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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='#Idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <li><a href='#examples'>Examples</a></li> <li><a href='#properties'>Properties</a></li> <li><a href='#related_entries'>Related entries</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="Idea">Idea</h2> <p>Recall (eg. from <a href="monad#RelationBetweenAdjunctionsAndMonads">here</a>) that every <a class="existingWikiWord" href="/nlab/show/right+adjoint+functor">right adjoint functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>⊣</mo><mi>G</mi><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>ℬ</mi><mo>→</mo><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">F\dashv G \,\colon\, \mathcal{B}\to\mathcal{A}</annotation></semantics></math> induces a <a class="existingWikiWord" href="/nlab/show/monad">monad</a> on <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">\mathcal{A}</annotation></semantics></math> whose <a class="existingWikiWord" href="/nlab/show/underlying">underlying</a> <a class="existingWikiWord" href="/nlab/show/endofunctor">endofunctor</a> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>∘</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">G\circ F</annotation></semantics></math>.</p> <p>The notion of the <em>codensity monad</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝕋</mi> <mi>G</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{T}^G</annotation></semantics></math> is a generalization of this construction to functors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo lspace="verythinmathspace">:</mo><mi>ℬ</mi><mo>→</mo><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">G \colon \mathcal{B}\to\mathcal{A}</annotation></semantics></math> that need not be <a class="existingWikiWord" href="/nlab/show/right+adjoints">right adjoints</a> but do at least admit a 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>Ran</mi> <mi>G</mi></msub><mi>G</mi></mrow><annotation encoding="application/x-tex">Ran_G G</annotation></semantics></math> along themselves, such that both constructions agree when <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 in fact a <a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjoint</a>.</p> <p>The name ‘codensity monad’ stems from the fact that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝕋</mi> <mi>G</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{T}^G</annotation></semantics></math> reduces to the <a class="existingWikiWord" href="/nlab/show/identity+functor">identity</a> monad iff <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo lspace="verythinmathspace">:</mo><mi>ℬ</mi><mo>→</mo><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">G \colon \mathcal{B}\to\mathcal{A}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/codense+functor">codense functor</a>. Thus, in general, the codensity monad “measures the failure of <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> to be codense”.</p> <p>The same idea applies to <a class="existingWikiWord" href="/nlab/show/2-categories">2-categories</a> or <a class="existingWikiWord" href="/nlab/show/bicategories">bicategories</a> more general than <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a>: codensity monads can be defined whenever suitable right Kan extensions exist.</p> <h2 id="definition">Definition</h2> <p> <div class='num_defn' id='codensity_monad'> <h6>Definition</h6> <p><strong>(codensity monad)</strong> <br /> Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo lspace="verythinmathspace">:</mo><mi>ℬ</mi><mo>→</mo><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">G \colon \mathcal{B}\to\mathcal{A}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/functor">functor</a> whose <a href="Kan+extension#Pointwise">pointwise</a> 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>Ran</mi> <mi>G</mi></msub><mi>G</mi><mspace width="thinmathspace"></mspace><mo>≡</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">(</mo><msup><mi>T</mi> <mi>G</mi></msup><mo>,</mo><mspace width="thickmathspace"></mspace><mi>α</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ran_G G \,\equiv\, (T^G,\;\alpha)</annotation></semantics></math> along itself exists, with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><msup><mi>T</mi> <mi>G</mi></msup><mo>∘</mo><mi>G</mi><mo>⇒</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">\alpha \,\colon\, T^G \circ G \Rightarrow G</annotation></semantics></math> denoting the corresponding <a class="existingWikiWord" href="/nlab/show/universal+construction">universal</a> <a class="existingWikiWord" href="/nlab/show/2-morphism">2-morphism</a> on the <a class="existingWikiWord" href="/nlab/show/underlying">underlying</a> functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>T</mi> <mi>G</mi></msup><mo lspace="verythinmathspace">:</mo><mi>𝒜</mi><mo>→</mo><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">T^G \colon \mathcal{A}\to\mathcal{A}</annotation></semantics></math>.