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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> <h4 id="higher_category_theory">Higher category theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></li> <li><a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a></li> </ul> <h2 id="basic_concepts">Basic concepts</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/k-morphism">k-morphism</a>, <a class="existingWikiWord" href="/nlab/show/coherence">coherence</a></li> <li><a class="existingWikiWord" href="/nlab/show/looping+and+delooping">looping and delooping</a></li> <li><a class="existingWikiWord" href="/nlab/show/stabilization">looping and suspension</a></li> </ul> <h2 id="basic_theorems">Basic theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a>-theorem</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/delooping+hypothesis">delooping hypothesis</a>-theorem</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/periodic+table">periodic table</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stabilization+hypothesis">stabilization hypothesis</a>-theorem</p> </li> <li> <p><a class="existingWikiWord" href="/michaelshulman/show/exactness+hypothesis">exactness hypothesis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/holographic+principle+of+higher+category+theory">holographic principle</a></p> </li> </ul> <h2 id="applications">Applications</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/applications+of+%28higher%29+category+theory">applications of (higher) category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+category+theory+and+physics">higher category theory and physics</a></p> </li> </ul> <h2 id="models">Models</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/%28n%2Cr%29-category">(n,r)-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Theta-space">Theta-space</a></li> <li><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-category">∞-category</a>/<a class="existingWikiWord" href="/nlab/show/%E2%88%9E-category">∞-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category">(∞,n)-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/n-fold+complete+Segal+space">n-fold complete Segal space</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C2%29-category">(∞,2)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/quasi-category">quasi-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/algebraic+quasi-category">algebraic quasi-category</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/simplicially+enriched+category">simplicially enriched category</a></li> <li><a class="existingWikiWord" href="/nlab/show/complete+Segal+space">complete Segal space</a></li> <li><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C0%29-category">(∞,0)-category</a>/<a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/algebraic+Kan+complex">algebraic Kan complex</a></li> <li><a class="existingWikiWord" href="/nlab/show/simplicial+T-complex">simplicial T-complex</a></li> </ul> </li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2CZ%29-category">(∞,Z)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/n-category">n-category</a> = (n,n)-category <ul> <li><a class="existingWikiWord" href="/nlab/show/2-category">2-category</a>, <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-category">(2,1)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/1-category">1-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/0-category">0-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/%28-1%29-category">(-1)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/%28-2%29-category">(-2)-category</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/n-poset">n-poset</a> = <a class="existingWikiWord" href="/nlab/show/n-poset">(n-1,n)-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/poset">poset</a> = <a class="existingWikiWord" href="/nlab/show/%280%2C1%29-category">(0,1)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/2-poset">2-poset</a> = <a class="existingWikiWord" href="/nlab/show/%281%2C2%29-category">(1,2)-category</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/n-groupoid">n-groupoid</a> = (n,0)-category <ul> <li><a class="existingWikiWord" href="/nlab/show/2-groupoid">2-groupoid</a>, <a class="existingWikiWord" href="/nlab/show/3-groupoid">3-groupoid</a></li> </ul> </li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/categorification">categorification</a>/<a class="existingWikiWord" href="/nlab/show/decategorification">decategorification</a></li> <li><a class="existingWikiWord" href="/nlab/show/geometric+definition+of+higher+category">geometric definition of higher category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a></li> <li><a class="existingWikiWord" href="/nlab/show/quasi-category">quasi-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/simplicial+model+for+weak+%E2%88%9E-categories">simplicial model for weak ∞-categories</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/complicial+set">complicial