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1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> bicategory (changes) </h1> <div class="navigation"> <span class="skipNav"><a href='#navEnd'>Skip the Navigation Links</a> | </span> <span style="display:inline-block; width: 0.3em;"></span> <a href="/nlab/show/diff/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/8951/#Item_23" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <p class="show_diff"> Showing changes from revision #65 to #66: <ins class="diffins">Added</ins> | <del class="diffdel">Removed</del> | <del class="diffmod">Chan</del><ins class="diffmod">ged</ins> </p> <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'> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/2-category+theory'>2-category theory</a></strong></p> <p><strong>Definitions</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/2-category'>2-category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/strict+2-category'>strict 2-category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/bicategory'>bicategory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/2-functor'>2-functor</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/pseudofunctor'>pseudofunctor</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/lax+functor'>lax functor</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/equivalence+of+2-categories'>equivalence of 2-categories</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/strict+2-natural+transformation'>2-natural transformation</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/lax+natural+transformation'>lax natural transformation</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/icon'>icon</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/modification'>modification</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/fully+faithful+morphism'>fully faithful morphism</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/faithful+morphism'>faithful morphism</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/conservative+morphism'>conservative morphism</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/pseudomonic+morphism'>pseudomonic morphism</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/discrete+morphism'>discrete morphism</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/adjunction'>adjunction</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/mate'>mate</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/monad'>monad</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cartesian+object'>cartesian object</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fibration+in+a+2-category'>fibration in a 2-category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/2-limit'>2-limit</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/2-pullback'>2-pullback</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/comma+object'>comma object</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/inserter'>inserter</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/inverter'>inverter</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/equifier'>equifier</a></p> </li> </ul> <p><strong>Structures on 2-categories</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/2-monad'>2-monad</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/lax-idempotent+2-monad'>lax-idempotent 2-monad</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/2-monad'>pseudomonad</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/pseudoalgebra+for+a+2-monad'>pseudoalgebra for a 2-monad</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/monoidal+bicategory'>monoidal 2-category</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/cartesian+bicategory'>cartesian bicategory</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Gray+tensor+product'>Gray tensor product</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/2-category+equipped+with+proarrows'>proarrow equipment</a></p> </li> </ul> </div> <h4 id='higher_category_theory'>Higher category theory</h4> <div class='hide'> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/higher+category+theory'>higher category theory</a></strong></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/category+theory'>category theory</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/homotopy+theory'>homotopy theory</a></li> </ul> <h2 id='basic_concepts'>Basic concepts</h2> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/k-morphism'>k-morphism</a>, <a class='existingWikiWord' href='/nlab/show/diff/coherence+law'>coherence</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/looping'>looping and delooping</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/stabilization'>looping and suspension</a></li> </ul> <h2 id='basic_theorems'>Basic theorems</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+hypothesis'>homotopy hypothesis</a>-theorem</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/delooping+hypothesis'>delooping hypothesis</a>-theorem</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/periodic+table'>periodic table</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/stabilization+hypothesis'>stabilization hypothesis</a>-theorem</p> </li> <li> <p><a class='existingWikiWord' href='/michaelshulman/show/diff/exactness+hypothesis' title='michaelshulman'>exactness hypothesis</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/applications+of+%28higher%29+category+theory'>applications of (higher) category theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/%28n%2Cr%29-category'>(n,r)-category</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Theta-space'>Theta-space</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/infinity-category'>∞-category</a>/<a class='existingWikiWord' href='/nlab/show/diff/infinity-category'>∞-category</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2Cn%29-category'>(∞,n)-category</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/n-fold+complete+Segal+space'>n-fold complete Segal space</a></li> </ul> </li> <li><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C2%29-category'>(∞,2)-category</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category'>(∞,1)-category</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/quasi-category'>quasi-category</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/algebraic+quasi-category'>algebraic quasi-category</a></li> </ul> </li> <li><a class='existingWikiWord' href='/nlab/show/diff/simplicially+enriched+category'>simplicially enriched category</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/complete+Segal+space'>complete Segal space</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/model+category'>model category</a></li> </ul> </li> <li><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C0%29-category'>(∞,0)-category</a>/<a class='existingWikiWord' href='/nlab/show/diff/infinity-groupoid'>∞-groupoid</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Kan+complex'>Kan complex</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/algebraic+Kan+complex'>algebraic Kan complex</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/simplicial+T-complex'>simplicial T-complex</a></li> </ul> </li> </ul> </li> <li><a class='existingWikiWord' href='/nlab/show/diff/%28%E2%88%9E%2CZ%29-category'>(∞,Z)-category</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/n-category'>n-category</a> = (n,n)-category <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/2-category'>2-category</a>, <a class='existingWikiWord' href='/nlab/show/diff/%282%2C1%29-category'>(2,1)-category</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/1-category'>1-category</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/0-category'>0-category</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/%28-1%29-category'>(-1)-category</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/%28-2%29-category'>(-2)-category</a></li> </ul> </li> <li><a class='existingWikiWord' href='/nlab/show/diff/n-poset'>n-poset</a> = <a class='existingWikiWord' href='/nlab/show/diff/n-poset'>(n-1,n)-category</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/partial+order'>poset</a> = <a class='existingWikiWord' href='/nlab/show/diff/%280%2C1%29-category'>(0,1)-category</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/2-poset'>2-poset</a> = <a class='existingWikiWord' href='/nlab/show/diff/2-poset'>(1,2)-category</a></li> </ul> </li> <li><a class='existingWikiWord' href='/nlab/show/diff/n-groupoid'>n-groupoid</a> = (n,0)-category <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/2-groupoid'>2-groupoid</a>, <a class='existingWikiWord' href='/nlab/show/diff/3-groupoid'>3-groupoid</a></li> </ul> </li> </ul> </li> <li><a class='existingWikiWord' href='/nlab/show/diff/vertical+categorification'>categorification</a>/<a class='existingWikiWord' href='/nlab/show/diff/decategorification'>decategorification</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/geometric+definition+of+higher+categories'>geometric definition of higher category</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Kan+complex'>Kan complex</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/quasi-category'>quasi-category</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/simplicial+model+for+weak+omega-categories'>simplicial model for weak ∞-categories</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/complicial+set'>complicial set</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/weak+complicial+set'>weak complicial set</a></li> </ul> </li> </ul> </li> <li><a class='existingWikiWord' href='/nlab/show/diff/algebraic+definition+of+higher+categories'>algebraic definition of higher category</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/bicategory'>bicategory</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/bigroupoid'>bigroupoid</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/tricategory'>tricategory</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/tetracategory'>tetracategory</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/strict+omega-category'>strict ∞-category</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/Batanin+omega-category'>Batanin ∞-category</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/Trimble+n-category'>Trimble ∞-category</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/Grothendieck-Maltsiniotis+infinity-category'>Grothendieck-Maltsiniotis ∞-categories</a></li> </ul> </li> <li><a class='existingWikiWord' href='/nlab/show/diff/stable+homotopy+theory'>stable homotopy theory</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/symmetric+monoidal+category'>symmetric monoidal category</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/symmetric+monoidal+%28infinity%2C1%29-category'>symmetric monoidal (∞,1)-category</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/stable+%28infinity%2C1%29-category'>stable (∞,1)-category</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/dg-category'>dg-category</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/A-infinity-category'>A-∞ category</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/triangulated+category'>triangulated category</a></li> </ul> </li> </ul> </li> </ul> <h2 id='morphisms'>Morphisms</h2> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/k-morphism'>k-morphism</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/2-morphism'>2-morphism</a></li> </ul> </li> <li><a class='existingWikiWord' href='/nlab/show/diff/transfor'>transfor</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/natural+transformation'>natural transformation</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/modification'>modification</a></li> </ul> </li> </ul> <h2 id='functors'>Functors</h2> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/functor'>functor</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/2-functor'>2-functor</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/pseudofunctor'>pseudofunctor</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/lax+functor'>lax functor</a></li> </ul> </li> <li><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-functor'>(∞,1)-functor</a></li> </ul> <h2 id='universal_constructions'>Universal constructions</h2> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/2-limit'>2-limit</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/adjoint+%28infinity%2C1%29-functor'>(∞,1)-adjunction</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-Kan+extension'>(∞,1)-Kan extension</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/%28%E2%88%9E%2C1%29-limit'>(∞,1)-limit</a></li> </ul> </li> <li><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%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/diff/cosmic+cube'>cosmic cube</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/k-tuply+monoidal+n-category'>k-tuply monoidal n-category</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/strict+omega-category'>strict ∞-category</a>, <a class='existingWikiWord' href='/nlab/show/diff/strict+omega-groupoid'>strict ∞-groupoid</a></li> </ul> </li> <li><a class='existingWikiWord' href='/nlab/show/diff/stable+%28infinity%2C1%29-category'>stable (∞,1)-category</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-topos'>(∞,1)-topos</a></li> </ul> <h2 id='1categorical_presentations'>1-categorical presentations</h2> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/homotopical+category'>homotopical category</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/model+category'>model category theory</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/enriched+category+theory'>enriched category theory</a></li> </ul> </div> </div> </div> <h1 id='bicategories'>Bicategories</h1> <div class='maruku_toc'><ul><li><a href='#idea'>Idea</a></li><li><a href='#definition'>Definition</a><ul><li><a href='#detailedDefn'>Details</a></li></ul></li><li><a href='#examples'>Examples</a></li><li><a href='#Coherence'>Coherence theorems</a></li><li><a href='#terminology'>Terminology</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>bicategory</strong> is a particular <a class='existingWikiWord' href='/nlab/show/diff/algebraic+definition+of+higher+categories'>algebraic</a> notion of <em>weak <a class='existingWikiWord' href='/nlab/show/diff/2-category'>2-category</a></em> (in fact, the earliest to be formulated, and still the one in most common use). The idea is that a bicategory is a category <em><a class='existingWikiWord' href='/nlab/show/diff/weak+enrichment'>weakly enriched</a></em> over <a class='existingWikiWord' href='/nlab/show/diff/Cat'>Cat</a>: the <a class='existingWikiWord' href='/nlab/show/diff/hom-object'>hom-objects</a> of a bicategory are <a class='existingWikiWord' href='/nlab/show/diff/hom-category'>hom-categories</a>, but the associativity and unity laws of <a class='existingWikiWord' href='/nlab/show/diff/enriched+category'>enriched categories</a> hold only up to coherent isomorphism.</p> <p>For information on <a class='existingWikiWord' href='/nlab/show/diff/morphism'>morphisms</a> of bicategories, see <a class='existingWikiWord' href='/nlab/show/diff/pseudofunctor'>pseudofunctor</a>.</p> <h2 id='definition'>Definition</h2> <p>A <strong>bicategory</strong> <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math> consists of</p> <ul> <li>A <a class='existingWikiWord' href='/nlab/show/diff/collection'>collection</a> of <strong><a class='existingWikiWord' href='/nlab/show/diff/object'>objects</a></strong> <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>,</mo><mi>…</mi></mrow><annotation encoding='application/x-tex'>x,y,z,\dots</annotation></semantics></math>, also called <strong><math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>0</mn></mrow><annotation encoding='application/x-tex'>0</annotation></semantics></math>-cells</strong>;</li> <li>For each pair of <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>0</mn></mrow><annotation encoding='application/x-tex'>0</annotation></semantics></math>-cells <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow><annotation encoding='application/x-tex'>x,y</annotation></semantics></math>, a <a class='existingWikiWord' href='/nlab/show/diff/category'>category</a> <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi><mo stretchy='false'>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>B(x,y)</annotation></semantics></math>, whose objects are called <strong><a class='existingWikiWord' href='/nlab/show/diff/morphism'>morphisms</a></strong> or <strong><math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>1</mn></mrow><annotation encoding='application/x-tex'>1</annotation></semantics></math>-cells</strong> and whose morphisms are called <strong><a class='existingWikiWord' href='/nlab/show/diff/2-morphism'>2-morphisms</a></strong> or <strong><math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>2</mn></mrow><annotation encoding='application/x-tex'>2</annotation></semantics></math>-cells</strong>;</li> <li>For each <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>0</mn></mrow><annotation encoding='application/x-tex'>0</annotation></semantics></math>-cell <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math>, a distinguished <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>1</mn></mrow><annotation encoding='application/x-tex'>1</annotation></semantics></math>-cell <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mn>1</mn> <mi>x</mi></msub><mo>∈</mo><mi>B</mi><mo stretchy='false'>(</mo><mi>x</mi><mo>,</mo><mi>x</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>1_x\in B(x,x)</annotation></semantics></math> called the <strong><a class='existingWikiWord' href='/nlab/show/diff/identity+morphism'>identity morphism</a></strong> or <strong>identity <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>1</mn></mrow><annotation encoding='application/x-tex'>1</annotation></semantics></math>-cell</strong> at <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math>;</li> <li>For each triple of <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>0</mn></mrow><annotation encoding='application/x-tex'>0</annotation></semantics></math>-cells <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi></mrow><annotation encoding='application/x-tex'>x,y,z</annotation></semantics></math>, a functor <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>∘</mo><mo lspace='verythinmathspace'>:</mo><mi>B</mi><mo stretchy='false'>(</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy='false'>)</mo><mo>×</mo><mi>B</mi><mo stretchy='false'>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy='false'>)</mo><mo>→</mo><mi>B</mi><mo stretchy='false'>(</mo><mi>x</mi><mo>,</mo><mi>z</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>{\circ}\colon B(y,z)\times B(x,y) \to B(x,z)</annotation></semantics></math> called <strong><a class='existingWikiWord' href='/nlab/show/diff/horizontal+composition'>horizontal composition</a></strong>;</li> <li>For each pair of <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>0</mn></mrow><annotation encoding='application/x-tex'>0</annotation></semantics></math>-cells <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow><annotation encoding='application/x-tex'>x,y</annotation></semantics></math>, <a class='existingWikiWord' href='/nlab/show/diff/natural+isomorphism'>natural isomorphisms</a> called <strong><a class='existingWikiWord' href='/nlab/show/diff/unitor'>unitors</a></strong>: <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mo>(</mo><mtable columnalign='right center left' displaystyle='false' rowspacing='0.5ex'><mtr><mtd><mi>f</mi></mtd> <mtd><mo>↦</mo></mtd> <mtd><mi>f</mi><mo>∘</mo><msub><mn>1</mn> <mi>x</mi></msub></mtd></mtr> <mtr><mtd><mi>θ</mi></mtd> <mtd><mo>↦</mo></mtd> <mtd><mi>θ</mi><mo>∘</mo><msub><mn>1</mn> <mrow><msub><mn>1</mn> <mi>x</mi></msub></mrow></msub></mtd></mtr></mtable><mo>)</mo></mrow><mo>≅</mo><msub><mi>id</mi> <mrow><mi>B</mi><mo stretchy='false'>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy='false'>)</mo></mrow></msub><mo>≅</mo><mrow><mo>(</mo><mtable columnalign='right center left' displaystyle='false' rowspacing='0.5ex'><mtr><mtd><mi>f</mi></mtd> <mtd><mo>↦</mo></mtd> <mtd><msub><mn>1</mn> <mi>y</mi></msub><mo>∘</mo><mi>f</mi></mtd></mtr> <mtr><mtd><mi>θ</mi></mtd> <mtd><mo>↦</mo></mtd> <mtd><msub><mn>1</mn> <mrow><msub><mn>1</mn> <mi>y</mi></msub></mrow></msub><mo>∘</mo><mi>θ</mi></mtd></mtr></mtable><mo>)</mo></mrow><mo>:</mo><mi>B</mi><mo stretchy='false'>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy='false'>)</mo><mo>→</mo><mi>B</mi><mo stretchy='false'>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\left( \begin{array}{rcl} f&\mapsto&f \circ 1_x\\ \theta&\mapsto&\theta \circ 1_{1_x} \end{array} \right) \cong id_{B(x,y)} \cong \left( \begin{array}{rcl} f&\mapsto&1_y\circ f\\ \theta&\mapsto&1_{1_y} \circ \theta \end{array} \right):B(x,y)\rightarrow B(x,y)</annotation></semantics></math></li> <li>For each quadruple of <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>0</mn></mrow><annotation encoding='application/x-tex'>0</annotation></semantics></math>-cells <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>w</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi></mrow><annotation encoding='application/x-tex'>w,x,y,z</annotation></semantics></math>, a natural isomorphism called the <strong><a class='existingWikiWord' href='/nlab/show/diff/associator'>associator</a></strong> between the two functors from <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi><mo stretchy='false'>(</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy='false'>)</mo><mo>×</mo><mi>B</mi><mo stretchy='false'>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy='false'>)</mo><mo>×</mo><mi>B</mi><mo stretchy='false'>(</mo><mi>w</mi><mo>,</mo><mi>x</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>B(y,z) \times B(x,y) \times B(w,x)</annotation></semantics></math> to <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi><mo stretchy='false'>(</mo><mi>w</mi><mo>,</mo><mi>z</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>B(w,z)</annotation></semantics></math> built out of <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_25' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>∘</mo></mrow><annotation encoding='application/x-tex'>{\circ}</annotation></semantics></math></li> </ul> <p>such that</p> <ul> <li>The <a class='existingWikiWord' href='/nlab/show/diff/pentagon+identity'>pentagon identity</a> is satisfied by the <a class='existingWikiWord' href='/nlab/show/diff/associator'>associators</a>;</li> <li>And the triangle identity is satisfied by the <a class='existingWikiWord' href='/nlab/show/diff/unitor'>unitors</a>.</li> </ul> <p>If there is exactly one <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>0</mn></mrow><annotation encoding='application/x-tex'>0</annotation></semantics></math>-cell, say <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>*</mo></mrow><annotation encoding='application/x-tex'>*</annotation></semantics></math>, then the definition is exactly the same as a monoidal structure on the category <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi><mo stretchy='false'>(</mo><mo>*</mo><mo>,</mo><mo>*</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>B(*,*)</annotation></semantics></math>. This is one of the motivating examples behind the <a class='existingWikiWord' href='/nlab/show/diff/delooping+hypothesis'>delooping hypothesis</a> and the general notion of <a class='existingWikiWord' href='/nlab/show/diff/k-tuply+monoidal+n-category'>k-tuply monoidal n-category</a>.</p> <h3 id='detailedDefn'>Details</h3> <p>Here we spell out the above definition in full detail. Compare to the <a href='/nlab/show/strict+2-category#detailedDefn'>detailed definition of strict <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_29' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>2</mn></mrow><annotation encoding='application/x-tex'>2</annotation></semantics></math>-category</a>, which is written in the same style but is simpler.</p> <p>A bicategory <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_30' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math> consists of</p> <ul> <li>a collection <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Ob</mi><mi>B</mi></mrow><annotation encoding='application/x-tex'>Ob B</annotation></semantics></math> or <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_32' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Ob</mi> <mi>B</mi></msub></mrow><annotation encoding='application/x-tex'>Ob_B</annotation></semantics></math> of <em>objects</em> or <em><math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_33' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>0</mn></mrow><annotation encoding='application/x-tex'>0</annotation></semantics></math>-cells</em>,</li> <li>for each object <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_34' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>a</mi></mrow><annotation encoding='application/x-tex'>a</annotation></semantics></math> and object <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_35' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>b</mi></mrow><annotation encoding='application/x-tex'>b</annotation></semantics></math>, a collection <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_36' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi><mo stretchy='false'>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>B(a,b)</annotation></semantics></math> or <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_37' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Hom</mi> <mi>B</mi></msub><mo stretchy='false'>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Hom_B(a,b)</annotation></semantics></math> of <em>morphisms</em> or <em><math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_38' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>1</mn></mrow><annotation encoding='application/x-tex'>1</annotation></semantics></math>-cells</em> <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_39' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>a</mi><mo>→</mo><mi>b</mi></mrow><annotation encoding='application/x-tex'>a \to b</annotation></semantics></math>, and</li> <li>for each object <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_40' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>a</mi></mrow><annotation encoding='application/x-tex'>a</annotation></semantics></math>, object <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_41' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>b</mi></mrow><annotation encoding='application/x-tex'>b</annotation></semantics></math>, morphism <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_42' xmlns='http://www.w3.org/1998/Math/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>, and morphism <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_43' xmlns='http://www.w3.org/1998/Math/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 \to b</annotation></semantics></math>, a collection <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_44' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi><mo stretchy='false'>(</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>B(f,g)</annotation></semantics></math> or <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_45' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>2</mn><msub><mi>Hom</mi> <mi>B</mi></msub><mo stretchy='false'>(</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>2Hom_B(f,g)</annotation></semantics></math> of <em><math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_46' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>2</mn></mrow><annotation encoding='application/x-tex'>2</annotation></semantics></math>-morphisms</em> or <em><math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_47' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>2</mn></mrow><annotation encoding='application/x-tex'>2</annotation></semantics></math>-cells</em> <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_48' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>⇒</mo><mi>g</mi></mrow><annotation encoding='application/x-tex'>f \Rightarrow g</annotation></semantics></math> or <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_49' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>⇒</mo><mi>g</mi><mo lspace='verythinmathspace'>:</mo><mi>a</mi><mo>→</mo><mi>b</mi></mrow><annotation encoding='application/x-tex'>f \Rightarrow g\colon a \to b</annotation></semantics></math>,</li> </ul> <p>equipped with</p> <ul> <li>for each object <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_50' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>a</mi></mrow><annotation encoding='application/x-tex'>a</annotation></semantics></math>, an <em>identity</em> <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_51' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mn>1</mn> <mi>a</mi></msub><mo lspace='verythinmathspace'>:</mo><mi>a</mi><mo>→</mo><mi>a</mi></mrow><annotation encoding='application/x-tex'>1_a\colon a \to a</annotation></semantics></math> or <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_52' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mo lspace='0em' rspace='thinmathspace'>id</mo> <mi>a</mi></msub><mo lspace='verythinmathspace'>:</mo><mi>a</mi><mo>→</mo><mi>a</mi></mrow><annotation encoding='application/x-tex'>\id_a\colon a \to a</annotation></semantics></math>,</li> <li>for each <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_53' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi></mrow><annotation encoding='application/x-tex'>a,b,c</annotation></semantics></math>, <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_54' xmlns='http://www.w3.org/1998/Math/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>, and <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_55' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>g</mi><mo lspace='verythinmathspace'>:</mo><mi>b</mi><mo>→</mo><mi>c</mi></mrow><annotation encoding='application/x-tex'>g\colon b \to c</annotation></semantics></math>, a <em>composite</em> <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_56' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>;</mo><mi>g</mi><mo lspace='verythinmathspace'>:</mo><mi>a</mi><mo>→</mo><mi>c</mi></mrow><annotation encoding='application/x-tex'>f ; g\colon a \to c</annotation></semantics></math> or <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_57' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>g</mi><mo>∘</mo><mi>f</mi><mo lspace='verythinmathspace'>:</mo><mi>a</mi><mo>→</mo><mi>c</mi></mrow><annotation encoding='application/x-tex'>g \circ f\colon a \to c</annotation></semantics></math>,</li> <li>for each <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_58' xmlns='http://www.w3.org/1998/Math/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>, an <em>identity</em> or <em><math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_59' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>2</mn></mrow><annotation encoding='application/x-tex'>2</annotation></semantics></math>-identity</em> <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_60' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mn>1</mn> <mi>f</mi></msub><mo lspace='verythinmathspace'>:</mo><mi>f</mi><mo>⇒</mo><mi>f</mi></mrow><annotation encoding='application/x-tex'>1_f\colon f \Rightarrow f</annotation></semantics></math> or <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_61' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mo lspace='0em' rspace='thinmathspace'>Id</mo> <mi>f</mi></msub><mo lspace='verythinmathspace'>:</mo><mi>f</mi><mo>→</mo><mi>f</mi></mrow><annotation encoding='application/x-tex'>\Id_f\colon f \to f</annotation></semantics></math>,</li> <li>for each <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_62' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi><mo>,</mo><mi>h</mi><mo lspace='verythinmathspace'>:</mo><mi>a</mi><mo>→</mo><mi>b</mi></mrow><annotation encoding='application/x-tex'>f,g,h\colon a \to b</annotation></semantics></math>, <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_63' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>η</mi><mo lspace='verythinmathspace'>:</mo><mi>f</mi><mo>⇒</mo><mi>g</mi></mrow><annotation encoding='application/x-tex'>\eta\colon f \Rightarrow g</annotation></semantics></math>, and <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_64' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>θ</mi><mo lspace='verythinmathspace'>:</mo><mi>g</mi><mo>⇒</mo><mi>h</mi></mrow><annotation encoding='application/x-tex'>\theta\colon g \Rightarrow h</annotation></semantics></math>, a <em>vertical composite</em> <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_65' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>θ</mi><mo>•</mo><mi>η</mi><mo lspace='verythinmathspace'>:</mo><mi>f</mi><mo>⇒</mo><mi>h</mi></mrow><annotation encoding='application/x-tex'>\theta \bullet \eta\colon f \Rightarrow h</annotation></semantics></math>,</li> <li>for each <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_66' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi></mrow><annotation encoding='application/x-tex'>a,b,c</annotation></semantics></math>, <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_67' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi><mo lspace='verythinmathspace'>:</mo><mi>a</mi><mo>→</mo><mi>b</mi></mrow><annotation encoding='application/x-tex'>f,g\colon a \to b</annotation></semantics></math>, <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_68' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>h</mi><mo lspace='verythinmathspace'>:</mo><mi>b</mi><mo>→</mo><mi>c</mi></mrow><annotation encoding='application/x-tex'>h\colon b \to c</annotation></semantics></math>, and <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_69' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>η</mi><mo lspace='verythinmathspace'>:</mo><mi>f</mi><mo>⇒</mo><mi>g</mi></mrow><annotation encoding='application/x-tex'>\eta\colon f \Rightarrow g</annotation></semantics></math>, a <em>left whiskering</em> <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_70' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>h</mi><mo>◃</mo><mi>η</mi><mo lspace='verythinmathspace'>:</mo><mi>h</mi><mo>∘</mo><mi>f</mi><mo>⇒</mo><mi>h</mi><mo>∘</mo><mi>g</mi></mrow><annotation encoding='application/x-tex'>h \triangleleft \eta \colon h \circ f \Rightarrow h \circ g</annotation></semantics></math>,</li> <li>for each <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_71' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi></mrow><annotation encoding='application/x-tex'>a,b,c</annotation></semantics></math>, <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_72' xmlns='http://www.w3.org/1998/Math/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>, <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_73' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>g</mi><mo>,</mo><mi>h</mi><mo lspace='verythinmathspace'>:</mo><mi>b</mi><mo>→</mo><mi>c</mi></mrow><annotation encoding='application/x-tex'>g,h\colon b \to c</annotation></semantics></math>, and <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_74' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>η</mi><mo lspace='verythinmathspace'>:</mo><mi>g</mi><mo>⇒</mo><mi>h</mi></mrow><annotation encoding='application/x-tex'>\eta\colon g \Rightarrow h</annotation></semantics></math>, a <em>right whiskering</em> <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_75' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>η</mi><mo>▹</mo><mi>f</mi><mo lspace='verythinmathspace'>:</mo><mi>g</mi><mo>∘</mo><mi>f</mi><mo>⇒</mo><mi>h</mi><mo>∘</mo><mi>f</mi></mrow><annotation encoding='application/x-tex'>\eta \triangleright f\colon g \circ f \Rightarrow h \circ f</annotation></semantics></math>,</li> <li>for each <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_76' xmlns='http://www.w3.org/1998/Math/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>, a <em>left unitor</em> <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_77' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>λ</mi> <mi>f</mi></msub><mo lspace='verythinmathspace'>:</mo><msub><mo lspace='0em' rspace='thinmathspace'>id</mo> <mi>b</mi></msub><mo>∘</mo><mi>f</mi><mo>⇒</mo><mi>f</mi></mrow><annotation encoding='application/x-tex'>\lambda_f\colon \id_b \circ f \Rightarrow f</annotation></semantics></math>, and an <em>inverse left unitor</em> <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_78' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mover><mi>λ</mi><mo stretchy='false'>¯</mo></mover> <mi>f</mi></msub><mo lspace='verythinmathspace'>:</mo><mi>f</mi><mo>⇒</mo><msub><mo lspace='0em' rspace='thinmathspace'>id</mo> <mi>b</mi></msub><mo>∘</mo><mi>f</mi></mrow><annotation encoding='application/x-tex'>\bar{\lambda}_f\colon f \Rightarrow \id_b \circ f</annotation></semantics></math>,</li> <li>for each <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_79' xmlns='http://www.w3.org/1998/Math/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>, a <em>right unitor</em> <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_80' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ρ</mi> <mi>f</mi></msub><mo lspace='verythinmathspace'>:</mo><mi>f</mi><mo>∘</mo><msub><mo lspace='0em' rspace='thinmathspace'>id</mo> <mi>a</mi></msub><mo>⇒</mo><mi>f</mi></mrow><annotation encoding='application/x-tex'>\rho_f\colon f \circ \id_a \Rightarrow f</annotation></semantics></math> and an <em>inverse right unitor</em> <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_81' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mover><mi>ρ</mi><mo stretchy='false'>¯</mo></mover> <mi>f</mi></msub><mo lspace='verythinmathspace'>:</mo><mi>f</mi><mo>⇒</mo><mi>f</mi><mo>∘</mo><msub><mo lspace='0em' rspace='thinmathspace'>id</mo> <mi>a</mi></msub></mrow><annotation encoding='application/x-tex'>\bar{\rho}_f\colon f \Rightarrow f \circ \id_a</annotation></semantics></math>, and</li> <li>for each <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_82' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>a</mi><mover><mo>→</mo><mi>f</mi></mover><mi>b</mi><mover><mo>→</mo><mi>g</mi></mover><mi>c</mi><mover><mo>→</mo><mi>h</mi></mover><mi>d</mi></mrow><annotation encoding='application/x-tex'>a \overset{f}\to b \overset{g}\to c \overset{h}\to d</annotation></semantics></math>, an <em>associator</em> <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_83' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>α</mi> <mrow><mi>h</mi><mo>,</mo><mi>g</mi><mo>,</mo><mi>f</mi></mrow></msub><mo lspace='verythinmathspace'>:</mo><mo stretchy='false'>(</mo><mi>h</mi><mo>∘</mo><mi>g</mi><mo stretchy='false'>)</mo><mo>∘</mo><mi>f</mi><mo>⇒</mo><mi>h</mi><mo>∘</mo><mo stretchy='false'>(</mo><mi>g</mi><mo>∘</mo><mi>f</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\alpha_{h,g,f}\colon (h \circ g) \circ f \Rightarrow h \circ (g \circ f)</annotation></semantics></math> and an <em>inverse associator</em> <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_84' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mover><mi>α</mi><mo stretchy='false'>¯</mo></mover> <mrow><mi>h</mi><mo>,</mo><mi>g</mi><mo>,</mo><mi>f</mi></mrow></msub><mo lspace='verythinmathspace'>:</mo><mi>h</mi><mo>∘</mo><mo stretchy='false'>(</mo><mi>g</mi><mo>∘</mo><mi>f</mi><mo stretchy='false'>)</mo><mo>⇒</mo><mo stretchy='false'>(</mo><mi>h</mi><mo>∘</mo><mi>g</mi><mo stretchy='false'>)</mo><mo>∘</mo><mi>f</mi></mrow><annotation encoding='application/x-tex'>\bar{\alpha}_{h,g,f}\colon h \circ (g \circ f) \Rightarrow (h \circ g) \circ f</annotation></semantics></math>,</li> </ul> <p>such that</p> <ul> <li>for each <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_85' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>η</mi><mo lspace='verythinmathspace'>:</mo><mi>f</mi><mo>⇒</mo><mi>g</mi><mo lspace='verythinmathspace'>:</mo><mi>a</mi><mo>→</mo><mi>b</mi></mrow><annotation encoding='application/x-tex'>\eta\colon f \Rightarrow g\colon a \to b</annotation></semantics></math>, the vertical composites <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_86' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>η</mi><mo>•</mo><msub><mo lspace='0em' rspace='thinmathspace'>Id</mo> <mi>f</mi></msub></mrow><annotation encoding='application/x-tex'>\eta \bullet \Id_f</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_87' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mo lspace='0em' rspace='thinmathspace'>Id</mo> <mi>g</mi></msub><mo>•</mo><mi>η</mi></mrow><annotation encoding='application/x-tex'>\Id_g \bullet \eta</annotation></semantics></math> both equal <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_88' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>η</mi></mrow><annotation encoding='application/x-tex'>\eta</annotation></semantics></math>,</li> <li>for each <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_89' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mover><mo>⇒</mo><mi>η</mi></mover><mi>g</mi><mover><mo>⇒</mo><mi>θ</mi></mover><mi>h</mi><mover><mo>⇒</mo><mi>ι</mi></mover><mi>i</mi><mo