</p> <p>The <em>codensity monad</em> of <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 the <a class="existingWikiWord" href="/nlab/show/monad">monad</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>𝕋</mi> <mi>G</mi></msup><mo>≔</mo><mo maxsize="1.2em" minsize="1.2em">⟨</mo><mspace width="thickmathspace"></mspace><msup><mi>T</mi> <mi>G</mi></msup><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>𝒜</mi><mo>→</mo><mi>𝒜</mi><mo>,</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msup><mi>η</mi> <mi>G</mi></msup><mo lspace="verythinmathspace">:</mo><msub><mi>id</mi> <mi>𝒜</mi></msub><mo>⇒</mo><msup><mi>T</mi> <mi>G</mi></msup><mo>,</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msup><mi>μ</mi> <mi>G</mi></msup><mo lspace="verythinmathspace">:</mo><msup><mi>T</mi> <mi>G</mi></msup><mo>∘</mo><msup><mi>T</mi> <mi>G</mi></msup><mo>⇒</mo><msup><mi>T</mi> <mi>G</mi></msup><mspace width="thickmathspace"></mspace><mo maxsize="1.2em" minsize="1.2em">⟩</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \mathbb{T}^G \coloneqq \big\langle \; T^G \,\colon\, \mathcal{A} \to\mathcal{A} ,\;\;\; \eta^G \colon id_\mathcal{A}\Rightarrow T^G ,\;\;\; \mu^G \colon T^G\circ T^G \Rightarrow T^G \; \big\rangle \,, </annotation></semantics></math></div> <p>where</p> <ul> <li> <p>the carrier functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝕋</mi> <mi>G</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{T}^G</annotation></semantics></math> is given by the end</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>𝕋</mi> <mi>G</mi></msup><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mo>∫</mo> <mrow><mi>B</mi><mo>∈</mo><mi>ℬ</mi></mrow></msub><mi>𝒜</mi><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>GB</mi><mo stretchy="false">)</mo><mo>⋔</mo><mi>GB</mi></mrow><annotation encoding="application/x-tex"> \mathbb{T}^G(A) = \int_{B \in \mathcal{B}} \mathcal{A}(A, GB) \pitchfork GB </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⋔</mo></mrow><annotation encoding="application/x-tex">\pitchfork</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/power">power</a>, i.e. repeated product.</p> </li> <li> <p>the <em><a class="existingWikiWord" href="/nlab/show/monad+unit">monad unit</a></em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>η</mi> <mi>G</mi></msup><mo lspace="verythinmathspace">:</mo><msub><mi>id</mi> <mi>𝒜</mi></msub><mo>⇒</mo><msup><mi>T</mi> <mi>G</mi></msup></mrow><annotation encoding="application/x-tex">\eta^G \colon id_\mathcal{A}\Rightarrow T^G</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a> given by the <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>T</mi> <mi>G</mi></msup><mo>,</mo><mspace width="thickmathspace"></mspace><mi>α</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(T^G,\;\alpha)</annotation></semantics></math> with respect to the pair <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>id</mi> <mi>𝒜</mi></msub><mo>,</mo><mspace width="thickmathspace"></mspace><msub><mn>1</mn> <mi>G</mi></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">(id_\mathcal{A},\;1_G)\;</annotation></semantics></math>,</p> </li> <li> <p>the <em>monad multiplication</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>μ</mi> <mi>G</mi></msup><mo lspace="verythinmathspace">:</mo><msup><mi>T</mi> <mi>G</mi></msup><mo>∘</mo><msup><mi>T</mi> <mi>G</mi></msup><mo>⇒</mo><msup><mi>T</mi> <mi>G</mi></msup></mrow><annotation encoding="application/x-tex">\mu^G \colon T^G\circ T^G\Rightarrow T^G</annotation></semantics></math> results from the <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>T</mi> <mi>G</mi></msup><mo>,</mo><mspace width="thickmathspace"></mspace><mi>α</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(T^G,\;\alpha)</annotation></semantics></math> with respect to the pair <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>T</mi> <mi>G</mi></msup><mo>∘</mo><msup><mi>T</mi> <mi>G</mi></msup><mo>,</mo><mspace width="thickmathspace"></mspace><mi>α</mi><mo>∘</mo><mo stretchy="false">(</mo><msub><mn>1</mn> <mrow><msup><mi>T</mi> <mi>G</mi></msup></mrow></msub><mo>*</mo><mi>α</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(T^G\circ T^G,\;\alpha\circ (1_{T^G}\ast\alpha))</annotation></semantics></math>.</p> </li> </ul> <p></p> </div> </p> <p>Concerning existence, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Ran</mi> <mi>G</mi></msub><mi>G</mi></mrow><annotation encoding="application/x-tex">Ran_G G</annotation></semantics></math> exists for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo lspace="verythinmathspace">:</mo><mi>ℬ</mi><mo>→</mo><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">G \colon \mathcal{B}\to\mathcal{A}</annotation></semantics></math>, e.g. when <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">\mathcal{B}</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/small+category">small</a> and <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">\mathcal{A}</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/complete+category">complete</a>.