set</a></li> <li><a class="existingWikiWord" href="/nlab/show/weak+complicial+set">weak complicial set</a></li> </ul> </li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/algebraic+definition+of+higher+category">algebraic definition of higher category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/bicategory">bicategory</a></li> <li><a class="existingWikiWord" href="/nlab/show/bigroupoid">bigroupoid</a></li> <li><a class="existingWikiWord" href="/nlab/show/tricategory">tricategory</a></li> <li><a class="existingWikiWord" href="/nlab/show/tetracategory">tetracategory</a></li> <li><a class="existingWikiWord" href="/nlab/show/strict+%E2%88%9E-category">strict ∞-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/Batanin+%E2%88%9E-category">Batanin ∞-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/Trimble+n-category">Trimble ∞-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/Grothendieck-Maltsiniotis+%E2%88%9E-categories">Grothendieck-Maltsiniotis ∞-categories</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a></li> <li><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28%E2%88%9E%2C1%29-category">symmetric monoidal (∞,1)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/dg-category">dg-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+category">A-∞ category</a></li> <li><a class="existingWikiWord" href="/nlab/show/triangulated+category">triangulated category</a></li> </ul> </li> </ul> </li> </ul> <h2 id="morphisms">Morphisms</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/k-morphism">k-morphism</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/2-morphism">2-morphism</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/transfor">transfor</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a></li> <li><a class="existingWikiWord" href="/nlab/show/modification">modification</a></li> </ul> </li> </ul> <h2 id="functors">Functors</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/functor">functor</a></li> <li><a class="existingWikiWord" href="/nlab/show/2-functor">2-functor</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/pseudofunctor">pseudofunctor</a></li> <li><a class="existingWikiWord" href="/nlab/show/lax+functor">lax functor</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a></li> </ul> <h2 id="universal_constructions">Universal constructions</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/2-limit">2-limit</a></li> <li><a class="existingWikiWord" href="/nlab/show/adjoint+%28%E2%88%9E%2C1%29-functor">(∞,1)-adjunction</a></li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Kan+extension">(∞,1)-Kan extension</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/limit+in+a+quasi-category">(∞,1)-limit</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Grothendieck+construction">(∞,1)-Grothendieck construction</a></li> </ul> <h2 id="extra_properties_and_structure">Extra properties and structure</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/cosmic+cube">cosmic cube</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/k-tuply+monoidal+n-category">k-tuply monoidal n-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/strict+%E2%88%9E-category">strict ∞-category</a>, <a class="existingWikiWord" href="/nlab/show/strict+%E2%88%9E-groupoid">strict ∞-groupoid</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a></li> </ul> <h2 id="1categorical_presentations">1-categorical presentations</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical category</a></li> <li><a class="existingWikiWord" href="/nlab/show/model+category">model category theory</a></li> <li><a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a></li> </ul> </div></div> </div> </div> <h1 id="companion_pairs">Companion pairs</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_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>A <strong>companion pair</strong> in a <a class="existingWikiWord" href="/nlab/show/double+category">double category</a> is a way of saying that a <a class="existingWikiWord" href="/nlab/show/loose+arrow">loose arrow</a> and a <a class="existingWikiWord" href="/nlab/show/tight+morphism">tight morphism</a> are “<a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphic</a>”, even though they do not live in the same <a class="existingWikiWord" href="/nlab/show/1-category">1-category</a>/<a class="existingWikiWord" href="/nlab/show/2-category">2-category</a>.</p> <p>A <strong>connection pair</strong> in a <a class="existingWikiWord" href="/nlab/show/double+category">double category</a> is a <a class="existingWikiWord" href="/nlab/show/strict+2-functor">strictly 2-functorial</a> choice of companion pairs for every tight morphism.</p> <h2 id="definition">Definition</h2> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">f\colon A\to B</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/tight+morphism">tight morphism</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>↛</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">g\colon A \nrightarrow B</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/loose+arrow">loose arrow</a> in a <a class="existingWikiWord" href="/nlab/show/double+category">double category</a>. 