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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"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Strict -categories</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="2category_theory">2-Category theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/2-category+theory">2-category theory</a></strong></p> <p><strong>Definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-category">2-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/strict+2-category">strict 2-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bicategory">bicategory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+bicategory">enriched bicategory</a></p> </li> </ul> <p><strong>Transfors between 2-categories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-functor">2-functor</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/pseudofunctor">pseudofunctor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/lax+functor">lax functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivalence+of+2-categories">equivalence of 2-categories</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-natural+transformation">2-natural transformation</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/lax+natural+transformation">lax natural transformation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/icon">icon</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/modification">modification</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yoneda+lemma+for+bicategories">Yoneda lemma for bicategories</a></p> </li> </ul> <p><strong>Morphisms in 2-categories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fully+faithful+morphism">fully faithful morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/faithful+morphism">faithful morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conservative+morphism">conservative morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pseudomonic+morphism">pseudomonic morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/discrete+morphism">discrete morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/eso+morphism">eso morphism</a></p> </li> </ul> <p><strong>Structures in 2-categories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mate">mate</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monad">monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cartesian+object">cartesian object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fibration+in+a+2-category">fibration in a 2-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/codiscrete+cofibration">codiscrete cofibration</a></p> </li> </ul> <p><strong>Limits in 2-categories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-limit">2-limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-pullback">2-pullback</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/comma+object">comma object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/inserter">inserter</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/inverter">inverter</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equifier">equifier</a></p> </li> </ul> <p><strong>Structures on 2-categories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-monad">2-monad</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/lax-idempotent+2-monad">lax-idempotent 2-monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pseudomonad">pseudomonad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pseudoalgebra+for+a+2-monad">pseudoalgebra for a 2-monad</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+2-category">monoidal 2-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/cartesian+bicategory">cartesian bicategory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gray+tensor+product">Gray tensor product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/proarrow+equipment">proarrow equipment</a></p> </li> </ul> </div></div> <h4 id="higher_category_theory">Higher category theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></li> <li><a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a></li> </ul> <h2 id="basic_concepts">Basic concepts</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/k-morphism">k-morphism</a>, <a class="existingWikiWord" href="/nlab/show/coherence">coherence</a></li> <li><a class="existingWikiWord" href="/nlab/show/looping+and+delooping">looping and delooping</a></li> <li><a class="existingWikiWord" href="/nlab/show/stabilization">looping and suspension</a></li> </ul> <h2 id="basic_theorems">Basic theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a>-theorem</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/delooping+hypothesis">delooping hypothesis</a>-theorem</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/periodic+table">periodic table</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stabilization+hypothesis">stabilization hypothesis</a>-theorem</p> </li> <li> <p><a class="existingWikiWord" href="/michaelshulman/show/exactness+hypothesis">exactness hypothesis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/holographic+principle+of+higher+category+theory">holographic principle</a></p> </li> </ul> <h2 id="applications">Applications</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/applications+of+%28higher%29+category+theory">applications of (higher) category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+category+theory+and+physics">higher category theory and physics</a></p> </li> </ul> <h2 id="models">Models</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/%28n%2Cr%29-category">(n,r)-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Theta-space">Theta-space</a></li> <li><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-category">∞-category</a>/<a class="existingWikiWord" href="/nlab/show/%E2%88%9E-category">∞-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category">(∞,n)-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/n-fold+complete+Segal+space">n-fold complete Segal space</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C2%29-category">(∞,2)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/quasi-category">quasi-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/algebraic+quasi-category">algebraic quasi-category</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/simplicially+enriched+category">simplicially enriched category</a></li> <li><a class="existingWikiWord" href="/nlab/show/complete+Segal+space">complete Segal space</a></li> <li><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C0%29-category">(∞,0)-category</a>/<a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/algebraic+Kan+complex">algebraic Kan complex</a></li> <li><a class="existingWikiWord" href="/nlab/show/simplicial+T-complex">simplicial T-complex</a></li> </ul> </li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2CZ%29-category">(∞,Z)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/n-category">n-category</a> = (n,n)-category <ul> <li><a