</p> <p>In this circumstance, when <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">\mathcal{B}</annotation></semantics></math> is small and <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">\mathcal{A}</annotation></semantics></math> is complete, then the codensity monad is equivalently the one that <a href="monad#RelationToAdjunctionsAndMonadicity">arises</a> from the <a class="existingWikiWord" href="/nlab/show/adjoint+functor">adjunction</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi><munderover><mo>⊥</mo><munder><mo>⟵</mo><mrow></mrow></munder><mover><mo>⟶</mo><mrow><mi>hom</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>G</mi><mo stretchy="false">)</mo></mrow></mover></munderover><mo stretchy="false">[</mo><mi>ℬ</mi><mo>,</mo><mi>Set</mi><msup><mo stretchy="false">]</mo> <mi>op</mi></msup><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \mathcal{A} \underoverset {\underset{}{\longleftarrow}} {\overset{hom(-,G)}{\longrightarrow}} {\bot} [\mathcal{B},Set]^{op} \,, </annotation></semantics></math></div> <p>where</p> <ul> <li> <p>the <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>hom</mi><mo stretchy="false">(</mo><mtext>-</mtext><mo>,</mo><mi>G</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>𝒜</mi><mo>→</mo><mo stretchy="false">[</mo><mi>ℬ</mi><mo>,</mo><mi>Set</mi><msup><mo stretchy="false">]</mo> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">hom(\text{-},G) \,\colon\, \mathcal{A} \to [\mathcal{B},Set]^{op}</annotation></semantics></math> takes any <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> to the <a class="existingWikiWord" href="/nlab/show/hom-functor">hom-functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mi>𝒜</mi></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>a</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>G</mi><mo stretchy="false">(</mo><mtext>-</mtext><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mo lspace="verythinmathspace">:</mo><mi>ℬ</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">Hom_{\mathcal{A}}\big(a, \,G(\text{-})\big) \colon \mathcal{B}\to Set</annotation></semantics></math>,</p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjoint</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>ℬ</mi><mo>,</mo><mi>Set</mi><msup><mo stretchy="false">]</mo> <mi>op</mi></msup><mo>→</mo><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">[\mathcal{B},Set]^{op}\to \mathcal{A}</annotation></semantics></math> is the unique <a class="existingWikiWord" href="/nlab/show/preserved+limit">limit-preserving</a> functor from the <a class="existingWikiWord" href="/nlab/show/free+completion">free completion</a> 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">\mathcal{B}</annotation></semantics></math> to <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">\mathcal{A}</annotation></semantics></math> which agrees on <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">\mathcal{B}</annotation></semantics></math> with <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>.</p> </li> </ul> <p>(See also <a class="existingWikiWord" href="/nlab/show/nerve+and+realization">nerve and realization</a>; the description of the adjunction above is a formal dual of a nerve-realization adjunction, and gives the right Kan extension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Ran</mi> <mi>G</mi></msub><mi>G</mi></mrow><annotation encoding="application/x-tex">Ran_G G</annotation></semantics></math> as a pointwise Kan extension. In the pointwise setting, <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 codense if and only if the left adjoint is <a class="existingWikiWord" href="/nlab/show/full+and+faithful">full and faithful</a>.)</p> <p>Even if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Ran</mi> <mi>G</mi></msub><mi>G</mi></mrow><annotation encoding="application/x-tex">Ran_G G</annotation></semantics></math> (assuming it exists) is not a pointwise Kan extension, Def. <a class="maruku-ref" href="#codensity_monad"></a> indeed defines a <a class="existingWikiWord" href="/nlab/show/monad">monad</a>. The proof may be given generally for any <a class="existingWikiWord" href="/nlab/show/2-category">2-category</a> in which the right Kan extension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Ran</mi> <mi>G</mi></msub><mi>G</mi></mrow><annotation encoding="application/x-tex">Ran_G G</annotation></semantics></math> exists for a 1-cell <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>:</mo><mi>ℬ</mi><mo>→</mo><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">G: \mathcal{B} \to \mathcal{A}</annotation></semantics></math>.</p> <p> <div class='num_theorem'> <h6>Theorem</h6> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Ran</mi> <mi>G</mi></msub><mi>G</mi></mrow><annotation encoding="application/x-tex">Ran_G G</annotation></semantics></math>, with the unit <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>η</mi> <mi>G</mi></msup></mrow><annotation encoding="application/x-tex">\eta^G</annotation></semantics></math> and multiplication <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>μ</mi> <mi>G</mi></msup></mrow><annotation encoding="application/x-tex">\mu^G</annotation></semantics></math>, is a monad.