref" href="#eq:PlainDecategorification">(1)</a> factors as</p> <div class="maruku-equation" id="eq:PlainDecategorificationFactoredThroughCore"><span class="maruku-eq-number">(2)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>K</mi><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>Cat</mi><mover><mo>⟶</mo><mi>Core</mi></mover><mi>Grpd</mi><mover><mo>⟶</mo><mrow><msub><mi>τ</mi> <mn>0</mn></msub></mrow></mover><mi>Set</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> K \,\colon\, Cat \overset{Core}{\longrightarrow} Grpd \overset{\tau_0}{\longrightarrow} Set \,, </annotation></semantics></math></div> <p>where</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Core</mi></mrow><annotation encoding="application/x-tex">Core</annotation></semantics></math> assigns the “<a class="existingWikiWord" href="/nlab/show/core">core</a>” of a category, namely the maximal <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a> inside it, hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Core</mi></mrow><annotation encoding="application/x-tex">Core</annotation></semantics></math> “discards” all non-invertible morphisms;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\tau_0</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/0-truncation">0-truncation</a> functor which turns a <a class="existingWikiWord" href="/nlab/show/1-groupoid">1-groupoid</a> into its <a class="existingWikiWord" href="/nlab/show/0-groupoid">0-groupoid</a> of <a class="existingWikiWord" href="/nlab/show/connected+components">connected components</a>.</p> </li> </ol> <p>It may be interesting to notice here that:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/core">Core</a> is <em><a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjoint</a></em> to the <a class="existingWikiWord" href="/nlab/show/fully+faithful+2-functor">embedding</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Grpd</mi><mo>↪</mo><mi>Cat</mi></mrow><annotation encoding="application/x-tex">Grpd \hookrightarrow Cat</annotation></semantics></math>, hence is a <a class="existingWikiWord" href="/nlab/show/coreflective+subcategory">co-reflection</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/0-truncation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <msub><mi>τ</mi> <mn>0</mn></msub> </mrow> <annotation encoding="application/x-tex">\tau_0</annotation> </semantics> </math></a> is <em><a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a></em> to the <a class="existingWikiWord" href="/nlab/show/fully+faithful+functor">embedding</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Set</mi><mo>↪</mo><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">Set \hookrightarrow Grpd</annotation></semantics></math>, hence is a <a class="existingWikiWord" href="/nlab/show/reflective+subcategory">reflection</a>.</p> </li> </ul> <h3 id="DefinitionForHigherCategories">For higher categories</h3> <p>For any sensible notion of <a class="existingWikiWord" href="/nlab/show/higher+categories">higher categories</a> one will have corresponding analogs of the <a class="existingWikiWord" href="/nlab/show/core">core</a>- and the <a class="existingWikiWord" href="/nlab/show/truncation">truncation</a>-operations used in <a class="maruku-eqref" href="#eq:PlainDecategorificationFactoredThroughCore">(2)</a>, which allows to define decategorification of higher categories.</p> <p>For example, for <a class="existingWikiWord" href="/nlab/show/2-categories">2-categories</a> there are the evident notions of <em><a class="existingWikiWord" href="/nlab/show/core+in+a+2-category">core in a 2-category</a></em>.</p> <p>More generally, for any of the models of <a class="existingWikiWord" href="/nlab/show/%28infinity%2Cn%29-categories"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mo stretchy="false">(</mo> <mn>∞</mn> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <annotation encoding="application/x-tex">(\infty,n)</annotation> </semantics> </math>-categories</a> we have a <a class="existingWikiWord" href="/nlab/show/coreflection">coreflection</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo><mi>Cat</mi><munderover><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>⊥</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow><munder><mo>⟵</mo><mrow><msub><mi>Core</mi> <mi>n</mi></msub></mrow></munder><mo>↪</mo></munderover><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mi>Cat</mi></mrow><annotation encoding="application/x-tex"> (\infty,n) Cat \underoverset {\underset{Core_n}{\longleftarrow}} {\hookrightarrow} {\;\; \bot \;\;} (\infty,n+1)Cat </annotation></semantics></math></div> <p>which together with the <a class="existingWikiWord" href="/nlab/show/truncated+object+in+an+%28infinity%2C1%29-category"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>m</mi> </mrow> <annotation encoding="application/x-tex">m</annotation> </semantics> </math>-truncation</a>-operation to <a class="existingWikiWord" href="/nlab/show/homotopy+n-types">homotopy <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>m</mi> </mrow> <annotation encoding="application/x-tex">m</annotation> </semantics> </math>-types</a> yields towers of higher decategorification functors</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mi>Cat</mi><mover><mo>⟶</mo><mrow><msub><mi>Core</mi> <mi>n</mi></msub></mrow></mover><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo><mi>Cat</mi><mo>→</mo><mi>⋯</mi><mo>→</mo><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mi>Cat</mi><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mn>∞</mn><mi>Grpd</mi><mover><mo>⟶</mo><mrow><msub><mi>τ</mi> <mi>m</mi></msub></mrow></mover><mi>m</mi><mi>Grpd</mi><mo>→</mo><mi>⋯</mi><mo>→</mo><mn>0</mn><mi>Grpd</mi><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mi>Set</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> (\infty,n+1)Cat \overset{Core_n}{\longrightarrow} (\infty,n)Cat \to \cdots \to (\infty,0)Cat \;\simeq\; \infty Grpd \overset{ \tau_m }{\longrightarrow} m Grpd \to \cdots \to 0 Grpd \;\simeq\; Set \,, </annotation></semantics></math></div> <p>any stage of which may reasonably be addressed as an intermediate stage of higher decategorification.