class="existingWikiWord" href="/nlab/show/2-category">2-category</a>, <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-category">(2,1)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/1-category">1-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/0-category">0-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/%28-1%29-category">(-1)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/%28-2%29-category">(-2)-category</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/n-poset">n-poset</a> = <a class="existingWikiWord" href="/nlab/show/n-poset">(n-1,n)-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/poset">poset</a> = <a class="existingWikiWord" href="/nlab/show/%280%2C1%29-category">(0,1)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/2-poset">2-poset</a> = <a class="existingWikiWord" href="/nlab/show/%281%2C2%29-category">(1,2)-category</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/n-groupoid">n-groupoid</a> = (n,0)-category <ul> <li><a class="existingWikiWord" href="/nlab/show/2-groupoid">2-groupoid</a>, <a class="existingWikiWord" href="/nlab/show/3-groupoid">3-groupoid</a></li> </ul> </li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/categorification">categorification</a>/<a class="existingWikiWord" href="/nlab/show/decategorification">decategorification</a></li> <li><a class="existingWikiWord" href="/nlab/show/geometric+definition+of+higher+category">geometric definition of higher category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a></li> <li><a class="existingWikiWord" href="/nlab/show/quasi-category">quasi-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/simplicial+model+for+weak+%E2%88%9E-categories">simplicial model for weak ∞-categories</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/complicial+set">complicial set</a></li> <li><a class="existingWikiWord" href="/nlab/show/weak+complicial+set">weak complicial set</a></li> </ul> </li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/algebraic+definition+of+higher+category">algebraic definition of higher category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/bicategory">bicategory</a></li> <li><a class="existingWikiWord" href="/nlab/show/bigroupoid">bigroupoid</a></li> <li><a class="existingWikiWord" href="/nlab/show/tricategory">tricategory</a></li> <li><a class="existingWikiWord" href="/nlab/show/tetracategory">tetracategory</a></li> <li><a class="existingWikiWord" href="/nlab/show/strict+%E2%88%9E-category">strict ∞-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/Batanin+%E2%88%9E-category">Batanin ∞-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/Trimble+n-category">Trimble ∞-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/Grothendieck-Maltsiniotis+%E2%88%9E-categories">Grothendieck-Maltsiniotis ∞-categories</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a></li> <li><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28%E2%88%9E%2C1%29-category">symmetric monoidal (∞,1)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/dg-category">dg-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+category">A-∞ category</a></li> <li><a class="existingWikiWord" href="/nlab/show/triangulated+category">triangulated category</a></li> </ul> </li> </ul> </li> </ul> <h2 id="morphisms">Morphisms</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/k-morphism">k-morphism</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/2-morphism">2-morphism</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/transfor">transfor</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a></li> <li><a class="existingWikiWord" href="/nlab/show/modification">modification</a></li> </ul> </li> </ul> <h2 id="functors">Functors</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/functor">functor</a></li> <li><a class="existingWikiWord" href="/nlab/show/2-functor">2-functor</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/pseudofunctor">pseudofunctor</a></li> <li><a class="existingWikiWord" href="/nlab/show/lax+functor">lax functor</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a></li> </ul> <h2 id="universal_constructions">Universal constructions</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/2-limit">2-limit</a></li> <li><a class="existingWikiWord" href="/nlab/show/adjoint+%28%E2%88%9E%2C1%29-functor">(∞,1)-adjunction</a></li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Kan+extension">(∞,1)-Kan extension</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/limit+in+a+quasi-category">(∞,1)-limit</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Grothendieck+construction">(∞,1)-Grothendieck construction</a></li> </ul> <h2 id="extra_properties_and_structure">Extra properties and structure</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/cosmic+cube">cosmic cube</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/k-tuply+monoidal+n-category">k-tuply monoidal n-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/strict+%E2%88%9E-category">strict ∞-category</a>, <a class="existingWikiWord" href="/nlab/show/strict+%E2%88%9E-groupoid">strict ∞-groupoid</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a></li> </ul> <h2 id="1categorical_presentations">1-categorical presentations</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical category</a></li> <li><a class="existingWikiWord" href="/nlab/show/model+category">model category theory</a></li> <li><a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a></li> </ul> </div></div> </div> </div> <h1 id="strict_categories">Strict <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-categories</h1> <div class='maruku_toc'> <ul> <li><a href='#Idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <ul> <li><a href='#DefinitionStrictTwoCategories'>Strict 2-categories</a></li> <li><a href='#strict_2groupoids'>Strict 2-groupoids</a></li> <li><a href='#details'>Details</a></li> <li><a href='#detailedDefn'>More details</a></li> </ul> <li><a href='#AsSesquicategories'>2-categories as sesquicategories</a></li> <li><a href='#remarks'>Remarks</a></li> <li><a href='#in_dependent_type_theory'>In dependent type theory</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#history'>History</a></li> <li><a href='#References'>References</a></li> </ul> </div> <h2 id="Idea">Idea</h2> <p>The concept of <em>strict 2-categories</em> is the simplest generalization of that of <a class="existingWikiWord" href="/nlab/show/categories">categories</a> to the <em><a class="existingWikiWord" href="/nlab/show/n-categories"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>n</mi> </mrow> <annotation encoding="application/x-tex">n</annotation> </semantics> </math>-categories</a></em> of <a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a>. It is a one-step <a class="existingWikiWord" href="/nlab/show/vertical+categorification">categorification</a> of the concept of a <em><a class="existingWikiWord" href="/nlab/show/category">category</a></em> with strict choice of structure morphisms.