</p> </div> </p> <p> <div class='proof'> <h6>Proof</h6> <p>The <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a> of the <a class="existingWikiWord" href="/nlab/show/Kan+extension">Kan extension</a> states that for any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>:</mo><mi>𝒜</mi><mo>→</mo><mi>ℬ</mi></mrow><annotation encoding="application/x-tex">H: \mathcal{A} \to \mathcal{B}</annotation></semantics></math>, there is a natural bijection</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>hom</mi><mo stretchy="false">(</mo><mi>H</mi><mo>,</mo><msup><mi>T</mi> <mi>G</mi></msup><mo stretchy="false">)</mo><mo>≅</mo><mi>hom</mi><mo stretchy="false">(</mo><mi>H</mi><mi>G</mi><mo>,</mo><mi>G</mi><mo stretchy="false">)</mo><mo>;</mo></mrow><annotation encoding="application/x-tex">\hom(H, T^G) \cong \hom(H G, G);</annotation></semantics></math></div> <p>let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ε</mi><mo>:</mo><msup><mi>T</mi> <mi>G</mi></msup><mi>G</mi><mo>→</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">\varepsilon: T^G G \to G</annotation></semantics></math> be the 2-cell corresponding to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mn>1</mn> <mrow><msup><mi>T</mi> <mi>G</mi></msup></mrow></msub><mo>∈</mo><mi>hom</mi><mo stretchy="false">(</mo><msup><mi>T</mi> <mi>G</mi></msup><mo>,</mo><msup><mi>T</mi> <mi>G</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">1_{T^G} \in \hom(T^G, T^G)</annotation></semantics></math>. 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class="maruku-mathml"><semantics><mrow><msup><mi>η</mi> <mi>G</mi></msup><mo lspace="verythinmathspace">:</mo><mn>1</mn><mo>→</mo><msup><mi>T</mi> <mi>G</mi></msup></mrow><annotation encoding="application/x-tex">\eta^G\colon 1 \to T^G</annotation></semantics></math> is defined so that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>ε</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>η</mi> <mi>G</mi></msup><mi>G</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mn>1</mn> <mi>G</mi></msub></mrow><annotation encoding="application/x-tex">(\varepsilon) (\eta^G G) = 1_G</annotation></semantics></math>, and the 2-cell <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>μ</mi> <mi>G</mi></msup><mo lspace="verythinmathspace">:</mo><msup><mi>T</mi> <mi>G</mi></msup><msup><mi>T</mi> <mi>G</mi></msup><mo>→</mo><msup><mi>T</mi> 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transform="matrix(0.996095,0,0,-0.996095,133.650603,44.066235)"></path> <path style="fill:none;stroke-width:0.47818;stroke-linecap:round;stroke-linejoin:round;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -2.487774 2.870324 C -2.032873 1.148757 -1.021109 0.333072 -0.00150243 -0.000261266 C -1.021109 -0.333594 -2.032873 -1.14928 -2.487774 -2.870847 " transform="matrix(0.996095,0,0,-0.996095,130.708528,78.964584)"></path> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#2Jty0XCbCkjQ1Sei2wl9YpABlk4=-glyph1-2" x="93.117516" y="91.055012"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#2Jty0XCbCkjQ1Sei2wl9YpABlk4=-glyph2-1" x="100.721454" y="87.552494"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#2Jty0XCbCkjQ1Sei2wl9YpABlk4=-glyph1-4" x="108.433718" y="91.055012"></use> </g> <path style="fill:none;stroke-width:0.47818;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 41.672294 -35.035431 L 96.887926 -35.035431 " transform="matrix(0.996095,0,0,-0.996095,133.650603,44.066235)"></path> <path style="fill:none;stroke-width:0.47818;stroke-linecap:round;stroke-linejoin:round;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -2.485491 2.870324 C -2.030589 1.148757 -1.018825 0.333072 0.000781357 -0.000261266 C -1.018825 -0.333594 -2.030589 -1.14928 -2.485491 -2.870847 " transform="matrix(0.996095,0,0,-0.996095,230.397659,78.964584)"></path> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#2Jty0XCbCkjQ1Sei2wl9YpABlk4=-glyph1-4" x="200.463658" y="86.738934"></use> </g> </g> </svg> <p>where commutativity of the triangle comes from how we introduced <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>η</mi> 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style="fill:none;stroke-width:0.47818;stroke-linecap:round;stroke-linejoin:round;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -2.484812 2.867737 C -2.034062 1.147047 -1.018894 0.335696 0.000193933 -0.00138682 C -1.018894 -0.33455 -2.034062 -1.145901 -2.484812 -2.870511 " transform="matrix(0.996602,0,0,-0.996602,258.175588,80.53768)"></path> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#WP5WUqXPl75PI4YMA38PYnc45iM=-glyph1-3" x="223.37241" y="88.315881"></use> </g> </g> </svg> <p>and finally finish the proof by observing that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ε</mi></mrow><annotation encoding="application/x-tex">\varepsilon</annotation></semantics></math> coequalizes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>μ</mi> <mi>G</mi></msup><mi>G</mi></mrow><annotation encoding="application/x-tex">\mu^G G</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>T</mi> <mi>G</mi></msup><mi>ε</mi></mrow><annotation encoding="application/x-tex">T^G \varepsilon</annotation></semantics></math>.