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href="/nlab/show/Kan+extension">Kan extension</a></p> </li> </ul> </li> </ul> <h2 id="sidebar_theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+construction">Grothendieck construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+functor+theorem">adjoint functor theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monadicity+theorem">monadicity theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+lifting+theorem">adjoint lifting theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gabriel-Ulmer+duality">Gabriel-Ulmer duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freyd-Mitchell+embedding+theorem">Freyd-Mitchell embedding theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relation+between+type+theory+and+category+theory">relation between type theory and category theory</a></p> </li> </ul> <h2 id="sidebar_extensions">Extensions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf+and+topos+theory">sheaf and topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a></p> </li> </ul> <h2 id="sidebar_applications">Applications</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/applications+of+%28higher%29+category+theory">applications of (higher) category theory</a></li> </ul> <div> <p> <a href="/nlab/edit/category+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="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='#Idea'>Idea</a></li> <li><a href='#Statement'>Statement</a></li> <ul> <li><a href='#AsEquations'>As equations</a></li> <li><a href='#AsDiagrams'>As diagrams</a></li> <li><a href='#AsStringDiagrams'>As string diagrams</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="Idea">Idea</h2> <p>The <strong>triangle identities</strong> or <strong>zigzag identities</strong> are identities characterized by the <a class="existingWikiWord" href="/nlab/show/unit+of+an+adjunction">unit and counit</a> of an <a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a>, such as a <a class="existingWikiWord" href="/nlab/show/adjoint+pair">pair</a> of <a class="existingWikiWord" href="/nlab/show/adjoint+functors">adjoint functors</a>. These identities <em>define</em>, equivalently, the nature of adjunction (<a href="adjoint+functor#GeneralAdjunctsInTermsOfAdjunctionUnitCounit">this prop.</a>).</p> <h2 id="Statement">Statement</h2> <p>Consider:</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo>,</mo><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}, \mathcal{D}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/pair">pair</a> of <a class="existingWikiWord" href="/nlab/show/categories">categories</a>, or, generally, of <a class="existingWikiWord" href="/nlab/show/objects">objects</a> in a given <a class="existingWikiWord" href="/nlab/show/2-category">2-category</a>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo lspace="verythinmathspace">:</mo><mi>𝒞</mi><mo>→</mo><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">L \colon \mathcal{C} \to \mathcal{D}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo lspace="verythinmathspace">:</mo><mi>𝒟</mi><mo>→</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">R \colon \mathcal{D} \to \mathcal{C}</annotation></semantics></math> a pair of <a class="existingWikiWord" href="/nlab/show/functors">functors</a> between these, or generally <a class="existingWikiWord" href="/nlab/show/1-morphisms">1-morphisms</a> in the ambient <a class="existingWikiWord" href="/nlab/show/2-category">2-category</a>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo lspace="verythinmathspace">:</mo><msub><mi>id</mi> <mi>𝒞</mi></msub><mo>⇒</mo><mi>R</mi><mo>∘</mo><mi>L</mi></mrow><annotation encoding="application/x-tex">\eta \colon id_{\mathcal{C}} \Rightarrow R \circ L</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mo lspace="verythinmathspace">:</mo><mi>L</mi><mo>∘</mo><mi>R</mi><mo>⇒</mo><msub><mi>id</mi> <mi>𝒟</mi></msub></mrow><annotation encoding="application/x-tex">\epsilon \colon L \circ R \Rightarrow id_{\mathcal{D}}</annotation></semantics></math> two <a class="existingWikiWord" href="/nlab/show/natural+transformations">natural transformations</a> or, generally <a class="existingWikiWord" href="/nlab/show/2-morphisms">2-morphisms</a>.</p> </li> </ol> <p>This data is called a <em><a class="existingWikiWord" href="/nlab/show/adjoint+pair">pair</a> of <a class="existingWikiWord" href="/nlab/show/adjoint+functors">adjoint functors</a></em> (generally: an <em><a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a></em>) if the <em>triangle identities</em> are satisfied, which may be expressed in any of the following equivalent ways:</p> <ol> <li> <p><em><a href="#AsEquations">As equations</a></em></p> </li> <li> <p><em><a href="#AsDiagrams">As diagrams</a></em></p> </li> <li> <p><em><a href="#AsStringDiagrams">As string diagrams</a></em></p> </li> </ol> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> <h3 id="AsEquations">As equations</h3> <p>As <a class="existingWikiWord" href="/nlab/show/equations">equations</a>, the triangle identities read</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>ϵ</mi><mi>L</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>∘</mo><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>L</mi><mi>η</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><msub><mi>id</mi> <mi>L</mi></msub></mrow><annotation encoding="application/x-tex"> \big( \epsilon L \big) \circ \big( L \eta \big) \;=\; id_L </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>R</mi><mi>ϵ</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>∘</mo><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>η</mi><mi>R</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><msub><mi>id</mi> <mi>R</mi></msub></mrow><annotation encoding="application/x-tex"> \big( R \epsilon \big) \circ \big( \eta R \big) \;=\; id_R </annotation></semantics></math></div> <p>Here juxtaposition denotes the <a