</p> <p>More concretely, a strict 2-category is a <a class="existingWikiWord" href="/nlab/show/directed+n-graph">directed 2-graph</a> equipped with <a class="existingWikiWord" href="/nlab/show/horizontal+composition">horizontal-</a> and <a class="existingWikiWord" href="/nlab/show/vertical+composition">vertical composition</a> of adjacent 1-cells (<a class="existingWikiWord" href="/nlab/show/1-morphisms">1-morphisms</a>) and 2-cells (<a class="existingWikiWord" href="/nlab/show/2-morphisms">2-morphisms</a>), respectively, which is strictly <a class="existingWikiWord" href="/nlab/show/unitality">unital</a> and <a class="existingWikiWord" href="/nlab/show/associativity">associative</a> in both directions, and such that both types of composition are compatible (the “<a class="existingWikiWord" href="/nlab/show/interchange+law">interchange law</a>”).</p> <p>A quick way of making this precise, is to say that strict 2-categories are <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a>-<a class="existingWikiWord" href="/nlab/show/enriched+categories">enriched categories</a>, see <a href="#DefinitionStrictTwoCategories">below</a>.</p> <p>The term <em>2-category</em> implicitly refers to a <a class="existingWikiWord" href="/nlab/show/geometric+shapes+for+higher+structures">globular</a> structure. By contrast, <a class="existingWikiWord" href="/nlab/show/double+category">double categories</a> are based on <a class="existingWikiWord" href="/nlab/show/cubes">cubes</a> instead. The two notions are closely related, however: every strict 2-category gives rise to several strict double categories, and every double category has several <a class="existingWikiWord" href="/nlab/show/underlying">underlying</a> 2-categories.</p> <p>Notice that <em><a class="existingWikiWord" href="/nlab/show/double+category">double category</a></em> is another term for <em><a class="existingWikiWord" href="/nlab/show/n-fold+category">2-fold category</a></em>. Strict 2-categories may be identified with those strict 2-fold/double categories whose category of vertical morphisms is <a class="existingWikiWord" href="/nlab/show/discrete+category">discrete</a>, or those whose category of horizontal morphisms is discrete.</p> <p>(And similarly, strict globular <a class="existingWikiWord" href="/nlab/show/n-category">n-categories</a> may be identified with those <a class="existingWikiWord" href="/nlab/show/n-fold+category">n-fold categories</a> for which all cube faces “in one direction” are discrete. A similar statement for weak <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-categories is to be expected, but little seems to be known about this.)</p> <h2 id="definition">Definition</h2> <h3 id="DefinitionStrictTwoCategories">Strict 2-categories</h3> <p>A <em>strict 2-category</em>, often called simply a <em>2-category</em>, is a <a class="existingWikiWord" href="/nlab/show/enriched+category">category enriched over</a> <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a>, where <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a> is regarded as the <a class="existingWikiWord" href="/nlab/show/1-category">1-category</a> of <a class="existingWikiWord" href="/nlab/show/strict+categories">strict categories</a> with <a class="existingWikiWord" href="/nlab/show/functors">functors</a> between them and equipped with the <a class="existingWikiWord" href="/nlab/show/cartesian+monoidal+category">cartesian monoidal</a>-<a class="existingWikiWord" href="/nlab/show/structure">structure</a> given by forming <a class="existingWikiWord" href="/nlab/show/product+categories">product categories</a>.</p> <h3 id="strict_2groupoids">Strict 2-groupoids</h3> <p>Similarly, a <a class="existingWikiWord" href="/nlab/show/strict+2-groupoid">strict 2-groupoid</a> is a <a class="existingWikiWord" href="/nlab/show/enriched+groupoid">groupoid enriched</a> over the <a class="existingWikiWord" href="/nlab/show/1-category">1-category</a> <a class="existingWikiWord" href="/nlab/show/Grpd">Grpd</a>. This is also called a <em><a class="existingWikiWord" href="/nlab/show/geometric+shapes+for+higher+structures">globular</a> strict 2-groupoid</em>, to emphasise the underlying geometry.</p> <p>The category of strict 2-groupoids is <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalent</a> to the category of <a class="existingWikiWord" href="/nlab/show/crossed+modules">crossed modules</a> over groupoids. It is also equivalent to the category of (strict) <a class="existingWikiWord" href="/nlab/show/double+groupoids">double groupoids</a> “with connections”.</p> <p>They are also special cases of strict globular omega-groupoids, and the category of these is equivalent to the category of <a class="existingWikiWord" href="/nlab/show/crossed+complex">crossed complex</a>es.</p> <h3 id="details">Details</h3> <p>Working out the meaning of <em><a class="existingWikiWord" href="/nlab/show/Cat">Cat</a>-<a class="existingWikiWord" href="/nlab/show/enriched+category">enriched category</a></em> (<a href="#DefinitionStrictTwoCategories">above</a>), we find that a strict 2-category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> is given by</p> <ul> <li> <p>a <a class="existingWikiWord" href="/nlab/show/class">class</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ob</mi><mi>K</mi></mrow><annotation encoding="application/x-tex">ob K</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/objects">objects</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>,</mo><mi>…</mi></mrow><annotation encoding="application/x-tex">a,b,c,\ldots</annotation></semantics></math>, together with</p> </li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/hom-category">hom-category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(a,b)</annotation></semantics></math> for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>,</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">a,b</annotation></semantics></math>, and</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/functors">functors</a></p> <ul> <li> <p>(<a class="existingWikiWord" href="/nlab/show/identity+morphisms">identity morphisms</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mn>1</mn> <mi>a</mi></msub><mo>:</mo><mstyle mathvariant="bold"><mn>1</mn></mstyle><mo>→</mo><mi>K</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">1_a : \mathbf{1} \to K(a,a)</annotation></semantics></math></p> </li> <li> <p>(<a class="existingWikiWord" href="/nlab/show/composition">composition</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>comp</mi><mo>:</mo><mi>K</mi><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>×</mo><mi>K</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo>→</mo><mi>K</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">comp : K(b,c) \times K(a,b) \to K(a,c)</annotation></semantics></math></p> </li> </ul> <p>for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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>, satisfying <a class="existingWikiWord" href="/nlab/show/associativity">associativity</a> and <a class="existingWikiWord" href="/nlab/show/unitality">unitality</a> <a class="existingWikiWord" href="/nlab/show/axioms">axioms</a> (as given at <em><a class="existingWikiWord" href="/nlab/show/enriched+category">enriched category</a></em>).