</p> </div> </p> <h2 id="examples">Examples</h2> <p> <div class='num_remark'> <h6>Example</h6> <p>Every monad that is <a href="monad#RelationBetweenAdjunctionsAndMonads">induced</a> by an adjunction <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo>⊣</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">L \dashv R</annotation></semantics></math> is the codensity monad 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, every <a class="existingWikiWord" href="/nlab/show/enriched+monad">enriched monad</a> is a codensity monad (via its <a class="existingWikiWord" href="/nlab/show/Kleisli+category">Kleisli category</a>).</p> </div> </p> <p> <div class='num_remark'> <h6>Example</h6> <p>Let <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> be an object in a <a class="existingWikiWord" href="/nlab/show/closed+category">closed 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>. Then the <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>-enriched codensity monad of the <a class="existingWikiWord" href="/nlab/show/constant+functor">constant functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>:</mo><mn>1</mn><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">d : 1 \to C</annotation></semantics></math> is the <strong>double dualization monad</strong> associated to <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>, given by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>d</mi> <mrow><msup><mi>d</mi> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msup></mrow></msup></mrow><annotation encoding="application/x-tex">d^{d^{(-)}}</annotation></semantics></math>.</p> <p>More conceptually, the codensity monad construction may be seen as a generalisation of the double dualisation construction analogous to the generalisation from <a class="existingWikiWord" href="/nlab/show/algebras+for+a+monad">algebras for a monad</a> to <a class="existingWikiWord" href="/nlab/show/modules+over+a+monad">modules over a monad</a> (the latter is the perspective that is most natural 2-categorically).</p> </div> </p> <p> <div class='num_remark'> <h6>Example</h6> <p>The <a class="existingWikiWord" href="/nlab/show/Giry+monad">Giry monad</a> (as well as a finitely additive version) arise as codensity monads of forgetful functors from subcategories of the category of <a class="existingWikiWord" href="/nlab/show/convex+sets">convex sets</a> to the category of <a class="existingWikiWord" href="/nlab/show/measurable+spaces">measurable spaces</a> (<a href="#Avery14">Avery 14</a>).</p> </div> </p> <p> <div class='num_remark'> <h6>Example</h6> <p>The codensity monad of the inclusion <a class="existingWikiWord" href="/nlab/show/FinSet">FinSet</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">\hookrightarrow</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Set">Set</a> is the <a class="existingWikiWord" href="/nlab/show/ultrafilter">ultrafilter</a> monad. Its algebras are <a class="existingWikiWord" href="/nlab/show/compact+Hausdorff+spaces">compact Hausdorff spaces</a>.</p> </div> </p> <p> <div class='num_remark'> <h6>Example</h6> <p>The codensity monad of the inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>FinGrp</mi><mo>↪</mo></mrow><annotation encoding="application/x-tex">FinGrp \hookrightarrow</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Grp">Grp</a>, is the <a class="existingWikiWord" href="/nlab/show/profinite+completion+of+a+group">profinite completion</a> monad, whose algebras may be identified with <a class="existingWikiWord" href="/nlab/show/profinite+groups">profinite groups</a> – that is, <a class="existingWikiWord" href="/nlab/show/topological+groups">topological groups</a> whose underlying topological space is profinite (<a href="#Avery17">Avery 17, Proposition 2.7.10</a>).</p> </div> </p> <p> <div class='num_remark'> <h6>Example</h6> <p>The codensity monad of the inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>FinSet</mi><mo>→</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex">FinSet \to Top</annotation></semantics></math> computes the <a class="existingWikiWord" href="/nlab/show/Stone+spectrum">Stone spectrum</a> of the <a class="existingWikiWord" href="/nlab/show/Boolean+algebra">Boolean algebra</a> of <a class="existingWikiWord" href="/nlab/show/clopen+subsets">clopen subsets</a> of a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>. Its algebras are precisely the <a class="existingWikiWord" href="/nlab/show/Stone+spaces">Stone spaces</a>. (<a href="#Sipos">Sipoș, Theorem 2</a>).</p> </div> </p> <p> <div class='num_remark'> <h6>Example</h6> <p>The codensity monad of the inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>→</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex">N \to Top</annotation></semantics></math>, where <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> denotes the <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> of <a class="existingWikiWord" href="/nlab/show/Top">Top</a> consisting of arbitrary <a class="existingWikiWord" href="/nlab/show/small+products">small products</a> of the <a class="existingWikiWord" href="/nlab/show/Sierpinski+space">Sierpiński space</a>, is the <span class="newWikiWord">localic spectrum<a href="/nlab/new/localic+spectrum">?