class="existingWikiWord" href="/nlab/show/whiskering">whiskering</a> operation of <a class="existingWikiWord" href="/nlab/show/1-morphisms">1-morphisms</a> on <a class="existingWikiWord" href="/nlab/show/2-morphisms">2-morphisms</a>, as made more manifest in the diagrammatic unravelling of these expressions:</p> <h3 id="AsDiagrams">As diagrams</h3> <p>In terms of <a class="existingWikiWord" href="/nlab/show/diagrams">diagrams</a> in the <a class="existingWikiWord" href="/nlab/show/functor+categories">functor categories</a> this means</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>L</mi><mover><mo>⇒</mo><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>L</mi><mi>η</mi><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow></mover><mi>L</mi><mi>R</mi><mi>L</mi><mover><mo>⇒</mo><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>ϵ</mi><mi>L</mi><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow></mover><mi>L</mi><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>L</mi><mover><mo>⇒</mo><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msub><mi>id</mi> <mi>L</mi></msub><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow></mover><mi>L</mi></mrow><annotation encoding="application/x-tex"> L \overset{\;\;L\eta\;\;}{\Rightarrow} L R L \overset{\;\;\epsilon L\;\;}{\Rightarrow} L \;\; = \;\; L \overset{\;\;id_L\;\;}{\Rightarrow} L </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>R</mi><mover><mo>⇒</mo><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>η</mi><mi>R</mi><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow></mover><mi>R</mi><mi>L</mi><mi>R</mi><mover><mo>⇒</mo><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>R</mi><mi>ϵ</mi><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow></mover><mi>R</mi><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>R</mi><mover><mo>⇒</mo><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msub><mi>id</mi> <mi>R</mi></msub><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow></mover><mi>R</mi></mrow><annotation encoding="application/x-tex"> R \overset{\;\;\eta R\;\;}{\Rightarrow} R L R \overset{\;\;R\epsilon\;\;}{\Rightarrow} R \;\; = \;\; R \overset{\;\;id_R\;\;}{\Rightarrow} R </annotation></semantics></math></div> <p>In terms of diagrams of <a class="existingWikiWord" href="/nlab/show/2-morphisms">2-morphisms</a> in the ambient <a class="existingWikiWord" href="/nlab/show/2-category">2-category</a>, this looks as follows:</p> <svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="451.766" height="78.605" viewBox="0 0 451.766 78.605"> <defs> <g> <g id="x6mnvjSw05AA6F96gZ8rOb8dRaY=-glyph-0-0"> </g> <g id="x6mnvjSw05AA6F96gZ8rOb8dRaY=-glyph-0-1"> <path d="M 5.921875 -1.875 C 5.921875 -1.953125 5.875 -1.953125 5.8125 -1.953125 C 5.609375 -1.953125 5.3125 -1.78125 5.3125 -1.78125 C 5.0625 -1.625 5.015625 -1.546875 4.875 -1.34375 C 4.5 -0.78125 3.984375 -0.375 3.203125 -0.375 C 2.125 -0.375 1.15625 -1.140625 1.15625 -2.9375 C 1.15625 -4.015625 1.59375 -5.4375 2.21875 -6.390625 C 2.75 -7.140625 3.390625 -7.765625 4.625 -7.765625 C 5.078125 -7.765625 5.359375 -7.609375 5.359375 -7.15625 C 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8.762, 39.302)"></path> <path fill="none" stroke-width="0.478" stroke-linecap="round" stroke-linejoin="round" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M 72.780781 -10.201906 L 72.780781 -24.717531 " transform="matrix(1, 0, 0, -1, 6.77, 39.302)"></path> </svg> <p>where on the right the <a class="existingWikiWord" href="/nlab/show/identity+morphism">identity</a> <a class="existingWikiWord" href="/nlab/show/2-morphisms">2-morphisms</a> are left notationally implicit.</p> <p>If we leave the <a class="existingWikiWord" href="/nlab/show/identity+morphism">identity</a> <a class="existingWikiWord" href="/nlab/show/1-morphisms">1-morphisms</a> on the left notationally implicit, then we get the following suggestive form of the triangle identities:</p> <center> <img src="https://ncatlab.org/nlab/files/Adjointness.jpg" width="540" /> </center> <p>(taken from <em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+categories+and+toposes">geometry of physics – categories and toposes</a></em>).</p> <h3 id="AsStringDiagrams">As string diagrams</h3> <p>As <a class="existingWikiWord" href="/nlab/show/string+diagrams">string diagrams</a>, the triangle identities appear as the action of “pulling zigzags straight” (hence the name):</p> <div style="text-align:center"> <p><span style="margin-right: 50px"><img src="/nlab/files/adjunction-up-string.png" alt="String diagram of first zigzag identity (for 'Adjunction')" /></span><img src="/nlab/files/adjunction-down-string.png" alt="" /></p> </div> <p>With labels left implicit, this notation becomes very economical:</p> <div style="text-align:center"> <p><span style="margin-right: 50px"><img src="/nlab/files/adjunction-up-string-minimal.png" alt="Minimal string diagram of first zigzag identity (for 'Adjunction')" /></span><img src="/nlab/files/adjunction-down-string-minimal.png" alt="Minimal string diagram of second zigzag identity (for 'Adjunction')" /></p> </div> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/unit+of+an+adjunction">unit of an adjunction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjunct">adjunct</a></p> </li> </ul> <h2 id="references">References</h2> <p>Textbook accounts include</p> <ul> <li id="Borceux94"><a class="existingWikiWord" href="/nlab/show/Francis+Borceux">Francis Borceux</a>, Theorem 3.1.5 and Diagram 3.3 in: <em>Basic Category Theory</em>, Vol. 1 of <em><a class="existingWikiWord" href="/nlab/show/Handbook+of+Categorical+Algebra">Handbook of Categorical Algebra</a></em>, Cambridge University Press (1994)</li> </ul> <p>See the references at <em><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></em> for more.</p> </body></html> </div> <div class="revisedby"> <p> Last revised on June 22, 2023 at 16:20:30. 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