</p> </li> </ul> <p>As for ordinary (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Set</mi></mrow><annotation encoding="application/x-tex">Set</annotation></semantics></math>-enriched) <a class="existingWikiWord" href="/nlab/show/categories">categories</a>, an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∈</mo><mi>K</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f \in K(a,b)</annotation></semantics></math> is called a <em><a class="existingWikiWord" href="/nlab/show/1-morphism">1-morphism</a></em> or <em>1-cell</em> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math> and written <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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> as usual. But given <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi><mo>:</mo><mi>a</mi><mo>→</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">f,g:a\to b</annotation></semantics></math>, it is now possible to have non-trivial arrows <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>:</mo><mi>f</mi><mo>→</mo><mi>g</mi><mo>∈</mo><mi>K</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\alpha:f\to g \in K(a,b)</annotation></semantics></math>, called <em><a class="existingWikiWord" href="/nlab/show/2-morphisms">2-morphisms</a></em> or <em>2-cells</em> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> and written as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>:</mo><mi>f</mi><mo>⇒</mo><mi>g</mi></mrow><annotation encoding="application/x-tex">\alpha : f \Rightarrow g</annotation></semantics></math>. Because the hom-objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(a,b)</annotation></semantics></math> are by definition categories, 2-cells carry an associative and unital operation called <em>vertical composition</em>. The identities for this operation, of course, are the identity 2-cells <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mn>1</mn> <mi>f</mi></msub></mrow><annotation encoding="application/x-tex">1_f</annotation></semantics></math> given by the category structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(a,b)</annotation></semantics></math>.</p> <p>The functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>comp</mi></mrow><annotation encoding="application/x-tex">comp</annotation></semantics></math> gives us an operation of <em>horizontal</em> composition on 2-cells. Functoriality of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>comp</mi></mrow><annotation encoding="application/x-tex">comp</annotation></semantics></math> then says that given <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>:</mo><mi>f</mi><mo>⇒</mo><mi>g</mi><mo>:</mo><mi>a</mi><mo>→</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">\alpha : f \Rightarrow g : a\to b</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>β</mi><mo>:</mo><mi>f</mi><mo>′</mo><mo>⇒</mo><mi>g</mi><mo>′</mo><mo>:</mo><mi>b</mi><mo>→</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">\beta : f' \Rightarrow g' : b\to c</annotation></semantics></math>, the composite <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="0em" rspace="thinmathspace">comp</mo><mo stretchy="false">(</mo><mi>β</mi><mo>,</mo><mi>α</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\comp(\beta,\alpha)</annotation></semantics></math> is a 2-cell <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>β</mi><mi>α</mi><mo>:</mo><mi>f</mi><mo>′</mo><mi>f</mi><mo>⇒</mo><mi>g</mi><mo>′</mo><mi>g</mi><mo>:</mo><mi>a</mi><mo>→</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">\beta \alpha : f'f \Rightarrow g'g : a \to c</annotation></semantics></math>. Note that the boundaries of the composite 2-cell are the composites of the boundaries of the components.</p> <p>We also have the <em><a class="existingWikiWord" href="/nlab/show/interchange+law">interchange law</a></em> (also called <em>Godement law</em> or <em>middle 4 interchange law</em>): because <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>comp</mi></mrow><annotation encoding="application/x-tex">comp</annotation></semantics></math> is a functor it commutes with composition in the hom-categories, so we have (writing <a class="existingWikiWord" href="/nlab/show/vertical+composition">vertical composition</a> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∘</mo></mrow><annotation encoding="application/x-tex">\circ</annotation></semantics></math> and <a class="existingWikiWord" href="/nlab/show/horizontal+composition">horizontal composition</a> as juxtaposition):</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>β</mi><mo>′</mo><mo>∘</mo><mi>β</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>α</mi><mo>′</mo><mo>∘</mo><mi>α</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>β</mi><mo>′</mo><mi>α</mi><mo>′</mo><mo stretchy="false">)</mo><mo>∘</mo><mo stretchy="false">(</mo><mi>β</mi><mi>α</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (\beta' \circ \beta)(\alpha' \circ \alpha) = (\beta' \alpha') \circ (\beta \alpha) </annotation></semantics></math></div> <p>The <a class="existingWikiWord" href="/nlab/show/axioms">axioms</a> for <a class="existingWikiWord" href="/nlab/show/associativity">associativity</a> and <a class="existingWikiWord" href="/nlab/show/unitality">unitality</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>comp</mi></mrow><annotation encoding="application/x-tex">comp</annotation></semantics></math> ensure that <a class="existingWikiWord" href="/nlab/show/horizontal+composition">horizontal composition</a> behaves just like <a class="existingWikiWord" href="/nlab/show/composition">composition</a> of <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> in a <a class="existingWikiWord" href="/nlab/show/1-category">1-category</a>. In particular, the action of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>comp</mi></mrow><annotation encoding="application/x-tex">comp</annotation></semantics></math> on objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi></mrow><annotation encoding="application/x-tex">f,g</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/hom-categories">hom-categories</a> (i.e. 1-cells of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>) is the usual composite of morphisms.</p> <h3 id="detailedDefn">More details</h3> <p>(See also the section below on <a class="existingWikiWord" href="/nlab/show/sesquicategories">sesquicategories</a>, which provide a conceptual package for the <a class="existingWikiWord" href="/nlab/show/stuff%2C+structure%2C+property">stuff and structure</a> described below.)