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1.228223 0.0580587 2.387085 -0.000871476 C 1.22863 -0.0562287 -0.799973 -1.44674 -1.550868 -3.069162 " transform="matrix(-0.00352879,0.991184,0.991184,0.00352879,52.821787,68.118338)"></path> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#ER_3NZxidwtDiDKqHkEKchdJXVE=-glyph2-3" x="56.393241" y="46.683125"></use> </g> <g clip-path="url(#ER_3NZxidwtDiDKqHkEKchdJXVE=-clip3)" clip-rule="nonzero"> <path style="fill:none;stroke-width:2.79451;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 108.3633 25.368423 L 108.544576 -25.48363 " transform="matrix(0.991233,0,0,-0.991233,161.309342,42.856964)"></path> </g> <g clip-path="url(#ER_3NZxidwtDiDKqHkEKchdJXVE=-clip4)" clip-rule="nonzero"> <path style="fill:none;stroke-width:1.83812;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(100%,100%,100%);stroke-opacity:1;stroke-miterlimit:10;" d="M 108.3633 25.368423 L 108.544576 -25.48363 " transform="matrix(0.991233,0,0,-0.991233,161.309342,42.856964)"></path> </g> <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 -1.550863 3.070341 C -0.799969 1.447919 1.228634 0.057408 2.387076 -0.00189022 C 1.228227 -0.0568794 -0.798458 -1.448717 -1.553018 -3.069721 " transform="matrix(0.00352879,0.991184,0.991184,-0.00352879,268.903606,68.118338)"></path> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#ER_3NZxidwtDiDKqHkEKchdJXVE=-glyph2-4" x="272.302694" y="46.683125"></use> </g> </g> </svg> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><msub><mo>∘</mo> <mi>τ</mi></msub><mi>ψ</mi><mo>=</mo><msub><mi>id</mi> <mi>g</mi></msub></mrow><annotation encoding="application/x-tex">\phi \circ_\tau \psi = id_{g}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><msub><mo>∘</mo> <mi>λ</mi></msub><mi>ψ</mi><mo>=</mo><msub><mi>id</mi> <mi>f</mi></msub></mrow><annotation encoding="application/x-tex">\phi \circ_\lambda \psi = id_{f}</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∘</mo> <mi>τ</mi></msub></mrow><annotation encoding="application/x-tex">\circ_\tau</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∘</mo> <mi>λ</mi></msub></mrow><annotation encoding="application/x-tex">\circ_\lambda</annotation></semantics></math> denote, respectively, loose and tight composition of 2-cells.</p> <p>Given such a companion pair, we say that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> are <strong>companions</strong> of each other. A double category for which every tight morphism admits a companion may be called <strong>companionable</strong>.</p> <h2 id="examples">Examples</h2> <p> <div class='num_remark'> <h6>Example</h6> <p>In the double category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Sq</mi></mstyle><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{Sq}(K)</annotation></semantics></math> of squares (<a class="existingWikiWord" href="/nlab/show/quintets">quintets</a>) in any <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 companion pair is simply an invertible 2-cell between two parallel 1-morphisms 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>.</p> </div> </p> <p> <div class='num_remark'> <h6>Example</h6> <p>In the <a class="existingWikiWord" href="/nlab/show/double+category+of+algebras"> double category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>T</mi> </mrow> <annotation encoding="application/x-tex">T</annotation> </semantics> </math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mstyle mathvariant="bold"> <mi>Alg</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">\mathbf{Alg}</annotation> </semantics> </math> of algebras, lax morphisms, and colax morphisms</a> for a <a class="existingWikiWord" href="/nlab/show/2-monad">2-monad</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>, an arrow (of either sort) has a companion precisely when it is a <em>strong</em> (= pseudo) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>-morphism. This is important in the theory of <a class="existingWikiWord" href="/nlab/show/doctrinal+adjunction">doctrinal adjunction</a>.</p> </div> </p> <h2 id="properties">Properties</h2> <ul> <li> <p>The loose (or tight) dual of a companion pair is a <a class="existingWikiWord" href="/nlab/show/conjunction">conjunction</a>.</p> </li> <li> <p>Companion pairs (and conjunctions) have a <a class="existingWikiWord" href="/nlab/show/mate">mate</a> correspondence generalizing the calculus of mates in 2-categories.