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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="2category_theory">2-Category theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/2-category+theory">2-category theory</a></strong></p> <p><strong>Definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-category">2-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/strict+2-category">strict 2-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bicategory">bicategory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+bicategory">enriched bicategory</a></p> </li> </ul> <p><strong>Transfors between 2-categories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-functor">2-functor</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/pseudofunctor">pseudofunctor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/lax+functor">lax functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivalence+of+2-categories">equivalence of 2-categories</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-natural+transformation">2-natural transformation</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/lax+natural+transformation">lax natural transformation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/icon">icon</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/modification">modification</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yoneda+lemma+for+bicategories">Yoneda lemma for bicategories</a></p> </li> </ul> <p><strong>Morphisms in 2-categories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fully+faithful+morphism">fully faithful morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/faithful+morphism">faithful morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conservative+morphism">conservative morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pseudomonic+morphism">pseudomonic morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/discrete+morphism">discrete morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/eso+morphism">eso morphism</a></p> </li> </ul> <p><strong>Structures in 2-categories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mate">mate</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monad">monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cartesian+object">cartesian object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fibration+in+a+2-category">fibration in a 2-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/codiscrete+cofibration">codiscrete cofibration</a></p> </li> </ul> <p><strong>Limits in 2-categories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-limit">2-limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-pullback">2-pullback</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/comma+object">comma object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/inserter">inserter</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/inverter">inverter</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equifier">equifier</a></p> </li> </ul> <p><strong>Structures on 2-categories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-monad">2-monad</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/lax-idempotent+2-monad">lax-idempotent 2-monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pseudomonad">pseudomonad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pseudoalgebra+for+a+2-monad">pseudoalgebra for a 2-monad</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+2-category">monoidal 2-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/cartesian+bicategory">cartesian bicategory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gray+tensor+product">Gray tensor product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/proarrow+equipment">proarrow equipment</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> <ul> <li><a href='#general'>General</a></li> <li><a href='#naturality'>Naturality</a></li> </ul> <li><a href='#Examples'>Examples</a></li> <li><a href='#multivariable_mates'>Multi-variable mates</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="Definition">Definition</h2> <p> <div class='num_prop' id='MateBijection'> <h6>Proposition</h6> <p><strong>(mate bijection)</strong> <br /> Given a <a class="existingWikiWord" href="/nlab/show/2-category">2-category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>, <a class="existingWikiWord" href="/nlab/show/adjunction">adjoint pairs</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>η</mi><mo>,</mo><mi>ϵ</mi><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><mi>f</mi><mo>⊣</mo><mi>u</mi><mo lspace="verythinmathspace">:</mo><mi>b</mi><mo>→</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">(\eta,\epsilon) \colon f \dashv u \colon b \to a</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>η</mi><mo>′</mo><mo>,</mo><mi>ϵ</mi><mo>′</mo><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><mi>f</mi><mo>′</mo><mo>⊣</mo><mi>u</mi><mo>′</mo><mo lspace="verythinmathspace">:</mo><mi>b</mi><mo>′</mo><mo>→</mo><mi>a</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">(\eta',\epsilon') \colon f' \dashv u' \colon b' \to a'</annotation></semantics></math> , and <a class="existingWikiWord" href="/nlab/show/1-morphisms">1-cells</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo lspace="verythinmathspace">:</mo><mi>a</mi><mo>→</mo><mi>a</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">x \colon a \to a'</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo lspace="verythinmathspace">:</mo><mi>b</mi><mo>→</mo><mi>b</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">y \colon b \to b'</annotation></semantics></math>, there is a <a class="existingWikiWord" href="/nlab/show/bijection">bijection</a></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 stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>′</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo>′</mo><mi>x</mi><mo>,</mo><mi>y</mi><mi>f</mi><mo stretchy="false">)</mo><mo>≅</mo><mi>K</mi><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>a</mi><mo>′</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mi>u</mi><mo>,</mo><mi>u</mi><mo>′</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> K(a,b')(f' x,y f) \cong K(b,a')(x u,u' y) </annotation></semantics></math></div> <p>given by <a class="existingWikiWord" href="/nlab/show/pasting">pasting</a> with the <a class="existingWikiWord" href="/nlab/show/unit+of+an+adjunction">unit</a> of one adjunction and the <a class="existingWikiWord" href="/nlab/show/counit+of+an+adjunction">counit</a> of the other, i.e.:</p> <svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="285.183pt" height="64.922pt" viewBox="0 0 285.183 64.922" version="1.2"> <defs> <g> <symbol overflow="visible" 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<h6>Definition</h6> <p>The 2-cells <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi></mrow><annotation encoding="application/x-tex">\mu</annotation></semantics></math> in prop. <a class="maruku-ref" href="#MateBijection"></a> are called <strong>mates</strong> [<a href="#KellyStreet06">Kelly & Street (2006) p. 87</a>; <a href="#Leinster04">Leinster (2004), pp. 150</a>] (earlier: <em><a class="existingWikiWord" href="/nlab/show/conjugate+transformation+of+adjoints">conjugates</a></em>, <a href="#MacLane71">MacLane (1971), p. 98</a>, see Exp. <a class="maruku-ref" href="#ConjugateTransformationOfAdjoints"></a> below) with respect to the adjunctions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>⊣</mo><mi>u</mi></mrow><annotation encoding="application/x-tex">f \dashv u</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>′</mo><mo>⊣</mo><mi>u</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">f' \dashv u'</annotation></semantics></math> (and to the 1-cells <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> and <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>).</p> </div> </p> <h2 id="properties">Properties</h2> <h3 id="general">General</h3> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>Strict <a class="existingWikiWord" href="/nlab/show/2-functors">2-functors</a> preserve adjunctions and <a class="existingWikiWord" href="/nlab/show/pasting+diagrams">pasting diagrams</a>, so that if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo lspace="verythinmathspace">:</mo><mi>K</mi><mo>→</mo><mi>J</mi></mrow><annotation encoding="application/x-tex">F \colon K \to J</annotation></semantics></math> is a 2-functor and if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi></mrow><annotation encoding="application/x-tex">\mu</annotation></semantics></math> are mates wrt <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>⊣</mo><mi>u</mi></mrow><annotation encoding="application/x-tex">f \dashv u</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>′</mo><mo>⊣</mo><mi>u</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">f' \dashv u'</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mi>λ</mi></mrow><annotation encoding="application/x-tex">F \lambda</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mi>μ</mi></mrow><annotation encoding="application/x-tex">F \mu</annotation></semantics></math> are mates wrt <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mi>f</mi><mo>⊣</mo><mi>F</mi><mi>u</mi></mrow><annotation encoding="application/x-tex">F f \dashv F u</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mi>f</mi><mo>′</mo><mo>⊣</mo><mi>F</mi><mi>u</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">F f' \dashv F u'</annotation></semantics></math> in <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>.</p> </div> <div class="num_prop"> <h6 id="proposition_2">Proposition</h6> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo lspace="verythinmathspace">:</mo><mi>F</mi><mo>⇒</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">\alpha \colon F \Rightarrow G</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/2-natural+transformation">2-natural transformation</a>, then the naturality identities <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>α</mi> <mi>b</mi></msub><mo>∘</mo><mi>F</mi><mi>f</mi><mo>=</mo><mi>G</mi><mi>f</mi><mo>∘</mo><msub><mi>α</mi> <mi>a</mi></msub></mrow><annotation encoding="application/x-tex">\alpha_b \circ F f = G f \circ \alpha_a</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>α</mi> <mi>a</mi></msub><mo>∘</mo><mi>F</mi><mi>u</mi><mo>=</mo><mi>G</mi><mi>u</mi><mo>∘</mo><msub><mi>α</mi> <mi>b</mi></msub></mrow><annotation encoding="application/x-tex">\alpha_a \circ F u = G u \circ \alpha_b</annotation></semantics></math> are mates wrt <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mi>f</mi><mo>⊣</mo><mi>F</mi><mi>u</mi></mrow><annotation encoding="application/x-tex">F f \dashv F u</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mi>f</mi><mo>⊣</mo><mi>G</mi><mi>u</mi></mrow><annotation encoding="application/x-tex">G f \dashv G u</annotation></semantics></math>.