</p> <p>In even more detail, a strict <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> consists of <em>stuff</em>:</p> <ul> <li> <p>a collection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ob</mi><mi>K</mi></mrow><annotation encoding="application/x-tex">Ob K</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Ob</mi> <mi>K</mi></msub></mrow><annotation encoding="application/x-tex">Ob_K</annotation></semantics></math> of <em><a class="existingWikiWord" href="/nlab/show/objects">objects</a></em> or <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math>-cells</em>,</p> </li> <li> <p>for each object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math> and object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math>, a collection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(a,b)</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mi>K</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_K(a,b)</annotation></semantics></math> of <em><a class="existingWikiWord" href="/nlab/show/1-morphisms">1-morphisms</a></em> or <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math>-cells</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>→</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">a \to b</annotation></semantics></math>, and</p> </li> <li> <p>for each object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math>, object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math>, morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>a</mi><mo>→</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">f\colon a \to b</annotation></semantics></math>, and morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo lspace="verythinmathspace">:</mo><mi>a</mi><mo>→</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">g\colon a \to b</annotation></semantics></math>, a collection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(f,g)</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><msub><mi>Hom</mi> <mi>K</mi></msub><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">2 Hom_K(f,g)</annotation></semantics></math> of <em><a class="existingWikiWord" href="/nlab/show/2-morphisms">2-morphisms</a></em> or <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-cells</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>⇒</mo><mi>g</mi></mrow><annotation encoding="application/x-tex">f \Rightarrow g</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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>,</p> </li> </ul> <p>that is equipped with the following <em><a class="existingWikiWord" href="/nlab/show/structure">structure</a></em>:</p> <ul> <li> <p>for each object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math>, an <em><a class="existingWikiWord" href="/nlab/show/identity+morphism">identity</a></em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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>,</p> </li> <li> <p>for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>a</mi><mo>→</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">f\colon a \to b</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo lspace="verythinmathspace">:</mo><mi>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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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>,</p> </li> <li> <p>for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>a</mi><mo>→</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">f\colon a \to b</annotation></semantics></math>, an <em>identity</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 \Rightarrow f</annotation></semantics></math>,</p> </li> <li> <p>for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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><a class="existingWikiWord" href="/nlab/show/vertical+composition">vertical composite</a></em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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>,</p> </li> <li> <p>for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>a</mi><mo>→</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">f\colon a \to b</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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>left <a class="existingWikiWord" href="/nlab/show/whiskering">whiskering</a></em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 \triangleleft f\colon g \circ f \Rightarrow h \circ f</annotation></semantics></math>, and</p> </li> <li> <p>for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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>right <a class="existingWikiWord" href="/nlab/show/whiskering">whiskering</a></em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 \triangleright \eta \colon h \circ f \Rightarrow h \circ g</annotation></semantics></math>,</p> </li> </ul> <p>satisfying the following properties:</p> <ol> <li> <p>for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>a</mi><mo>→</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">f\colon a \to b</annotation></semantics></math>, the composites <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∘</mo><msub><mo lspace="0em" rspace="thinmathspace">id</mo> <mi>a</mi></msub></mrow><annotation encoding="application/x-tex">f \circ \id_a</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="0em" rspace="thinmathspace">id</mo> <mi>b</mi></msub><mo>∘</mo><mi>f</mi></mrow><annotation encoding="application/x-tex">\id_b \circ f</annotation></semantics></math> each equal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>,</p> </li> <li> <p>for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 composites <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><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">h \circ (g \circ f)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>h</mi><mo>∘</mo><mi>g</mi><mo stretchy="false">)</mo><mo>∘</mo><mi>f</mi></mrow><annotation encoding="application/x-tex">(h \circ g) \circ f</annotation></semantics></math> are equal,</p> </li> <li> <p>for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi></mrow><annotation encoding="application/x-tex">\eta</annotation></semantics></math>,</p> </li> <li> <p>for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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,</p> </li> <li> <p>for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 <a class="existingWikiWord" href="/nlab/show/whiskerings">whiskerings</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 \triangleleft f</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 \triangleright \Id_f</annotation></semantics></math> both equal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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>,</p> </li> <li> <p>for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 <a class="existingWikiWord" href="/nlab/show/whiskerings">whiskerings</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo>◃</mo><msub><mo lspace="0em" rspace="thinmathspace">id</mo> <mi>a</mi></msub></mrow><annotation encoding="application/x-tex">\eta \triangleleft \id_a</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="0em" rspace="thinmathspace">id</mo> <mi>b</mi></msub><mo>▹</mo><mi>η</mi></mrow><annotation encoding="application/x-tex">\id_b \triangleright \eta</annotation></semantics></math> equal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi></mrow><annotation encoding="application/x-tex">\eta</annotation></semantics></math>,</p> </li> <li> <p>for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>a</mi><mo>→</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">f\colon a \to b</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 \triangleleft f) \bullet (\eta \triangleleft f)</annotation></semantics></math> equals the <a class="existingWikiWord" href="/nlab/show/whiskering">whiskering</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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) \triangleleft f</annotation></semantics></math>,</p> </li> <li> <p>for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 \triangleright \theta) \bullet (i \triangleright \eta)</annotation></semantics></math> equals the <a class="existingWikiWord" href="/nlab/show/whiskering">whiskering</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 \triangleright (\theta \bullet \eta)</annotation></semantics></math>,</p> </li> <li> <p>for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 left <a class="existingWikiWord" href="/nlab/show/whiskerings">whiskerings</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</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">\eta \triangleleft (g \circ f)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>η</mi><mo>◃</mo><mi>g</mi><mo stretchy="false">)</mo><mo>◃</mo><mi>f</mi></mrow><annotation encoding="application/x-tex">(\eta \triangleleft g) \triangleleft f</annotation></semantics></math> are equal,</p> </li> <li> <p>for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 right <a class="existingWikiWord" href="/nlab/show/whiskerings">whiskerings</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>▹</mo><mo stretchy="false">(</mo><mi>h</mi><mo>▹</mo><mi>η</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">i \triangleright (h \triangleright \eta)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>i</mi><mo>∘</mo><mi>h</mi><mo stretchy="false">)</mo><mo>▹</mo><mi>η</mi></mrow><annotation encoding="application/x-tex">(i \circ h) \triangleright \eta</annotation></semantics></math> are equal,</p> </li> <li> <p>for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>a</mi><mo>→</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">f\colon a \to b</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 <a class="existingWikiWord" href="/nlab/show/whiskerings">whiskerings</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>▹</mo><mo stretchy="false">(</mo><mi>η</mi><mo>◃</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">i \triangleright (\eta \triangleleft f)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>i</mi><mo>▹</mo><mi>η</mi><mo stretchy="false">)</mo><mo>◃</mo><mi>f</mi></mrow><annotation encoding="application/x-tex">(i \triangleright \eta) \triangleleft f</annotation></semantics></math> are equal, and</p> </li> <li> <p>for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 \triangleright \eta) \bullet (\theta \triangleleft f)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>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 \triangleleft g) \bullet (h \triangleright \eta)</annotation></semantics></math> are equal.</p> </li> </ol> <p>The construction in the last axiom is the <em><a class="existingWikiWord" href="/nlab/show/horizontal+composition">horizontal composite</a></em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>θ</mi><mo>∘</mo><mi>η</mi><mo lspace="verythinmathspace">:</mo><mi>h</mi><mo>∘</mo><mi>f</mi><mo>→</mo><mi>i</mi><mo>∘</mo><mi>g</mi></mrow><annotation encoding="application/x-tex">\theta \circ \eta\colon h \circ f \to i \circ g</annotation></semantics></math>. It is possible (and probably more common) to take the <a class="existingWikiWord" href="/nlab/show/horizontal+composition">horizontal composite</a> as basic and the <a class="existingWikiWord" href="/nlab/show/whiskerings">whiskerings</a> as derived operations. This results in fewer, but more complicated, axioms.</p> <h2 id="AsSesquicategories">2-categories as sesquicategories</h2> <p>The fine-grained description in the <a href="#detailedDefn">previous subsection</a> can be concisely repackaged by saying that a 2-category is a <a class="existingWikiWord" href="/nlab/show/sesquicategory">sesquicategory</a> that satisfies the <a class="existingWikiWord" href="/nlab/show/interchange+axiom">interchange axiom</a>, i.e., the last axiom (12) which gives the <a class="existingWikiWord" href="/nlab/show/horizontal+composition">horizontal composition</a>-construction. This description is essentially patterned after the “five rules of functorial calculus” introduced by <a href="#Godement58">Godement (1958)</a> for the special case <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a>.</p> <p>So to say it again, but a little differently: a sesquicategory consists of a category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> (giving the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math>-cells and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math>-cells) together with a functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>K</mi> <mi>op</mi></msup><mo>×</mo><mi>K</mi><mo>⟶</mo><mi>Cat</mi></mrow><annotation encoding="application/x-tex"> K(-, -) \;\colon\; K^{op} \times K \longrightarrow Cat </annotation></semantics></math></div> <p>such that composing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(-, -)</annotation></semantics></math> with the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ob</mi><mo>:</mo><mi>Cat</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">ob: Cat \to Set</annotation></semantics></math> (the one sending a category to its set of objects) gives <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>hom</mi> <mi>K</mi></msub><mo>:</mo><msup><mi>K</mi> <mi>op</mi></msup><mo>×</mo><mi>K</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">\hom_K: K^{op} \times K \to Set</annotation></semantics></math>, the hom-functor for the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>. So: for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math>-cells <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>,</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">a, b</annotation></semantics></math>, the objects of the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(a, b)</annotation></semantics></math> are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math>-cells <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∈</mo><msub><mi>hom</mi> <mi>K</mi></msub><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f \in \hom_K(a, b)</annotation></semantics></math>. The morphisms of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(a, b)</annotation></semantics></math> are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-cells (with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math>-source <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math>-target <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math>). Composition within the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(a, b)</annotation></semantics></math> corresponds to <a class="existingWikiWord" href="/nlab/show/vertical+composition">vertical composition</a>.