lspace='verythinmathspace'>:</mo><mi>a</mi><mo>→</mo><mi>b</mi></mrow><annotation encoding='application/x-tex'>f \overset{\eta}\Rightarrow g \overset{\theta}\Rightarrow h \overset{\iota}\Rightarrow i\colon a \to b</annotation></semantics></math>, the vertical composites <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_90' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ι</mi><mo>•</mo><mo stretchy='false'>(</mo><mi>θ</mi><mo>•</mo><mi>η</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\iota \bullet (\theta \bullet \eta)</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_91' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>ι</mi><mo>•</mo><mi>θ</mi><mo stretchy='false'>)</mo><mo>•</mo><mi>η</mi></mrow><annotation encoding='application/x-tex'>(\iota \bullet \theta) \bullet \eta</annotation></semantics></math> are equal,</li> <li>for each <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_92' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>a</mi><mover><mo>→</mo><mi>f</mi></mover><mi>b</mi><mover><mo>→</mo><mi>g</mi></mover><mi>c</mi></mrow><annotation encoding='application/x-tex'>a \overset{f}\to b \overset{g}\to c</annotation></semantics></math>, the whiskerings <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_93' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mo lspace='0em' rspace='thinmathspace'>Id</mo> <mi>g</mi></msub><mo>▹</mo><mi>f</mi></mrow><annotation encoding='application/x-tex'>\Id_g \triangleright f</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_94' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>g</mi><mo>◃</mo><msub><mo lspace='0em' rspace='thinmathspace'>Id</mo> <mi>f</mi></msub></mrow><annotation encoding='application/x-tex'>g \triangleleft \Id_f</annotation></semantics></math> both equal <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_95' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mo lspace='0em' rspace='thinmathspace'>Id</mo> <mrow><mi>g</mi><mo>∘</mo><mi>f</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>\Id_{g \circ f }</annotation></semantics></math>,</li> <li>for each <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_96' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mover><mo>⇒</mo><mi>η</mi></mover><mi>g</mi><mover><mo>⇒</mo><mi>θ</mi></mover><mi>h</mi><mo lspace='verythinmathspace'>:</mo><mi>a</mi><mo>→</mo><mi>b</mi></mrow><annotation encoding='application/x-tex'>f \overset{\eta}\Rightarrow g \overset{\theta}\Rightarrow h\colon a \to b</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_97' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mo lspace='verythinmathspace'>:</mo><mi>b</mi><mo>→</mo><mi>c</mi></mrow><annotation encoding='application/x-tex'>i\colon b \to c</annotation></semantics></math>, the vertical composite <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_98' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>i</mi><mo>◃</mo><mi>θ</mi><mo stretchy='false'>)</mo><mo>•</mo><mo stretchy='false'>(</mo><mi>i</mi><mo>◃</mo><mi>η</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(i \triangleleft \theta) \bullet (i \triangleleft \eta)</annotation></semantics></math> equals the whiskering <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_99' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mo>◃</mo><mo stretchy='false'>(</mo><mi>θ</mi><mo>•</mo><mi>η</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>i \triangleleft (\theta \bullet \eta)</annotation></semantics></math>,</li> <li>for each <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_100' xmlns='http://www.w3.org/1998/Math/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> and <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_101' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>g</mi><mover><mo>⇒</mo><mi>η</mi></mover><mi>h</mi><mover><mo>⇒</mo><mi>θ</mi></mover><mi>i</mi><mo lspace='verythinmathspace'>:</mo><mi>b</mi><mo>→</mo><mi>c</mi></mrow><annotation encoding='application/x-tex'>g \overset{\eta}\Rightarrow h \overset{\theta}\Rightarrow i\colon b \to c</annotation></semantics></math>, the vertical composite <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_102' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>θ</mi><mo>▹</mo><mi>f</mi><mo stretchy='false'>)</mo><mo>•</mo><mo stretchy='false'>(</mo><mi>η</mi><mo>▹</mo><mi>f</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\theta \triangleright f) \bullet (\eta \triangleright f)</annotation></semantics></math> equals the whiskering <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_103' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>θ</mi><mo>•</mo><mi>η</mi><mo stretchy='false'>)</mo><mo>▹</mo><mi>f</mi></mrow><annotation encoding='application/x-tex'>(\theta \bullet \eta) \triangleright f</annotation></semantics></math>,</li> <li>for each <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_104' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>η</mi><mo lspace='verythinmathspace'>:</mo><mi>f</mi><mo>⇒</mo><mi>g</mi><mo lspace='verythinmathspace'>:</mo><mi>a</mi><mo>→</mo><mi>b</mi></mrow><annotation encoding='application/x-tex'>\eta\colon f \Rightarrow g\colon a \to b</annotation></semantics></math>, the vertical composites <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_105' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>λ</mi> <mi>g</mi></msub><mo>•</mo><mo stretchy='false'>(</mo><msub><mo lspace='0em' rspace='thinmathspace'>id</mo> <mi>b</mi></msub><mo>◃</mo><mi>η</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\lambda_g \bullet (\id_b \triangleleft \eta)</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_106' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>η</mi><mo>•</mo><msub><mi>λ</mi> <mi>f</mi></msub></mrow><annotation encoding='application/x-tex'>\eta \bullet \lambda_f</annotation></semantics></math> are equal,</li> <li>for each <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_107' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>η</mi><mo lspace='verythinmathspace'>:</mo><mi>f</mi><mo>⇒</mo><mi>g</mi><mo lspace='verythinmathspace'>:</mo><mi>a</mi><mo>→</mo><mi>b</mi></mrow><annotation encoding='application/x-tex'>\eta\colon f \Rightarrow g\colon a \to b</annotation></semantics></math>, the vertical composites <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_108' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ρ</mi> <mi>g</mi></msub><mo>•</mo><mo stretchy='false'>(</mo><mi>η</mi><mo>▹</mo><msub><mo lspace='0em' rspace='thinmathspace'>id</mo> <mi>a</mi></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\rho_g \bullet (\eta \triangleright \id_a)</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_109' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>η</mi><mo>•</mo><msub><mi>ρ</mi> <mi>f</mi></msub></mrow><annotation encoding='application/x-tex'>\eta \bullet \rho_f</annotation></semantics></math> are equal,</li> <li>for each <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_110' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>a</mi><mover><mo>→</mo><mi>f</mi></mover><mi>b</mi><mover><mo>→</mo><mi>g</mi></mover><mi>c</mi></mrow><annotation encoding='application/x-tex'>a \overset{f}\to b \overset{g}\to c</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_111' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>η</mi><mo lspace='verythinmathspace'>:</mo><mi>h</mi><mo>⇒</mo><mi>i</mi><mo lspace='verythinmathspace'>:</mo><mi>c</mi><mo>→</mo><mi>d</mi></mrow><annotation encoding='application/x-tex'>\eta\colon h \Rightarrow i\colon c \to d</annotation></semantics></math>, the vertical composites <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_112' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mover><mi>α</mi><mo stretchy='false'>¯</mo></mover> <mrow><mi>i</mi><mo>,</mo><mi>g</mi><mo>,</mo><mi>f</mi></mrow></msub><mo>•</mo><mo stretchy='false'>(</mo><mi>η</mi><mo>▹</mo><mo stretchy='false'>(</mo><mi>g</mi><mo>∘</mo><mi>f</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\bar{\alpha}_{i,g,f} \bullet (\eta \triangleright (g \circ f))</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_113' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mo stretchy='false'>(</mo><mi>η</mi><mo>▹</mo><mi>g</mi><mo stretchy='false'>)</mo><mo>▹</mo><mi>f</mi><mo stretchy='false'>)</mo><mo>•</mo><msub><mover><mi>α</mi><mo stretchy='false'>¯</mo></mover> <mrow><mi>h</mi><mo>,</mo><mi>g</mi><mo>,</mo><mi>f</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>((\eta \triangleright g) \triangleright f) \bullet \bar{\alpha}_{h,g,f}</annotation></semantics></math> are equal,</li> <li>for each <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_114' xmlns='http://www.w3.org/1998/Math/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>, <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_115' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>η</mi><mo lspace='verythinmathspace'>:</mo><mi>g</mi><mo>⇒</mo><mi>h</mi><mo lspace='verythinmathspace'>:</mo><mi>b</mi><mo>→</mo><mi>c</mi></mrow><annotation encoding='application/x-tex'>\eta\colon g \Rightarrow h\colon b \to c</annotation></semantics></math>, and <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_116' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mo lspace='verythinmathspace'>:</mo><mi>c</mi><mo>→</mo><mi>d</mi></mrow><annotation encoding='application/x-tex'>i\colon c \to d</annotation></semantics></math>, the vertical composites <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_117' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mover><mi>α</mi><mo stretchy='false'>¯</mo></mover> <mrow><mi>i</mi><mo>,</mo><mi>h</mi><mo>,</mo><mi>f</mi></mrow></msub><mo>•</mo><mo stretchy='false'>(</mo><mi>i</mi><mo>◃</mo><mo stretchy='false'>(</mo><mi>η</mi><mo>▹</mo><mi>f</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\bar{\alpha}_{i,h,f} \bullet (i \triangleleft (\eta \triangleright f))</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_118' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mo stretchy='false'>(</mo><mi>i</mi><mo>◃</mo><mi>η</mi><mo stretchy='false'>)</mo><mo>▹</mo><mi>f</mi><mo stretchy='false'>)</mo><mo>•</mo><msub><mover><mi>α</mi><mo stretchy='false'>¯</mo></mover> <mrow><mi>i</mi><mo>,</mo><mi>g</mi><mo>,</mo><mi>f</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>((i \triangleleft \eta) \triangleright f) \bullet \bar{\alpha}_{i,g,f}</annotation></semantics></math> are equal,</li> <li>for each <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_119' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>η</mi><mo lspace='verythinmathspace'>:</mo><mi>f</mi><mo>⇒</mo><mi>g</mi><mo lspace='verythinmathspace'>:</mo><mi>a</mi><mo>→</mo><mi>b</mi></mrow><annotation encoding='application/x-tex'>\eta\colon f \Rightarrow g\colon a \to b</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_120' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>b</mi><mover><mo>→</mo><mi>h</mi></mover><mi>c</mi><mover><mo>→</mo><mi>i</mi></mover><mi>d</mi></mrow><annotation encoding='application/x-tex'>b \overset{h}\to c \overset{i}\to d</annotation></semantics></math>, the vertical composites <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_121' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mover><mi>α</mi><mo stretchy='false'>¯</mo></mover> <mrow><mi>i</mi><mo>,</mo><mi>h</mi><mo>,</mo><mi>g</mi></mrow></msub><mo>•</mo><mo stretchy='false'>(</mo><mi>i</mi><mo>◃</mo><mo stretchy='false'>(</mo><mi>h</mi><mo>◃</mo><mi>η</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\bar{\alpha}_{i,h,g} \bullet (i \triangleleft (h \triangleleft \eta))</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_122' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mo stretchy='false'>(</mo><mi>i</mi><mo>∘</mo><mi>h</mi><mo stretchy='false'>)</mo><mo>◃</mo><mi>η</mi><mo stretchy='false'>)</mo><mo>•</mo><msub><mover><mi>α</mi><mo stretchy='false'>¯</mo></mover> <mrow><mi>i</mi><mo>,</mo><mi>h</mi><mo>,</mo><mi>f</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>((i \circ h) \triangleleft \eta) \bullet \bar{\alpha}_{i,h,f}</annotation></semantics></math> are equal,</li> <li>for each <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_123' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>η</mi><mo lspace='verythinmathspace'>:</mo><mi>f</mi><mo>⇒</mo><mi>g</mi><mo lspace='verythinmathspace'>:</mo><mi>a</mi><mo>→</mo><mi>b</mi></mrow><annotation encoding='application/x-tex'>\eta\colon f \Rightarrow g\colon a \to b</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_124' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>θ</mi><mo lspace='verythinmathspace'>:</mo><mi>h</mi><mo>⇒</mo><mi>i</mi><mo lspace='verythinmathspace'>:</mo><mi>b</mi><mo>→</mo><mi>c</mi></mrow><annotation encoding='application/x-tex'>\theta\colon h \Rightarrow i\colon b \to c</annotation></semantics></math>, the vertical composites <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_125' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>i</mi><mo>◃</mo><mi>η</mi><mo stretchy='false'>)</mo><mo>•</mo><mo stretchy='false'>(</mo><mi>θ</mi><mo>▹</mo><mi>f</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(i \triangleleft \eta) \bullet (\theta \triangleright f)</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_126' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>θ</mi><mo>▹</mo><mi>g</mi><mo stretchy='false'>)</mo><mo>•</mo><mo stretchy='false'>(</mo><mi>h</mi><mo>◃</mo><mi>η</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\theta \triangleright g) \bullet (h \triangleleft \eta)</annotation></semantics></math> are equal,</li> <li>for each <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_127' xmlns='http://www.w3.org/1998/Math/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>, the vertical composites <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_128' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>λ</mi> <mi>f</mi></msub><mo>•</mo><msub><mover><mi>λ</mi><mo stretchy='false'>¯</mo></mover> <mi>f</mi></msub><mo lspace='verythinmathspace'>:</mo><mi>f</mi><mo>⇒</mo><mi>f</mi></mrow><annotation encoding='application/x-tex'>\lambda_f \bullet \bar{\lambda}_f\colon f \Rightarrow f</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_129' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mover><mi>λ</mi><mo stretchy='false'>¯</mo></mover> <mi>f</mi></msub><mo>•</mo><msub><mi>λ</mi> <mi>f</mi></msub><mo lspace='verythinmathspace'>:</mo><msub><mo lspace='0em' rspace='thinmathspace'>id</mo> <mi>b</mi></msub><mo>∘</mo><mi>f</mi><mo>⇒</mo><msub><mo lspace='0em' rspace='thinmathspace'>id</mo> <mi>b</mi></msub><mo>∘</mo><mi>f</mi></mrow><annotation encoding='application/x-tex'>\bar{\lambda}_f \bullet \lambda_f\colon \id_b \circ f \Rightarrow \id_b \circ f</annotation></semantics></math> equal the appropriate identity <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_130' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>2</mn></mrow><annotation encoding='application/x-tex'>2</annotation></semantics></math>-morphisms,</li> <li>for each <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_131' xmlns='http://www.w3.org/1998/Math/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>, the vertical composites <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_132' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ρ</mi> <mi>f</mi></msub><mo>•</mo><msub><mover><mi>ρ</mi><mo stretchy='false'>¯</mo></mover> <mi>f</mi></msub><mo lspace='verythinmathspace'>:</mo><mi>f</mi><mo>⇒</mo><mi>f</mi></mrow><annotation encoding='application/x-tex'>\rho_f \bullet \bar{\rho}_f\colon f \Rightarrow f</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_133' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mover><mi>ρ</mi><mo stretchy='false'>¯</mo></mover> <mi>f</mi></msub><mo>•</mo><msub><mi>ρ</mi> <mi>f</mi></msub><mo lspace='verythinmathspace'>:</mo><mi>f</mi><mo>∘</mo><msub><mo lspace='0em' rspace='thinmathspace'>id</mo> <mi>a</mi></msub><mo>⇒</mo><mi>f</mi><mo>∘</mo><msub><mo lspace='0em' rspace='thinmathspace'>id</mo> <mi>a</mi></msub></mrow><annotation encoding='application/x-tex'>\bar{\rho}_f \bullet \rho_f\colon f \circ \id_a \Rightarrow f \circ \id_a</annotation></semantics></math> equal the appropriate identity <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_134' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>2</mn></mrow><annotation encoding='application/x-tex'>2</annotation></semantics></math>-morphisms,</li> <li>for each <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_135' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>a</mi><mover><mo>→</mo><mi>f</mi></mover><mi>b</mi><mover><mo>→</mo><mi>g</mi></mover><mi>c</mi><mover><mo>→</mo><mi>h</mi></mover><mi>d</mi></mrow><annotation encoding='application/x-tex'>a \overset{f}\to b \overset{g}\to c \overset{h}\to d</annotation></semantics></math>, the vertical composites <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_136' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mover><mi>α</mi><mo stretchy='false'>¯</mo></mover> <mrow><mi>h</mi><mo>,</mo><mi>g</mi><mo>,</mo><mi>f</mi></mrow></msub><mo>•</mo><msub><mi>α</mi> <mrow><mi>h</mi><mo>,</mo><mi>g</mi><mo>,</mo><mi>f</mi></mrow></msub><mo lspace='verythinmathspace'>:</mo><mo stretchy='false'>(</mo><mi>h</mi><mo>∘</mo><mi>g</mi><mo stretchy='false'>)</mo><mo>∘</mo><mi>f</mi><mo>⇒</mo><mo stretchy='false'>(</mo><mi>h</mi><mo>∘</mo><mi>g</mi><mo stretchy='false'>)</mo><mo>∘</mo><mi>f</mi></mrow><annotation encoding='application/x-tex'>\bar{\alpha}_{h,g,f} \bullet \alpha_{h,g,f}\colon (h \circ g) \circ f \Rightarrow (h \circ g) \circ f</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_137' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>α</mi> <mrow><mi>h</mi><mo>,</mo><mi>g</mi><mo>,</mo><mi>f</mi></mrow></msub><mo>•</mo><msub><mover><mi>α</mi><mo stretchy='false'>¯</mo></mover> <mrow><mi>h</mi><mo>,</mo><mi>g</mi><mo>,</mo><mi>f</mi></mrow></msub><mo lspace='verythinmathspace'>:</mo><mi>h</mi><mo>∘</mo><mo stretchy='false'>(</mo><mi>g</mi><mo>∘</mo><mi>f</mi><mo stretchy='false'>)</mo><mo>⇒</mo><mi>h</mi><mo>∘</mo><mo stretchy='false'>(</mo><mi>g</mi><mo>∘</mo><mi>f</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\alpha_{h,g,f} \bullet \bar{\alpha}_{h,g,f}\colon h \circ (g \circ f) \Rightarrow h \circ (g \circ f)</annotation></semantics></math> equal the appropriate identity <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_138' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>2</mn></mrow><annotation encoding='application/x-tex'>2</annotation></semantics></math>-morphisms,</li> <li>for each <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_139' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>a</mi><mover><mo>→</mo><mi>f</mi></mover><mi>b</mi><mover><mo>→</mo><mi>g</mi></mover><mi>c</mi></mrow><annotation encoding='application/x-tex'>a \overset{f}\to b \overset{g}\to c</annotation></semantics></math>, the vertical composite <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_140' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msub><mi>ρ</mi> <mi>g</mi></msub><mo>▹</mo><mi>f</mi><mo stretchy='false'>)</mo><mo>•</mo><msub><mover><mi>α</mi><mo stretchy='false'>¯</mo></mover> <mrow><mi>g</mi><mo>,</mo><msub><mo lspace='0em' rspace='thinmathspace'>id</mo> <mi>b</mi></msub><mo>,</mo><mi>f</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>(\rho_g \triangleright f) \bullet \bar{\alpha}_{g,\id_b,f}</annotation></semantics></math> equals the whiskering <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_141' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>g</mi><mo>◃</mo><msub><mi>λ</mi> <mi>f</mi></msub></mrow><annotation encoding='application/x-tex'>g \triangleleft \lambda_f</annotation></semantics></math>, and</li> <li>for each <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_142' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>a</mi><mover><mo>→</mo><mi>f</mi></mover><mi>b</mi><mover><mo>→</mo><mi>g</mi></mover><mi>c</mi><mover><mo>→</mo><mi>h</mi></mover><mi>d</mi><mover><mo>→</mo><mi>i</mi></mover><mi>e</mi></mrow><annotation encoding='application/x-tex'>a \overset{f}\to b \overset{g}\to c \overset{h}\to d \overset{i}\to e</annotation></semantics></math>, the vertical composites <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_143' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mo stretchy='false'>(</mo><msub><mover><mi>α</mi><mo stretchy='false'>¯</mo></mover> <mrow><mi>i</mi><mo>,</mo><mi>h</mi><mo>,</mo><mi>g</mi></mrow></msub><mo>▹</mo><mi>f</mi><mo stretchy='false'>)</mo><mo>•</mo><msub><mover><mi>α</mi><mo stretchy='false'>¯</mo></mover> <mrow><mi>i</mi><mo>,</mo><mi>h</mi><mo>∘</mo><mi>g</mi><mo>,</mo><mi>f</mi></mrow></msub><mo stretchy='false'>)</mo><mo>•</mo><mo stretchy='false'>(</mo><mi>i</mi><mo>◃</mo><msub><mover><mi>α</mi><mo stretchy='false'>¯</mo></mover> <mrow><mi>h</mi><mo>,</mo><mi>g</mi><mo>,</mo><mi>f</mi></mrow></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>((\bar{\alpha}_{i,h,g} \triangleright f) \bullet \bar{\alpha}_{i,h \circ g,f}) \bullet (i \triangleleft \bar{\alpha}_{h,g,f})</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_144' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mover><mi>α</mi><mo stretchy='false'>¯</mo></mover> <mrow><mi>i</mi><mo>∘</mo><mi>h</mi><mo>,</mo><mi>g</mi><mo>,</mo><mi>f</mi></mrow></msub><mo>•</mo><msub><mover><mi>α</mi><mo stretchy='false'>¯</mo></mover> <mrow><mi>i</mi><mo>,</mo><mi>h</mi><mo>,</mo><mi>g</mi><mo>∘</mo><mi>f</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>\bar{\alpha}_{i \circ h,g,f}\bullet \bar{\alpha}_{i,h,g \circ f} </annotation></semantics></math> are equal.