</p> </div> <h3 id="naturality">Naturality</h3> <p>There are two <a class="existingWikiWord" href="/nlab/show/double+categories">double categories</a> with objects those of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>, vertical arrows <a class="existingWikiWord" href="/nlab/show/adjoint+pairs">adjoint pairs</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> and horizontal arrows 1-cells of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>. In one the 2-cells are those of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda</annotation></semantics></math> above, while in the other they are those of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi></mrow><annotation encoding="application/x-tex">\mu</annotation></semantics></math>. It is easily shown, as in Kelly–Street, that the triangle identities and the definition of composition of adjoints make these two double categories isomorphic. So for any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> there is a double category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Adj</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Adj(K)</annotation></semantics></math>, defined up to isomorphism as above but with mate-pairs in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> as 2-cells.</p> <p>What this means is that, for example, the mate of a square coming from a <a class="existingWikiWord" href="/nlab/show/pasting+diagram">pasting diagram</a> is given by pasting the mates of the individual 2-cells (whenever this makes sense).</p> <p>In the double category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Adj</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Adj(K)</annotation></semantics></math>, every vertical arrow has both a <a class="existingWikiWord" href="/nlab/show/companion">companion</a> (the left adjoint) and a <a class="existingWikiWord" href="/nlab/show/conjoint">conjoint</a> (the right adjoint). (In fact, in some sense it is the universal double category constructed from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> with this property.) Therefore, it is equivalent to a <a class="existingWikiWord" href="/nlab/show/2-category+equipped+with+proarrows">2-category equipped with proarrows</a>. More explicitly, there is a forgetful functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo lspace="verythinmathspace">:</mo><msub><mi>Adj</mi> <mi>V</mi></msub><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo><mo>→</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">L \colon Adj_V(K) \to K</annotation></semantics></math> from the 2-category of objects, adjunctions and mate-pairs in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> that sends an adjunction <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>⊣</mo><mi>u</mi></mrow><annotation encoding="application/x-tex">f \dashv u</annotation></semantics></math> 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>. It is <a class="existingWikiWord" href="/nlab/show/locally+fully+faithful+2-functor">locally fully faithful</a>, and moreover every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mi>f</mi></mrow><annotation encoding="application/x-tex">L f</annotation></semantics></math> has a right adjoint in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> by definition; this gives the more traditional definition of a proarrow equipment.</p> <h2 id="Examples">Examples</h2> <p> <div class='num_remark' id='ConjugateTransformationOfAdjoints'> <h6>Example</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/conjugate+transformation+of+adjoints">conjugate transformation of adjoints</a>)</strong> <br /> Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>⊣</mo><mi>U</mi><mo lspace="verythinmathspace">:</mo><mi>D</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">F \dashv U \colon D \to C</annotation></semantics></math> be an <a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a> in the <a class="existingWikiWord" href="/nlab/show/2-category">2-category</a> <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a>, i.e. a pair of <a class="existingWikiWord" href="/nlab/show/adjoint+functors">adjoint functors</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo lspace="verythinmathspace">:</mo><mo>*</mo><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">A \colon * \to C</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo lspace="verythinmathspace">:</mo><mo>*</mo><mo>→</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">X \colon * \to D</annotation></semantics></math> be <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">C</annotation></semantics></math> and <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> considered as <a class="existingWikiWord" href="/nlab/show/functors">functors</a> out of the <a class="existingWikiWord" href="/nlab/show/terminal+category">terminal category</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>. Then taking mates with respect to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>⊣</mo><mn>1</mn><mo lspace="verythinmathspace">:</mo><mo>*</mo><mo>→</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">1 \dashv 1 \colon * \to *</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>⊣</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">F \dashv U</annotation></semantics></math> yields the <a href="adjoint+functor#InTermsOfHomIsomorphism">hom-isomorphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo stretchy="false">(</mo><mi>F</mi><mi>A</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>X</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≅</mo><mspace width="thickmathspace"></mspace><mi>C</mi><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>U</mi><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> D(F A,\, X) \;\cong\; C(A,\, U X) </annotation></semantics></math></div> <p>and the pasting operations as above yield the notion of <a class="existingWikiWord" href="/nlab/show/conjugate+transformation+of+adjoints">conjugate transformation of adjoints</a>. (This is the original notion, due to <a href="#MacLane71">MacLane (1971), p. 98</a>)</p> <p>Moreover, the naturality of the mate correspondence yields <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">naturality</a> of the bijection.</p> </div> </p> <div class="num_example"> <h6 id="example">Example</h6> <p>If the ambient <a class="existingWikiWord" href="/nlab/show/2-category">2-category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> of a <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mo>⊗</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C}, \otimes)</annotation></semantics></math> in that</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>≃</mo><msub><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mo>⊗</mo></msub><mi>𝒞</mi></mrow><annotation encoding="application/x-tex"> K \simeq \mathbf{B}_\otimes \mathcal{C} </annotation></semantics></math></div> <p>then an <a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> is a pair of <a class="existingWikiWord" href="/nlab/show/dual+objects">dual objects</a> and the mate-construction is the construction of <a class="existingWikiWord" href="/nlab/show/dual+morphisms">dual morphisms</a> between <a class="existingWikiWord" href="/nlab/show/dualizable+objects">dualizable objects</a>.</p> </div> <div class="num_example"> <h6 id="example_2">Example</h6> <p>Suppose given a <a class="existingWikiWord" href="/nlab/show/commutative+square">commutative square</a> (up to <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>) of <a class="existingWikiWord" href="/nlab/show/functor">functor</a>s:</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><mover><mo>→</mo><mrow><msup><mi>f</mi> <mo>*</mo></msup></mrow></mover></mtd> <mtd></mtd></mtr> <mtr><mtd><msup><mo></mo><mrow><msup><mi>g</mi> <mo>*</mo></msup></mrow></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mrow><msup><mi>k</mi> <mo>*</mo></msup></mrow></msup></mtd></mtr> <mtr><mtd></mtd> <mtd><munder><mo>→</mo><mrow><msup><mi>h</mi> <mo>*</mo></msup></mrow></munder></mtd> <mtd></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{ & \overset{f^*}{\to} & \\ ^{g^*}\downarrow && \downarrow^{k^*}\\ & \underset{h^*}{\to} & } </annotation></semantics></math></div> <p>in which <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup></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><msup><mi>h</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">h^*</annotation></semantics></math> have <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a>s <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mo>!</mo></msub></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><msub><mi>h</mi> <mo>!</mo></msub></mrow><annotation encoding="application/x-tex">h_!</annotation></semantics></math>, respectively. (The classical example is a <a class="existingWikiWord" href="/nlab/show/Wirthm%C3%BCller+context">Wirthmüller context</a>.) Then the <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a> that makes the square commute</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>k</mi> <mo>*</mo></msup><msup><mi>f</mi> <mo>*</mo></msup><mo>→</mo><msup><mi>h</mi> <mo>*</mo></msup><msup><mi>g</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex"> k^* f^* \to h^* g^* </annotation></semantics></math></div> <p>has a mate</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>h</mi> <mo>!</mo></msub><msup><mi>k</mi> <mo>*</mo></msup><mo>→</mo><msup><mi>g</mi> <mo>*</mo></msup><msub><mi>f</mi> <mo>!</mo></msub></mrow><annotation encoding="application/x-tex"> h_! k^* \to g^* f_! </annotation></semantics></math></div> <p>defined as the composite</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>h</mi> <mo>!</mo></msub><msup><mi>k</mi> <mo>*</mo></msup><mover><mo>→</mo><mi>η</mi></mover><msub><mi>h</mi> <mo>!</mo></msub><msup><mi>k</mi> <mo>*</mo></msup><msup><mi>f</mi> <mo>*</mo></msup><msub><mi>f</mi> <mo>!</mo></msub><mover><mo>→</mo><mo>≅</mo></mover><msub><mi>h</mi> <mo>!</mo></msub><msup><mi>h</mi> <mo>*</mo></msup><msup><mi>g</mi> <mo>*</mo></msup><msub><mi>f</mi> <mo>!</mo></msub><mover><mo>→</mo><mi>ϵ</mi></mover><msup><mi>g</mi> <mo>*</mo></msup><msub><mi>f</mi> <mo>!</mo></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> h_! k^* \overset{\eta}{\to} h_! k^* f^* f_! \overset{\cong}{\to} h_! h^* g^* f_! \overset{\epsilon}{\to} g^* f_! \,. </annotation></semantics></math></div> <p>One says that the original square satisfies the <strong><a class="existingWikiWord" href="/nlab/show/Beck-Chevalley+condition">Beck-Chevalley condition</a></strong> if this mate is an <a class="existingWikiWord" href="/nlab/show/equivalence">equivalence</a>.