</p> <p>For each object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> and each morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>:</mo><mi>b</mi><mo>→</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">h: b \to c</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>, there is a functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>h</mi><mo stretchy="false">)</mo><mo>:</mo><mi>K</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo>→</mo><mi>K</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(a, h): K(a, b) \to K(a, c)</annotation></semantics></math>. This is <em>right <a class="existingWikiWord" href="/nlab/show/whiskering">whiskering</a></em>; it sends a <a class="existingWikiWord" href="/nlab/show/2-morphism">2-cell</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi></mrow><annotation encoding="application/x-tex">\eta</annotation></semantics></math> (a morphism of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(a, b)</annotation></semantics></math>) to a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>▹</mo><mi>η</mi></mrow><annotation encoding="application/x-tex">h \triangleright \eta</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(b, c)</annotation></semantics></math>. Similarly, for each object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math> and morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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>, there is a functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>:</mo><mi>K</mi><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>→</mo><mi>K</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(f, c): K(b, c) \to K(a, c)</annotation></semantics></math>. This is <em>left <a class="existingWikiWord" href="/nlab/show/whiskering">whiskering</a></em>; it sends a <a class="existingWikiWord" href="/nlab/show/2-morphism">2-cell</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi></mrow><annotation encoding="application/x-tex">\eta</annotation></semantics></math> (a morphism of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(b, c)</annotation></semantics></math>) to a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo>◃</mo><mi>f</mi></mrow><annotation encoding="application/x-tex">\eta \triangleleft f</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(a, c)</annotation></semantics></math>.</p> <p>The long list of compatibility properties enumerated in the previous subsection, all except the last, are concisely summarized in the definition of sesquicategory as recalled above. For example, property (8) just says that left <a class="existingWikiWord" href="/nlab/show/whiskering">whiskering</a> preserves <a class="existingWikiWord" href="/nlab/show/vertical+composition">vertical composition</a>, as it must since it is a <a class="existingWikiWord" href="/nlab/show/functor">functor</a> (a morphism in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cat</mi></mrow><annotation encoding="application/x-tex">Cat</annotation></semantics></math>).</p> <p>In summary, a sesquicategory consists of “stuff” and structure as described in the <a href="/nlab/show/strict+2-category#detailedDefn">previous subsection</a>, satisfying properties 1-11. A 2-category is then a sesquicategory that further satisfies the <a class="existingWikiWord" href="/nlab/show/interchange+axiom">interchange axiom</a> (12). Some further illumination of this point of view can be obtained by contemplating <a class="existingWikiWord" href="/nlab/show/string+diagrams">string diagrams</a> for 2-categories, where the <a class="existingWikiWord" href="/nlab/show/interchange+axiom">interchange axiom</a> corresponds to <a class="existingWikiWord" href="/nlab/show/isotopies">isotopies</a> of (planar, progressive) string diagrams during which the relative heights of nodes labeled by 2-cells are interchanged.</p> <h2 id="remarks">Remarks</h2> <ul> <li> <p>Strict 2-categories are the same as a <a class="existingWikiWord" href="/nlab/show/strict+omega-categories">strict omega-categories</a> which are trivial in degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">n \geq 3</annotation></semantics></math>.</p> </li> <li> <p>This is to be contrasted with a <em>weak 2-category</em> called a <a class="existingWikiWord" href="/nlab/show/bicategory">bicategory</a>. In a strict 2-category composition of 1-morphisms is strictly associative and composition with <a class="existingWikiWord" href="/nlab/show/identity+morphism">identity morphism</a>s strictly satisfies the required identity law. In a weak 2-category these laws may hold only up to coherent 2-morphisms.</p> </li> </ul> <h2 id="in_dependent_type_theory">In dependent type theory</h2> <p>In <a class="existingWikiWord" href="/nlab/show/dependent+type+theory">dependent type theory</a>, there are multiple notions of a “strict 2-category”, because there are multiple notions of a <a class="existingWikiWord" href="/nlab/show/category">category</a>:</p> <ul> <li>A strict 2-category could be a <a class="existingWikiWord" href="/nlab/show/precategory">precategory</a> enriched over the <a class="existingWikiWord" href="/nlab/show/cartesian+monoidal+category">cartesian monoidal category</a> <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a> of <a class="existingWikiWord" href="/nlab/show/strict+categories">strict categories</a></li> <li>A strict 2-category could be a <a class="existingWikiWord" href="/nlab/show/strict+category">strict category</a> enriched over the cartesian monoidal category Cat of strict categories</li> <li>A strict 2-category could be a <a class="existingWikiWord" href="/nlab/show/univalent+category">univalent category</a> enriched over the cartesian monoidal category Cat of strict categories</li> </ul> <p>The first definition is a naive translation of “strict 2-category” from set theory to dependent type theory, but in the absence of <a class="existingWikiWord" href="/nlab/show/axiom+K">axiom K</a> or a similar axiom, these strict 2-categories behave differently from the strict 2-categories as defined in <a class="existingWikiWord" href="/nlab/show/set+theory">set theory</a>. The second definition adds a <a class="existingWikiWord" href="/nlab/show/0-truncation">0-truncation</a> condition to the type of objects to ensure that the strict 2-categories actually behave like the strict 2-categories in set theory. The third definition satisfies the <a class="existingWikiWord" href="/nlab/show/principle+of+equivalence">principle of equivalence</a>: equality of objects is the same as isomorphism of strict categories, and ensures that strict 2-categories are <a class="existingWikiWord" href="/nlab/show/h-groupoids">h-groupoids</a>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/strict+2-groupoid">strict 2-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/strict+2-functor">strict 2-functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/strict+2-natural+transformation">strict 2-natural transformation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/strict+adjoint+2-functor">strict adjoint 2-functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/strict+2-group">strict 2-group</a></p> </li> </ul> <h2 id="history">History</h2> <p>As intimated above, the essential rules which abstractly govern the behavior of functors and natural transformations and their various compositions were made explicit by <a href="#Godement58">Godement (1958)</a>, in his “five rules of functorial calculus”. He did not however go as far as use these rules to define the abstract notion of 2-category; this step was taken later by <a href="#Bénabou65">Bénabou (1965)</a>. In any event, the primitive compositional operations in <a href="#Godement58">Godement (1958)</a> were what we call <em><a class="existingWikiWord" href="/nlab/show/vertical+composition">vertical composition</a></em> and <em><a class="existingWikiWord" href="/nlab/show/whiskering">whiskering</a></em>, with <a class="existingWikiWord" href="/nlab/show/horizontal+composition">horizontal composition</a> of <a class="existingWikiWord" href="/nlab/show/natural+transformations">natural transformations</a> being a derived operation (made unambiguous in the presence of the interchange axiom). Indeed, <a class="existingWikiWord" href="/nlab/show/horizontal+composition">horizontal composition</a> is often called the <em>Godement product</em>.