</li> </ul> <p>It is quite possible that there are errors or omissions in this list, although they should be easy to correct. The point is not that one would <em>want</em> to write out the definition in such elementary terms (although apparently I just did anyway) but rather that one <em>can</em>.</p> <h2 id='examples'>Examples</h2> <ul> <li> <p>Any <a class='existingWikiWord' href='/nlab/show/diff/strict+2-category'>strict 2-category</a> is a bicategory in which the unitors and associator are identities. This includes <a class='existingWikiWord' href='/nlab/show/diff/Cat'>Cat</a>, <a class='existingWikiWord' href='/nlab/show/diff/MonCat'>MonCat</a>, the algebras for any strict <a class='existingWikiWord' href='/nlab/show/diff/2-monad'>2-monad</a>, and so on, at least as classically conceived.</p> </li> <li> <p>A <a class='existingWikiWord' href='/nlab/show/diff/monoidal+category'>monoidal category</a> <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_145' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>M</mi></mrow><annotation encoding='application/x-tex'>M</annotation></semantics></math> may be regarded as a bicategory <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_146' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi><mi>M</mi></mrow><annotation encoding='application/x-tex'>B M</annotation></semantics></math> with a single object <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_147' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>•</mo></mrow><annotation encoding='application/x-tex'>\bullet</annotation></semantics></math>. The objects <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_148' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> of <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_149' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>M</mi></mrow><annotation encoding='application/x-tex'>M</annotation></semantics></math> become 1-cells <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_150' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>A</mi><mo stretchy='false'>]</mo><mo>:</mo><mo>•</mo><mo>→</mo><mo>•</mo></mrow><annotation encoding='application/x-tex'>[A]: \bullet \to \bullet</annotation></semantics></math> of <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_151' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi><mi>M</mi></mrow><annotation encoding='application/x-tex'>B M</annotation></semantics></math>; these are composed across the 0-cell <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_152' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>•</mo></mrow><annotation encoding='application/x-tex'>\bullet</annotation></semantics></math> using the definition <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_153' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>A</mi><mo stretchy='false'>]</mo><msub><mo>∘</mo> <mn>0</mn></msub><mo stretchy='false'>[</mo><mi>B</mi><mo stretchy='false'>]</mo><mo>=</mo><mo stretchy='false'>[</mo><mi>A</mi><mo>⊗</mo><mi>B</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[A] \circ_0 [B] = [A \otimes B]</annotation></semantics></math>, using the monoidal product <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_154' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>⊗</mo></mrow><annotation encoding='application/x-tex'>\otimes</annotation></semantics></math> of <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_155' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>M</mi></mrow><annotation encoding='application/x-tex'>M</annotation></semantics></math>. The identity 1-cell <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_156' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>•</mo><mo>→</mo><mo>•</mo></mrow><annotation encoding='application/x-tex'>\bullet \to \bullet</annotation></semantics></math> is <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_157' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>I</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[I]</annotation></semantics></math>, where <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_158' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>I</mi></mrow><annotation encoding='application/x-tex'>I</annotation></semantics></math> is the monoidal unit of <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_159' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>M</mi></mrow><annotation encoding='application/x-tex'>M</annotation></semantics></math>. The morphisms <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_160' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'>f: A \to B</annotation></semantics></math> become 2-cells <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_161' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>f</mi><mo stretchy='false'>]</mo><mo>:</mo><mo stretchy='false'>[</mo><mi>A</mi><mo stretchy='false'>]</mo><mo>→</mo><mo stretchy='false'>[</mo><mi>B</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[f]: [A] \to [B]</annotation></semantics></math> of <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_162' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi><mi>M</mi></mrow><annotation encoding='application/x-tex'>B M</annotation></semantics></math>. The associativity and unit constraints of the monoidal category <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_163' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>M</mi></mrow><annotation encoding='application/x-tex'>M</annotation></semantics></math> transfer straightforwardly to associativity and unit data of the bicategory <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_164' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi><mi>M</mi></mrow><annotation encoding='application/x-tex'>B M</annotation></semantics></math>. The construction is a special case of <a class='existingWikiWord' href='/nlab/show/diff/delooping'>delooping</a> (see there).</p> </li> <li> <p>Categories, <a class='existingWikiWord' href='/nlab/show/diff/anafunctor'>anafunctor</a>s, and natural transformations, which is a more appropriate definition of <a class='existingWikiWord' href='/nlab/show/diff/Cat'>Cat</a> in the absence of the <a class='existingWikiWord' href='/nlab/show/diff/axiom+of+choice'>axiom of choice</a>, form a bicategory that is not a strict 2-category. Indeed, without the axiom of choice, the proper notion of bicategory is <a class='existingWikiWord' href='/nlab/show/diff/anabicategory'>anabicategory</a>.</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/ring'>Rings</a>, <a class='existingWikiWord' href='/nlab/show/diff/bimodule'>bimodule</a>s, and bimodule homomorphisms are the prototype for many similar examples. Notably, we can generalize from rings to <a class='existingWikiWord' href='/nlab/show/diff/enriched+category'>enriched categories</a>.</p> </li> <li> <p>Objects, <a class='existingWikiWord' href='/nlab/show/diff/span'>span</a>s, and morphisms of spans in any category with <a class='existingWikiWord' href='/nlab/show/diff/pullback'>pullback</a>s also form a bicategory.</p> </li> <li> <p>The <a class='existingWikiWord' href='/nlab/show/diff/fundamental+2-groupoid'>fundamental 2-groupoid</a> of a space is a bicategory which is not necessarily strict (although it can be made strict fairly easily when the space is Hausdorff by quotienting by <a class='existingWikiWord' href='/nlab/show/diff/thin+homotopy'>thin homotopy</a>, see <a class='existingWikiWord' href='/nlab/show/diff/path+groupoid'>path groupoid</a> and <a class='existingWikiWord' href='/nlab/show/diff/fundamental+infinity-groupoid'>fundamental infinity-groupoid</a>). When the space is a CW-complex, there are easier and more computationally amenable equivalent strict 2-categories, such as that arising from the fundamental <a class='existingWikiWord' href='/nlab/show/diff/crossed+complex'>crossed complex</a>.</p> </li> </ul> <h2 id='Coherence'>Coherence theorems</h2> <p>One way to state the <a class='existingWikiWord' href='/nlab/show/diff/coherence+theorem'>coherence theorem</a> for bicategories is that every bicategory is <a class='existingWikiWord' href='/nlab/show/diff/equivalence+of+2-categories'>equivalent</a> to a <a class='existingWikiWord' href='/nlab/show/diff/strict+2-category'>strict 2-category</a>. This “<a class='existingWikiWord' href='/nlab/show/diff/rectification'>rectification</a>” is not obtained naively by forcing <a class='existingWikiWord' href='/nlab/show/diff/composition'>composition</a> to be <a class='existingWikiWord' href='/nlab/show/diff/associativity'>associative</a>, but (at least in one construction) by freely adding new composites which are strictly associative. Another way to state the coherence theorem is that every formal diagram of the constraints (<a class='existingWikiWord' href='/nlab/show/diff/associator'>associators</a> and <a class='existingWikiWord' href='/nlab/show/diff/unitor'>unitors</a>) <a class='existingWikiWord' href='/nlab/show/diff/commutative+diagram'>commutes</a>.