</p> </div> <h2 id="multivariable_mates">Multi-variable mates</h2> <p>There is a version of the mate correspondence that applies to <a class="existingWikiWord" href="/nlab/show/two-variable+adjunctions">two-variable adjunctions</a> and <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>-variable adjunctions; see <a href="#CGR">Cheng-Gurski-Riehl</a>.</p> <p>The relationship between two of the adjoints in a multivariable adjunction can be described as a <strong>parametrized adjunction</strong>: fixing the variables in each of the categories that appear in the domains of both adjoints, the pair of functors define an adjunction between the remaining two categories. Relative to the parametrized adjunctions that define a multivariable adjunction, the multivariable mates can be understood as <a href="https://golem.ph.utexas.edu/category/2012/11/parametrized_mates_and_multiva.html">parametrized mates</a>.</p> <h2 id="references">References</h2> <p>The example of <a class="existingWikiWord" href="/nlab/show/conjugate+transformation+of+adjoints">conjugate transformation of adjoints</a> (but without the terminology of “mates”)</p> <ul> <li id="MacLane71"><a class="existingWikiWord" href="/nlab/show/Saunders+MacLane">Saunders MacLane</a>, p. 98 of: <em><a class="existingWikiWord" href="/nlab/show/Categories+Work">Categories for the working mathematician</a></em>, Graduate texts in mathematics, Springer (1971) [<a href="https://link.springer.com/book/10.1007/978-1-4757-4721-8">doi:10.1007/978-1-4757-4721-8</a>]</li> </ul> <p>The explicit notion of mates may be officially due to</p> <ul> <li id="KellyStreet06"><a class="existingWikiWord" href="/nlab/show/Max+Kelly">Max Kelly</a>, <a class="existingWikiWord" href="/nlab/show/Ross+Street">Ross Street</a>, §2.2 (esp. p. 87) of: <em>Review of the elements of 2-categories</em>, in: G.M. Kelly (ed.), Category Seminar, Lecture Notes in Mathematics <strong>420</strong> (1974) [<a href="https://doi.org/10.1007/BFb0063101">doi:10.1007/BFb0063101</a>]</li> </ul> <p>and is reviewed in:</p> <ul> <li id="Leinster04"><a class="existingWikiWord" href="/nlab/show/Tom+Leinster">Tom Leinster</a>, Section 6.1, pp. 150 in: <em>Higher operads, higher categories</em>, London Math. Soc. Lec. Note Series <strong>298</strong>, Cambridge University Press (2004) [<a href="http://arxiv.org/abs/math.CT/0305049">math.CT/0305049</a>, <a href="https://doi.org/10.1017/CBO9780511525896">doi:10.1017/CBO9780511525896</a>]</li> </ul> <p>Further review and discussion:</p> <ul> <li id="CGR"> <p><a class="existingWikiWord" href="/nlab/show/Eugenia+Cheng">Eugenia Cheng</a>, <a class="existingWikiWord" href="/nlab/show/Nick+Gurski">Nick Gurski</a>, <a class="existingWikiWord" href="/nlab/show/Emily+Riehl">Emily Riehl</a>, <em>Multivariable adjunctions and mates</em>, J. K-Theory <strong>13</strong> (2014), 337–396, <a href="https://doi.org/10.1017/is013012007jkt250">doi:10.1017/is013012007jkt250</a>, <a href="http://arxiv.org/abs/1208.4520">arXiv:1208.4520</a>.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Emily+Riehl">Emily Riehl</a>, <em>Parametrized mates and multivariable adjunctions</em> <a href="https://golem.ph.utexas.edu/category/2012/11/parametrized_mates_and_multiva.html">blog post</a>.</p> </li> </ul> <p id="ReferencesGeneralizationtoBicategories"> Discussion in the generalization of <a class="existingWikiWord" href="/nlab/show/bicategories">bicategories</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Aaron+Lauda">Aaron Lauda</a>, §3 of <em>Frobenius algebras and ambidextrous adjunctions</em>, Theory and Applications of Categories, <strong>16</strong> 04 (2006) 84-122 &lbrack<a href="http://www.tac.mta.ca/tac/volumes/16/4/16-04abs.html">tac:16-04</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Richard+Garner">Richard Garner</a>, <a class="existingWikiWord" href="/nlab/show/Michael+Shulman">Michael Shulman</a>, around 13.7 of <em>Enriched categories as a free cocompletion</em>, Advances in Mathematics <strong>289</strong> (2016) Pages 1-94, <a href="https://doi.org/10.1016/j.aim.2015.11.012">doi:10.1016/j.aim.2015.11.012</a>, <a href="https://arxiv.org/abs/1301.3191">arXiv:1301.3191</a>.</p> </li> <li id="JohnsonYau20"> <p><a class="existingWikiWord" href="/nlab/show/Niles+Johnson">Niles Johnson</a>, <a class="existingWikiWord" href="/nlab/show/Donald+Yau">Donald Yau</a>, Def. 6.1.12 in: <em>2-Dimensional Categories</em>, Oxford University Press (2021) [<a href="http://arxiv.org/abs/2002.06055">arXiv:2002.06055</a>, <a href="https://oxford.universitypressscholarship.com/view/10.1093/oso/9780198871378.001.0001/oso-9780198871378">doi:10.1093/oso/9780198871378.001.0001</a>]</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on December 4, 2023 at 11:21:19. See the <a href="/nlab/history/mate" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/mate" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/1289/#Item_20">Discuss</a><span class="backintime"><a href="/nlab/revision/mate/25" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/mate" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/mate" accesskey="S" class="navlink" id="history" rel="nofollow">History (25 revisions)</a> <a href="/nlab/show/mate/cite" style="color: black">Cite</a> <a href="/nlab/print/mate" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/mate" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>