</p> <h2 id="ExtraStructure">Extra structure</h2> <p>If the category in question has <a class="existingWikiWord" href="/nlab/show/extra+structure">extra</a> <a class="existingWikiWord" href="/nlab/show/higher+structure">higher</a> <a class="existingWikiWord" href="/nlab/show/structure">structure</a>, then this is usually inherited in some decategorified form by its decategorification. For instance if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(C)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/monoid">monoid</a>.</p> <p>A famous example are <a class="existingWikiWord" href="/nlab/show/fusion+category">fusion categories</a> whose decategorifications are called <em><a class="existingWikiWord" href="/nlab/show/Verlinde+ring">Verlinde ring</a>s</em>.</p> <p>There may also be extra structure induced more directly on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(C)</annotation></semantics></math>. For instance the <a class="existingWikiWord" href="/nlab/show/K-theory">K-group</a> of an <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a> is the decategorification of its category of bounded <a class="existingWikiWord" href="/nlab/show/chain+complexes">chain complexes</a> and this inherits a group structure from the fact that this is a <a class="existingWikiWord" href="/nlab/show/triangulated+category">triangulated category</a> (a <a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a>) in which there is a notion of <a class="existingWikiWord" href="/nlab/show/fibration+sequence">homotopy exact sequences</a>.</p> <h2 id="Examples">Examples</h2> <ul> <li> <p>The decategorifications of <a class="existingWikiWord" href="/nlab/show/finite+sets">finite sets</a> and <a class="existingWikiWord" href="/nlab/show/finite+dimensional+vector+spaces">finite dimensional vector spaces</a> (over any <a class="existingWikiWord" href="/nlab/show/ground+field">ground field</a>) are <a class="existingWikiWord" href="/nlab/show/natural+numbers">natural numbers</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>FinSet</mi> <mrow><mo stretchy="false">/</mo><mo>∼</mo></mrow></msub><mo>≃</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex"> FinSet_{/\sim} \simeq \mathbb{N} </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>FinDimVect</mi> <mrow><mo stretchy="false">/</mo><mo>∼</mo></mrow></msub><mo>≃</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex"> FinDimVect_{/\sim} \simeq \mathbb{N} </annotation></semantics></math></div> <p>For instance the <a class="existingWikiWord" href="/nlab/show/rank-nullity+theorem">rank-nullity theorem</a> is the decategorification of the <a class="existingWikiWord" href="/nlab/show/splitting+lemma">splitting lemma</a> in the category <a class="existingWikiWord" href="/nlab/show/FinDimVect">FinDimVect</a>.</p> </li> <li> <p>The decategorification of the 2-category <a class="existingWikiWord" href="/nlab/show/Grpd">Grpd</a> of (small) <a class="existingWikiWord" href="/nlab/show/groupoids">groupoids</a> is equivalent to the <a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a> of <a class="existingWikiWord" href="/nlab/show/homotopy+1-types">homotopy 1-types</a>.</p> </li> <li> <p>The decategorification in the same sense of the 2-category of (<a class="existingWikiWord" href="/nlab/show/small+category">small</a>) <a class="existingWikiWord" href="/nlab/show/categories">categories</a> is equivalent to the full <a class="existingWikiWord" href="/nlab/show/homotopy+category+of+topological+spaces">homotopy category</a>. (explain…)</p> </li> </ul> <h2 id="parable">Parable</h2> <p>From John Baez, https://math.ucr.edu/home/baez/week121.html</p> <p>If one studies categorification one soon discovers an amazing fact: many deep-sounding results in mathematics are just categorifications of facts we learned in high school! There is a good reason for this. All along, we have been unwittingly “decategorifying” mathematics by pretending that categories are just sets. We “decategorify” a category by forgetting about the morphisms and pretending that isomorphic objects are equal. We are left with a mere set: the set of isomorphism classes of objects.</p> <p>To understand this, the following parable may be useful. Long ago, when shepherds wanted to see if two herds of sheep were isomorphic, they would look for an explicit isomorphism. In other words, they would line up both herds and try to match each sheep in one herd with a sheep in the other. But one day, along came a shepherd who invented decategorification. She realized one could take each herd and “count” it, setting up an isomorphism between it and some set of “numbers”, which were nonsense words like “one, two, three,…” specially designed for this purpose. By comparing the resulting numbers, she could show that two herds were isomorphic without explicitly establishing an isomorphism! In short, by decategorifying the category of finite sets, the set of natural numbers was invented.</p> <p>According to this parable, decategorification started out as a stroke of mathematical genius. Only later did it become a matter of dumb habit, which we are now struggling to overcome by means of categorification. While the historical reality is far more complicated, categorification really has led to tremendous progress in mathematics during the 20th century. For example, Noether revolutionized algebraic topology by emphasizing the importance of homology groups. Previous work had focused on Betti numbers, which are just the dimensions of the rational homology groups. As with taking the cardinality of a set, taking the dimension of a vector space is a process of decategorification, since two vector spaces are isomorphic if and only if they have the same dimension. Noether noted that if we work with homology groups rather than Betti numbers, we can solve more problems, because we obtain invariants not only of spaces, but also of maps.</p> </body></html> </div> <div class="revisedby"> <p> Last revised on November 13, 2023 at 07:40:41. 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