</p> <p>Note that <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_165' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding='application/x-tex'>n=2</annotation></semantics></math> is the greatest value of <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_166' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math> for which every weak <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_167' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-category is equivalent to a fully strict one; see <a class='existingWikiWord' href='/nlab/show/diff/semi-strict+infinity-category'>semi-strict infinity-category</a> and <a class='existingWikiWord' href='/nlab/show/diff/Gray-category'>Gray-category</a>.</p> <p>The <a class='existingWikiWord' href='/nlab/show/diff/proof'>proof</a> of the coherence theorem is basically the same as the proof of the <a class='existingWikiWord' href='/nlab/show/diff/coherence+and+strictification+for+monoidal+categories'>coherence theorem for monoidal categories</a>. An abstract approach can be found in <a href='#Power89'>Power 1989</a>. For a related statement see at <em><a class='existingWikiWord' href='/nlab/show/diff/Lack%27s+coherence+observation'>Lack's coherence theorem</a></em>.</p> <p>The <a class='existingWikiWord' href='/nlab/show/diff/rectification'>rectification</a> <a class='existingWikiWord' href='/nlab/show/diff/adjunction'>adjunction</a> between <a class='existingWikiWord' href='/nlab/show/diff/bicategory'>bicategories</a> and <a class='existingWikiWord' href='/nlab/show/diff/strict+2-category'>strict 2-categories</a> can be expressed in terms of a <del class='diffdel'><a class='existingWikiWord' href='/nlab/show/diff/coreflective+subcategory'>coreflective</a></del><del class='diffdel'> </del><span class='newWikiWord'>triadjunction<a href='/nlab/new/triadjunction'>?</a></span> between <a class='existingWikiWord' href='/nlab/show/diff/tricategory'>tricategories</a>; see <a href='#Campbell18'>Campbell</a>.</p> <p>\begin{tikzcd} {2\text{-}\mathrm{Cat}} & {\mathrm{Bicat}} \arrow[{name=0, anchor=center, inner sep=0}, {\mathrm{str}}, shift right=2, from=1-2, to=1-1] \arrow[{name=1, anchor=center, inner sep=0}, shift right=2, from=1-1, to=1-2] \arrow[\dashv{anchor=center, rotate=-90}, draw=none, from=0, to=1] \end{tikzcd}</p> <p>Consequently, for any bicategory <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_168' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math> and 2-category <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_169' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math>:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_170' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>2</mn><mtext>-</mtext><mi mathvariant='normal'>Cat</mi><mo stretchy='false'>(</mo><mi mathvariant='normal'>str</mi><mi>B</mi><mo>,</mo><mi>A</mi><mo stretchy='false'>)</mo><mo>≃</mo><mi mathvariant='normal'>Bicat</mi><mo stretchy='false'>(</mo><mi>B</mi><mo>,</mo><mi>A</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>2\text{-}\mathrm{Cat}(\mathrm{str} B, A) \simeq \mathrm{Bicat}(B, A)</annotation></semantics></math></div> <h2 id='terminology'>Terminology</h2> <p>Classically, “2-category” meant <a class='existingWikiWord' href='/nlab/show/diff/strict+2-category'>strict 2-category</a>, with “bicategory” used for the weak notion. This led to the more general use of the prefix “2-” for strict (that is, strictly <a class='existingWikiWord' href='/nlab/show/diff/Cat'>Cat</a>-enriched) notions and “bi-” for weak ones. For example, classically a “2-adjunction” means a Cat-enriched adjunction, consisting of two strict 2-functors <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_171' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mo>,</mo><mi>G</mi></mrow><annotation encoding='application/x-tex'>F,G</annotation></semantics></math> and a strictly Cat-natural isomorphism of categories <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_172' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi><mo stretchy='false'>(</mo><mi>F</mi><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy='false'>)</mo><mo>≅</mo><mi>C</mi><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>G</mi><mi>Y</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>D(F X, Y)\cong C(X, G Y)</annotation></semantics></math>, while a “biadjunction” means the weak version, consisting of two weak 2-functors and a pseudo natural equivalence <math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_173' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi><mo stretchy='false'>(</mo><mi>F</mi><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy='false'>)</mo><mo>≃</mo><mi>C</mi><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>G</mi><mi>Y</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>D(F X, Y)\simeq C(X, G Y)</annotation></semantics></math>. Similarly for “2-equivalence” and “biequivalence,” and “2-limit” and “bilimit.”</p> <p>We often use “2-category” to mean a strict or weak 2-category without prejudice, although we do still use “bicategory” to refer to the particular classical algebraic notion of weak 2-category. We try to avoid the more general use of “bi-” meaning “weak,” however. For one thing, it is confusing; a “biproduct” could mean a weak <a class='existingWikiWord' href='/nlab/show/diff/2-limit'>2-limit</a>, but it could also mean an object which is both a product and a coproduct (which happens quite frequently in <a class='existingWikiWord' href='/nlab/show/diff/additive+category'>additive categories</a>).</p> <p>Moreover, in most cases the prefix is unnecessary, since once we know we are working in a bicategory, there is usually no point in considering strict notions at all. Fully weak limits are really the only sensible ones to ask for in a bicategory, and likewise for fully weak adjunctions and equivalences. Even in a strict 2-category, while we might need to say “strict” sometimes to be clear, we don't need to say “<math class='maruku-mathml' display='inline' id='mathml_75671c74b34a9bc45aa345235e84a59ce3bb4b29_174' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>2</mn></mrow><annotation encoding='application/x-tex'>2</annotation></semantics></math>-”, since we know that we are not working in a mere category. (Max Kelly pushed this point.)</p> <p>When we do have a strict 2-category, however, other strict notions can be quite technically useful, even if our ultimate interest is in the weak ones. This is somewhat analogous to the use of strict structures to model weak ones in <a class='existingWikiWord' href='/nlab/show/diff/homotopy+theory'>homotopy theory</a>; see <a href='http://arxiv.org/abs/math/0702535'>here</a> and <a href='http://arxiv.org/abs/math/0607646'>here</a> for good introductions to this sort of thing.</p> <h2 id='related_concepts'>Related concepts</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/category'>category</a></p> </li> <li> <p><strong>bicategory</strong></p> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C2%29-category'>(infinity,2)-category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/double+bicategory'>double bicategory</a> (and at <a class='existingWikiWord' href='/nlab/show/diff/double+category#double_bicategories'>double category</a>)</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/tricategory'>tricategory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/tetracategory'>tetracategory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/n-category'>n-category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2Cn%29-category'>(infinity,n)-category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/pseudofunctor'>pseudofunctor</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/locally+univalent+bicategory'>locally univalent bicategory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/univalent+bicategory'>univalent bicategory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/monoidal+bicategory'>monoidal bicategory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/closed+bicategory'>closed bicategory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/bi-initial+object'>initial object in a bicategory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/bi-terminal+object'>terminal object in a bicategory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/2-Grothendieck+construction'>2-Grothendieck construction</a></p> </li> </ul> <p>Discussion about the use of the term “weak enrichment” above is at <em><a class='existingWikiWord' href='/nlab/show/diff/weak+enrichment'>weak enrichment</a></em>.</p> <h2 id='references'>References</h2> <p>See also the references at <em><a class='existingWikiWord' href='/nlab/show/diff/2-category'>2-category</a></em>.</p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Jean+B%C3%A9nabou'>Jean Bénabou</a>, <em>Introduction to Bicategories</em>, Lecture Notes in Mathematics <strong>47</strong> Springer (1967), pp.1-77 (<a href='http://dx.doi.org/10.1007/BFb0074299'>doi:10.1007/BFb0074299</a>)</p> </li> <li id='Power89'> <p><a class='existingWikiWord' href='/nlab/show/diff/John+Power'>A. John Power</a>, <em>A general coherence result.</em> J. Pure Appl. Algebra 57 (1989), no. 2, 165–173. <a href='http://dx.doi.org/10.1016/0022-4049%2889%2990113-8'>doi:10.1016/0022-4049(89)90113-8</a> <a href='http://www.ams.org/mathscinet-getitem?mr=985657'>MR0985657</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Saunders+Mac+Lane'>Saunders MacLane</a>, §XII.6 of: <em><a class='existingWikiWord' href='/nlab/show/diff/Categories+for+the+Working+Mathematician'>Categories for the Working Mathematician</a></em>, Graduate Texts in Mathematics <strong>5</strong> Springer (second ed. 1997) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]</p> </li> <li id='Campbell18'> <p><a class='existingWikiWord' href='/nlab/show/diff/Alexander+Campbell'>Alexander Campbell</a>, <em>How strict is strictification?</em>, <a href='https://arxiv.org/abs/1802.07538'>arxiv</a></p> </li> </ul> <p>Formalization in <a class='existingWikiWord' href='/nlab/show/diff/homotopy+type+theory'>homotopy type theory</a> (see also at <a class='existingWikiWord' href='/nlab/show/diff/univalent+category'>internal category in homotopy type theory</a>):</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Benedikt+Ahrens'>Benedikt Ahrens</a>, Dan Frumin, Marco Maggesi, Niels van der Weide, <em>Bicategories in Univalent Foundations</em> (<a href='https://arxiv.org/abs/1903.01152'>arXiv:1903.01152</a>)</li> </ul> <p> </p> <p> </p> </div> <div class="revisedby"> <p> Last revised on June 18, 2024 at 22:42:14. 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