</a></span> of the <a class="existingWikiWord" href="/nlab/show/frame">frame</a> of opens of a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>. Its algebras are precisely the <a class="existingWikiWord" href="/nlab/show/sober+spaces">sober spaces</a>. (<a href="#Sipos">Sipoș, Theorem 6</a>)</p> </div> </p> <p> <div class='num_remark'> <h6>Example</h6> <p>The codensity monad of the inclusion of <a class="existingWikiWord" href="/nlab/show/countable+sets">countable sets</a> in all sets, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ctbl</mi><mo>↪</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">Ctbl \hookrightarrow Set</annotation></semantics></math>, assigns to each set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> the set of ultrafilters on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> closed under countable intersections. This still holds for the inclusion of the full subcategory of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ctbl</mi></mrow><annotation encoding="application/x-tex">Ctbl</annotation></semantics></math> on the single set <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">\mathbb{N}</annotation></semantics></math>.</p> </div> </p> <p> <div class='num_remark'> <h6>Example</h6> <p>More generally, the codensity monad of the inclusion of sets of cardinality less than that of fixed <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>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Set</mi> <mrow><mo><</mo><mi>Y</mi></mrow></msub><mo>↪</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">Set_{\lt Y} \hookrightarrow Set</annotation></semantics></math>, assigns to each set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> the set of <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>-complete ultrafilters on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> </div> </p> <p> <div class='num_remark'> <h6>Example</h6> <p>For the codensity monad induced by the inclusion of <a class="existingWikiWord" href="/nlab/show/homotopy+types+with+finite+homotopy+groups">homotopy types with finite homotopy groups</a> into all <a class="existingWikiWord" href="/nlab/show/homotopy+types">homotopy types</a> see <a href="homotopy+type+with+finite+homotopy+groups#CodensityMonadOfInclusionIntoAllHomotopyTypes">there</a>.</p> </div> </p> <p> <div class='num_remark'> <h6>Example</h6> <p>The codensity monad induced by the <a class="existingWikiWord" href="/nlab/show/Yoneda+embedding">Yoneda embedding</a> is isomorphic to the monad induced by the <a class="existingWikiWord" href="/nlab/show/Isbell+adjunction">Isbell adjunction</a>.</p> </div> </p> <p> <div class='num_remark'> <h6>Example</h6> <p>In the bicategory <a class="existingWikiWord" href="/nlab/show/Rel">Rel</a>, the right Kan extension of a relation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">T: A \to C</annotation></semantics></math> along a relation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">R: A \to B</annotation></semantics></math> is the relation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo stretchy="false">/</mo><mi>R</mi><mo>:</mo><mi>B</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">T/R: B \to C</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>T</mi><mo stretchy="false">/</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">(b,c)\in T/R</annotation></semantics></math> iff <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∀</mo> <mrow><mi>a</mi><mo>:</mo><mi>A</mi></mrow></msub><mspace width="thickmathspace"></mspace><mi>R</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo>⇒</mo><mi>T</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\forall_{a: A}\; R(a, b) \Rightarrow T(a, c)</annotation></semantics></math>. In particular, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo stretchy="false">/</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">R/R</annotation></semantics></math> reduces to the identity relation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>id</mi> <mi>B</mi></msub></mrow><annotation encoding="application/x-tex">id_B</annotation></semantics></math> iff whenever <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>b</mi><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(b,b')</annotation></semantics></math> is such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∀</mo> <mrow><mi>a</mi><mo>:</mo><mi>A</mi></mrow></msub><mspace width="thickmathspace"></mspace><mi>R</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo>⇒</mo><mi>R</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\forall_{a:A}\; R(a,b)\Rightarrow R(a,b')</annotation></semantics></math> then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>=</mo><mi>b</mi><mo>′</mo><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">b=b'\,</annotation></semantics></math>, in other words, iff <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>R</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>b</mi><mo>⊆</mo><msup><mi>R</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>b</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">R^{-1}b\subseteq R^{-1}b'</annotation></semantics></math> implies <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>=</mo><mi>b</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">b=b'</annotation></semantics></math>. The codensity monad <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo stretchy="false">/</mo><mi>R</mi><mo>:</mo><mi>B</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">R/R: B \to B</annotation></semantics></math>, being a monad in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Rel</mi></mrow><annotation encoding="application/x-tex">Rel</annotation></semantics></math>, is a <a class="existingWikiWord" href="/nlab/show/preorder">preorder</a>. This construction frequently recurs; see for instance <a class="existingWikiWord" href="/nlab/show/specialization+order">specialization order</a> for a <a class="existingWikiWord" href="/nlab/show/topological+space">topology</a>.</p> <p></p> </div> </p> <h2 id="properties">Properties</h2> <p> <div class='num_prop'> <h6>Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>:</mo><mi>B</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">G : B \to A</annotation></semantics></math> be a functor admitting a codensity monad <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>T</mi> <mi>G</mi></msup></mrow><annotation encoding="application/x-tex">T^G</annotation></semantics></math> on <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>. The <a class="existingWikiWord" href="/nlab/show/Kleisli+category">Kleisli category</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>T</mi> <mi>G</mi></msup></mrow><annotation encoding="application/x-tex">T^G</annotation></semantics></math> has homs <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Kl</mi><mo stretchy="false">(</mo><msup><mi>T</mi> <mi>G</mi></msup><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>a</mi><mo>′</mo><mo stretchy="false">)</mo><mo>≅</mo><mo stretchy="false">[</mo><mi>B</mi><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">(</mo><mi>a</mi><mo>′</mo><mo>,</mo><mi>G</mi><mo>−</mo><mo stretchy="false">)</mo><mo>,</mo><mi>B</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>G</mi><mo>−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Kl(T^G)(a, a') \cong [B, Set](B(a', G-), B(a, G-))</annotation></semantics></math>.</p> </div> </p> <p>See <a href="#BC80">Bourn and Cordier 1980</a>, for instance.</p> <h2 id="related_entries">Related entries</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/codense+functor">codense functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dense">dense</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dense+functor">dense functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/shape+theory">shape theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kan+extension">Kan extension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+theory">algebraic theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ultrafilter">ultrafilter</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/idempotent+monad">idempotent monad</a></p> </li> </ul> <h2 id="references">References</h2> <p>One of the first references is</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Anders+Kock">Anders Kock</a>, <em>Continuous Yoneda Representations of a Small Category</em>, Preprint Aarhus University (1966). (<a href="http://home.math.au.dk/kock/CYRSC.pdf">pdf</a>)</li> </ul> <p>For the special case of double dualisation, see:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Anders+Kock">Anders Kock</a>. <em>On double dualization monads</em>, Mathematica Scandinavica 27.2 (1970): 151-165. (<a href="https://www.jstor.org/stable/24489892">JSTOR</a>)</li> </ul> <p>Overview:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Fred+Linton">Fred Linton</a>, <em>Codensity triples</em>, Section 8 in: <em>An outline of functorial semantics</em>, in <em><a class="existingWikiWord" href="/nlab/show/Seminar+on+Triples+and+Categorical+Homology+Theory">Seminar on Triples and Categorical Homology Theory</a></em>, Lecture Notes in Mathematics <strong>80</strong>, Springer (1969) 7-52 [<a href="https://doi.org/10.1007/BFb0083080">doi:10.1007/BFb0083080</a>]</p> </li> <li id="Leinster13"> <p><a class="existingWikiWord" href="/nlab/show/Tom+Leinster">Tom Leinster</a>, <em>Codensity and the Ultrafilter Monad</em> , TAC <strong>12</strong> no.13 (2013) pp.332-370. [<a href="http://www.tac.mta.ca/tac/volumes/28/13/28-13abs.html">tac:28-13</a>]</p> </li> </ul> <p>See also:</p> <ul> <li> <p>nCafé blog 2012: <em><a href="https://golem.ph.utexas.edu/category/2012/09/where_do_monads_come_from.html">Where do Monads come from?</a></em></p> </li> <li> <p>MO-discussion <a class="existingWikiWord" href="/nlab/show/Tim+Campion">Tim Campion</a>: <em><a href="https://mathoverflow.net/questions/220246/what-is-the-point-of-pointwise-kan-extensions">What is the point of pointwise Kan extensions?</a></em></p> </li> </ul> <p>Codensity monads arising from subcategory inclusions are studied in</p> <ul> <li id="diLiberti19"><a class="existingWikiWord" href="/nlab/show/Ivan+Di+Liberti">Ivan Di Liberti</a>, <em>Codensity: Isbell duality, pro-objects, compactness and accessibility</em>, arXiv:1910.01014 (2019). (<a href="https://arxiv.org/abs/1910.01014">abstract</a>)</li> </ul> <p>The role in shape theory is discussed in</p> <ul> <li> <p>Armin Frei, <em>On categorical shape theory</em> , Cah. Top. Géom. Diff. <strong>XVII</strong> no.3 (1976) pp.261-294. (<a href="http://www.numdam.org/item/?id=CTGDC_1976__17_3_261_0">numdam</a>)</p> </li> <li> <p>D. Bourn, J.