</p> </li> <li> <p>If every tight arrow in some double category <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> has a companion, then the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>↦</mo><mi>g</mi></mrow><annotation encoding="application/x-tex">f\mapsto g</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/pseudofunctor">pseudofunctor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mi>D</mi><mo>→</mo><mi>L</mi><mi>D</mi></mrow><annotation encoding="application/x-tex">T D \to L D</annotation></semantics></math> from the tight 2-category to the loose one, which is the identity on objects, and <a class="existingWikiWord" href="/nlab/show/locally+fully+faithful+2-functor">locally fully faithful</a> by the mate correspondence. A choice of companions that make this a strict 2-functor is called a <a class="existingWikiWord" href="/nlab/show/connection+on+a+double+category">connection</a> on <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> (an arbitrary choice of companions may be called a “pseudo-connection”). A double category with a connection is thereby equivalent to an <a class="existingWikiWord" href="/nlab/show/F-category">F-category</a>. If every tight arrow also has a <a class="existingWikiWord" href="/nlab/show/conjoint">conjoint</a>, then this makes <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> into a <a class="existingWikiWord" href="/nlab/show/proarrow+equipment">proarrow equipment</a>, or equivalently a <a class="existingWikiWord" href="/nlab/show/framed+bicategory">framed bicategory</a>.</p> </li> <li> <p>Companion pairs and mate-pairs of 2-cells between them in any double category <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> form a 2-category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Comp</mi><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Comp(D)</annotation></semantics></math>. The functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Comp</mi><mo lspace="verythinmathspace">:</mo><mi>DblCat</mi><mo>→</mo><mn>2</mn><mi>Cat</mi></mrow><annotation encoding="application/x-tex">Comp\colon DblCat \to 2Cat</annotation></semantics></math> is right adjoint to the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sq</mi><mo lspace="verythinmathspace">:</mo><mn>2</mn><mi>Cat</mi><mo>→</mo><mi>DblCat</mi></mrow><annotation encoding="application/x-tex">Sq\colon 2Cat \to DblCat</annotation></semantics></math> sending a 2-category to its double category of <a class="existingWikiWord" href="/nlab/show/quintet+construction">squares</a>.</p> </li> <li> <p>A double category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\mathbb{C}</annotation></semantics></math>, carried by the span of categories <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mn>0</mn></msub><mover><mo>←</mo><mi>s</mi></mover><msub><mi>C</mi> <mn>1</mn></msub><mover><mo>→</mo><mi>t</mi></mover><msub><mi>C</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">C_0 \xleftarrow{s} C_1 \xrightarrow{t} C_0</annotation></semantics></math> has all companions iff such span is a <a class="existingWikiWord" href="/nlab/show/two-sided+fibration">two-sided fibration</a> in the sense of Street. Indeed, consider a <a class="existingWikiWord" href="/nlab/show/tight+map">tight map</a> <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:A \to B</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\mathbb{C}</annotation></semantics></math>, thus a morphism in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">C_0</annotation></semantics></math>, and the unit <a class="existingWikiWord" href="/nlab/show/loose+arrow">loose arrow</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">U_A</annotation></semantics></math> of <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>: since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi></mrow><annotation encoding="application/x-tex">s</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/opfibration">opfibration</a>, we obtain a square <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo>:</mo><mo>=</mo><msub><mi mathvariant="normal">cocart</mi> <mi>h</mi></msub><mo>:</mo><msub><mi>U</mi> <mi>A</mi></msub><mo>⇒</mo><msub><mi>h</mi> <mo>*</mo></msub><msub><mi>U</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">\eta:=\mathrm{cocart}_h: U_A \Rightarrow h_* U_A</annotation></semantics></math>, with source (top boundary) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math> and target (bottom boundary) necessarily <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mn>1</mn> <mi>B</mi></msub></mrow><annotation encoding="application/x-tex">1_B</annotation></semantics></math> since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi></mrow><annotation encoding="application/x-tex">s</annotation></semantics></math>-cartesian maps are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math>-vertical in a two-sided fibration. Dually, we get a cartesian square <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ε</mi><mo>:</mo><mo>=</mo><msub><mi mathvariant="normal">cart</mi> <mi>h</mi></msub><mo>:</mo><msup><mi>h</mi> <mo>*</mo></msup><msub><mi>U</mi> <mi>B</mi></msub><mo>⇒</mo><msub><mi>U</mi> <mi>B</mi></msub></mrow><annotation encoding="application/x-tex">\varepsilon:=\mathrm{cart}_h : h^*U_B \Rightarrow U_B </annotation></semantics></math>. We claim <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi mathvariant="normal">cocart</mi> <mi>h</mi></msub></mrow><annotation encoding="application/x-tex">\mathrm{cocart}_h</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi mathvariant="normal">cart</mi> <mi>h</mi></msub></mrow><annotation encoding="application/x-tex">\mathrm{cart}_h</annotation></semantics></math> are, respectively, the unit and counit of a companionship between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>′</mo><mo>:</mo><mo>=</mo><msup><mi>h</mi> <mo>*</mo></msup><msub><mi>U</mi> <mi>B</mi></msub><mo>=</mo><msub><mi>h</mi> <mo>*</mo></msub><msub><mi>U</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">h':=h^*U_B = h_*U_A</annotation></semantics></math>. First, observe this latter equation holds since the identiy square of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math> factors as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ε</mi><mi>η</mi></mrow><annotation encoding="application/x-tex">\varepsilon \eta</annotation></semantics></math>, using either universal property. This proves also the first companionship equation. As for the fact <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo>⊙</mo><mi>ε</mi><mo>=</mo><msub><mn>1</mn> <mrow><mi>h</mi><mo>′</mo></mrow></msub></mrow><annotation encoding="application/x-tex">\eta \odot \varepsilon = 1_{h'}</annotation></semantics></math>, it follows again from the aforementioned factorization: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo>⊙</mo><mi>ε</mi></mrow><annotation encoding="application/x-tex">\eta \odot \varepsilon</annotation></semantics></math> is an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>s</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(s,t)</annotation></semantics></math>-vertical morphism which is thus isomorphic to the identity of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">h'</annotation></semantics></math> by uniqueness of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi></mrow><annotation encoding="application/x-tex">s</annotation></semantics></math>-vertical, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>s</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(s,t)</annotation></semantics></math>-vertical, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math>-vertical factorization of a morphism in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">C_1</annotation></semantics></math>. <em>Vice versa</em>, companions can be used to construct the above co/cartesian lifts, as shown in (<a href="#Shulman08">Shulman ‘08</a>).</p> </li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/retrocell">retrocell</a></li> <li><a class="existingWikiWord" href="/nlab/show/conjunction">conjunction</a></li> </ul> <h2 id="references">References</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Marco+Grandis">Marco Grandis</a> and <a class="existingWikiWord" href="/nlab/show/Robert+Pare">Robert Pare</a>, <em>Adjoints for double categories</em>, <a href="http://www.numdam.org/item/CTGDC_2004__45_3_193_0.pdf">NUMDAM</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Robert+Dawson">Robert Dawson</a> and <a class="existingWikiWord" href="/nlab/show/Robert+Pare">Robert Pare</a> and <a class="existingWikiWord" href="/nlab/show/Dorette+Pronk">Dorette Pronk</a>, <em>The Span construction</em>, <a href="http://www.tac.mta.ca/tac/volumes/24/13/24-13abs.html">TAC</a>.</p> </li> <li id="Shulman08"> <p><a class="existingWikiWord" href="/nlab/show/Michael+Shulman">Michael Shulman</a>, <em>Framed bicategories and monoidal fibrations</em>, <a href="http://www.tac.mta.ca/tac/volumes/20/18/20-18abs.html">TAC</a></p> </li> </ul> <p>This latter reference explains the relationship between companions to connection pairs and <strong>foldings</strong>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Ronnie+Brown">Ronnie Brown</a> and C.B. Spencer, <a href="http://www.numdam.org/item/CTGDC_1976__17_4_343_0">Double groupoids and crossed modules</a>, <em>Cahiers de Topologie et Géométrie Différentielle Catégoriques</em> <strong>17</strong> (1976), 343–362.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ronald+Brown">Ronald Brown</a> and Ghafar H. Mosa, <em>Double categories, 2-categories, thin structures and connections</em>, Theory and Application of Categories 5.7 (1999): 163-1757.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Thomas+M.+Fiore">Thomas M. Fiore</a>, <em>Pseudo Algebras and Pseudo Double Categories</em>, <em>Journal of Homotopy and Related Structures</em>, Volume 2, Number 2, pages 119-170, 2007. 51 pages.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Robert+Pare">Robert Pare</a>, <em>Seeing double</em>, Talk given at FMCS 2018, (<a href="https://www.mscs.dal.ca/~pare/FMCS2.pdf">pdf</a>)</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on October 30, 2024 at 11:55:25. 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