-M. Cordier, <em>Distributeurs et théorie de la forme</em>, Cah. Top. Géom. Diff. Cat. <strong>21</strong> no.2 (1980) pp.161-189. (<a href="http://archive.numdam.org/ARCHIVE/CTGDC/CTGDC_1980__21_2/CTGDC_1980__21_2_161_0/CTGDC_1980__21_2_161_0.pdf">pdf</a>)</p> </li> <li> <p>J.-M. Cordier, <a class="existingWikiWord" href="/nlab/show/Tim+Porter">T. Porter</a>, <em>Shape Theory: Categorical Methods of Approximation</em> , (1989), Mathematics and its Applications, Ellis Horwood. Reprinted Dover (2008).</p> </li> </ul> <p>The dual concept of a “model-induced cotriple”:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Harry+Applegate">Harry Applegate</a>, <a class="existingWikiWord" href="/nlab/show/Myles+Tierney">Myles Tierney</a>, <em>Categories with models</em>, in: <a class="existingWikiWord" href="/nlab/show/Beno+Eckmann">Beno Eckmann</a> (ed.) <em><a class="existingWikiWord" href="/nlab/show/Seminar+on+Triples+and+Categorical+Homology+Theory">Seminar on Triples and Categorical Homology Theory</a></em> Lecture Notes in Mathematics, <strong>80</strong>, Springer (1969) 156-244 [<a href="https://doi.org/10.1007/BFb0083086">doi:10.1007/BFb0083086</a>, <a href="https://link.springer.com/content/pdf/10.1007/BFb0083086">pdf</a>]</li> </ul> <p>On possible uses in <a class="existingWikiWord" href="/nlab/show/functional+programming">functional programming</a>:</p> <ul> <li>Ralf Hinze, <em>Kan extensions for program optimisation - Or: Art and Dan explain an old trick</em>, in: Jeremy Gibbons, Pablo Nogueira (eds.), <em>11th International Conference on Mathematics of Program Construction (MPC ‘12)</em>, LNCS <strong>7342</strong> Springer (2012) 324–362. (<a href="http://dx.doi.org/10.1007/978-3-642-31113-0_16">doi: 10.1007/978-3-642-31113-0_16</a>, <a href="https://www.cs.ox.ac.uk/ralf.hinze/publications/MPC12.pdf">pdf draft</a>)</li> </ul> <p>For a description of the <a class="existingWikiWord" href="/nlab/show/Giry+monad">Giry monad</a> and other <a class="existingWikiWord" href="/nlab/show/probability+monads">probability monads</a> as codensity monads, see</p> <ul> <li id="Avery14"> <p><a class="existingWikiWord" href="/nlab/show/Tom+Avery">Tom Avery</a>, <em>Codensity and the Giry monad</em>, Journal of Pure and Applied Algebra <strong>220</strong> 3 (2016) 1229-1251 [<a href="https://arxiv.org/abs/1410.4432">arXiv:1410.4432</a>, <a href="https://doi.org/10.1016/j.jpaa.2015.08.017">doi:10.1016/j.jpaa.2015.08.017</a>]</p> </li> <li> <p>Ruben Van Belle, <em>Probability monads as codensity monads</em>. Theory and Applications of Categories 38 (2022), 811–842, (<a href="http://tac.mta.ca/tac/volumes/38/21/38-21abs.html">tac</a>)</p> </li> </ul> <p>Other references include</p> <ul> <li id="BC80"> <p><a class="existingWikiWord" href="/nlab/show/Dominique+Bourn">Dominique Bourn</a> and Jean-Marc Cordier, <em>Distributeurs et théorie de la forme</em>, Cahiers de topologie et géométrie différentielle 21.2 (1980): 161-189.</p> </li> <li id="Avery17"> <p><a class="existingWikiWord" href="/nlab/show/Tom+Avery">Tom Avery</a>, <em>Structure and Semantics</em>, (<a href="https://arxiv.org/abs/1708.01050">arXiv:1708.01050</a>)</p> </li> <li> <p>C. Casacuberta, A. Frei, <em>Localizations as idempotent approximations to completions</em> , JPAA <strong>142</strong> (1999) no. 1 pp.25–33. (<a href="http://atlas.mat.ub.es/personals/casac/articles/cfre1.pdf">draft</a>)</p> </li> <li> <p>Yves Diers, <em>Complétion monadique</em> , Cah. Top. Géom. Diff. Cat. <strong>XVII</strong> no.4 (1976) pp.362-379. (<a href="http://www.numdam.org/item/?id=CTGDC_1976__17_4_363_0">numdam</a>)</p> </li> <li> <p>S. Katsumata, T. Sato, <a class="existingWikiWord" href="/nlab/show/Tarmo+Uustalu">T. Uustalu</a>, <em>Codensity lifting of monads and its dual</em> , arXiv:1810.07972 (2012). (<a href="https://arxiv.org/abs/1810.07972">abstract</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Joachim+Lambek">J. Lambek</a>, B. A. Rattray, <em>Localization and Codensity Triples</em> , Comm. Algebra <strong>1</strong> (1974) pp.145-164.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ji%C5%99%C3%AD+Ad%C3%A1mek">Jiří Adámek</a>, <span class="newWikiWord">Lurdes Sousa<a href="/nlab/new/Lurdes+Sousa">?</a></span>, <em>D-Ultrafilters and their Monads</em>, (<a href="https://arxiv.org/abs/1909.04950">arXiv:1909.04950</a>)</p> </li> <li id="Sipos"> <p><span class="newWikiWord">Andrei Sipoş<a href="/nlab/new/Andrei+Sipo%C5%9F">?</a></span>, <em>Codensity and Stone spaces</em>, Mathematica Slovaca, 68 no. 1, p. 57–70, (2018). <a href="https://doi.org/10.1515/ms-2017-0080">doi:10.1515/ms-2017-0080</a>, (<a href="https://arxiv.org/abs/1409.1370">arXiv:1409.1370</a>)</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on December 29, 2024 at 11:35:41. 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