</p> <p>A few years after introducing 2-categories, Bénabou introduced the more general notion of <em><a class="existingWikiWord" href="/nlab/show/bicategories">bicategories</a></em>.</p> <p>Literature references for the abstract notion of <a class="existingWikiWord" href="/nlab/show/sesquicategory">sesquicategory</a>, a structure in which vertical compositions and whiskerings are primitive, do not seem to be abundant, but they are mentioned for example in <a href="#Street96">Street (1996)</a> together with the observation that 2-categories are special types of sesquicategories (page 535).</p> <h2 id="References">References</h2> <p>The “five rules of functorial calculus” (<a href="#AsSesquicategories">above</a>) were formulated (as: <em>cinq règles de calcul fonctoriel</em>) in:</p> <ul> <li id="Godement58"><a class="existingWikiWord" href="/nlab/show/Roger+Godement">Roger Godement</a>, Appendice (pp. 269) of: <em>Topologie algébrique et theorie des faisceaux</em>, Actualités Sci. Ind. <strong>1252</strong>, Hermann, Paris (1958) [<a href="https://www.editions-hermann.fr/livre/topologie-algebrique-et-theorie-des-faisceaux-roger-godement">webpage</a>, <a class="existingWikiWord" href="/nlab/files/Godement-TopologieAlgebrique.pdf" title="pdf">pdf</a>]</li> </ul> <p>The definition of “<a class="existingWikiWord" href="/nlab/show/double+categories">double categories</a>” (of which, at least in hindsight, strict 2-categories are an immediate special case) is due to:</p> <ul> <li id="Ehresmann63"><a class="existingWikiWord" href="/nlab/show/Charles+Ehresmann">Charles Ehresmann</a>, <em>Catégories double et catégories structurées</em>, C.R. Acad. Paris 256 (1963) 1198-1201 [<a class="existingWikiWord" href="/nlab/files/Ehresmann-CategoriesDoubles.pdf" title="pdf">pdf</a>, <a href="https://gallica.bnf.fr/ark:/12148/bpt6k3208j/f1246">gallica</a>]</li> </ul> <p>also discussed (according to <a href="https://www.jstor.org/stable/2314770">reviewers</a> who have seen the text) in:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Charles+Ehresmann">Charles Ehresmann</a>, <em>Catégories et structures</em>, Dunod, Paris, (1965)</li> </ul> <p>However, Ehresmann did not isolate the notion of strict 2-categories as such.</p> <p>But apparently inspired by <a href="#Ehresmann63">Ehresmann (1963)</a> the actual definition of strict 2-categories is due to:</p> <ul> <li id="Bénabou65"> <p><a class="existingWikiWord" href="/nlab/show/Jean+B%C3%A9nabou">Jean Bénabou</a>, Example (5) of: <em>Catégories relatives</em>, C. R. Acad. Sci. Paris <strong>260</strong> (1965) 3824-3827 [<a href="https://gallica.bnf.fr/ark:/12148/bpt6k4019v/f37.item">gallica</a>]</p> <blockquote> <p>(conceived as <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a>-<a class="existingWikiWord" href="/nlab/show/enriched+categories">enriched categories</a> and under the name “2-categories”)</p> </blockquote> </li> <li id="Maranda65"> <p><a class="existingWikiWord" href="/nlab/show/Jean-Marie+Maranda">Jean-Marie Maranda</a>, Def. 1 in: <em>Formal categories</em>, Canadian Journal of Mathematics <strong>17</strong> (1965) 758-801 [<a href="https://doi.org/10.4153/CJM-1965-076-0">doi:10.4153/CJM-1965-076-0</a>, <a href="https://www.cambridge.org/core/services/aop-cambridge-core/content/view/A7C463460EB8CAC64C2CA340F870CF80/S0008414X00039729a.pdf/formal-categories.pdf">pdf</a>]</p> <blockquote> <p>(conceived as <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a>-<a class="existingWikiWord" href="/nlab/show/enriched+categories">enriched categories</a> and under the name “categories of the second type”)</p> </blockquote> </li> <li id="EK65"> <p><a class="existingWikiWord" href="/nlab/show/Samuel+Eilenberg">Samuel Eilenberg</a>, <a class="existingWikiWord" href="/nlab/show/G.+Max+Kelly">G. Max Kelly</a>, p. 425 of: <em>Closed Categories</em>, in: <a class="existingWikiWord" href="/nlab/show/Samuel+Eilenberg">S. Eilenberg</a>, <a class="existingWikiWord" href="/nlab/show/D.+K.+Harrison">D. K. Harrison</a>, <a class="existingWikiWord" href="/nlab/show/S.+MacLane">S. MacLane</a>, <a class="existingWikiWord" href="/nlab/show/H.+R%C3%B6hrl">H. Röhrl</a> (eds.): <em><a class="existingWikiWord" href="/nlab/show/Proceedings+of+the+Conference+on+Categorical+Algebra+-+La+Jolla+1965">Proceedings of the Conference on Categorical Algebra - La Jolla 1965</a></em>, Springer (1966) [<a href="https://doi.org/10.1007/978-3-642-99902-4">doi:10.1007/978-3-642-99902-4</a>]</p> <blockquote> <p>(expressed entirely in components and under the name <em>hypercategories</em>)</p> </blockquote> </li> </ul> <p>and the notion is invoked for various purposes (such as in speaking of <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a> as a 2-category) in:</p> <ul> <li id="Bénabou68"><a class="existingWikiWord" href="/nlab/show/Jean+B%C3%A9nabou">Jean Bénabou</a>, <em>Structures algébriques dans les catégories</em>, Cahiers de topologie et géométrie différentielle, <strong>10</strong> 1 (1968) 1-126 [<a href="http://www.numdam.org/item/?id=CTGDC_1968__10_1_1_0">doi:CTGDC_1968__10_1_1_0</a>]</li> </ul> <p>Exposition and review:</p> <ul> <li id="Street96"> <p><a class="existingWikiWord" href="/nlab/show/Ross+Street">Ross Street</a>, p. 535 in: <em>Categorical Structures</em>, in Handbook of Algebra Vol. 1 (ed. M. Hazewinkel), Elsevier Science, Amsterdam (1996) [<a href="https://doi.org/10.1016/S1570-7954(96)80019-2">doi:10.1016/S1570-7954(96)80019-2</a>, <a href="http://maths.mq.edu.au/~street/45.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Street-CategoricalStructures.pdf" title="pdf">pdf</a> <a href="https://shop.elsevier.com/books/handbook-of-algebra/hazewinkel/978-0-444-82212-3">ISBN:978-0-444-82212-3</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Saunders+MacLane">Saunders MacLane</a>, §XII.3 of: <em><a class="existingWikiWord" href="/nlab/show/Categories+for+the+Working+Mathematician">Categories for the Working Mathematician</a></em>, Graduate Texts in Mathematics <strong>5</strong> Springer (second ed. 1997) [<a href="https://link.springer.com/book/10.1007/978-1-4757-4721-8">doi:10.1007/978-1-4757-4721-8</a>]</p> </li> <li id="Lack10"> <p><a class="existingWikiWord" href="/nlab/show/Steve+Lack">Steve Lack</a>, <em>A 2-categories companion</em>, In: Baez J., May J. (eds.) <em><a class="existingWikiWord" href="/nlab/show/Towards+Higher+Categories">Towards Higher Categories</a></em>. The IMA Volumes in Mathematics and its Applications, vol 152. Springer 2010 (<a href="http://arxiv.org/abs/math.CT/0702535">arXiv:math.CT/0702535</a>, <a href="https://doi.org/10.1007/978-1-4419-1524-5_4">doi:10.1007/978-1-4419-1524-5_4</a>)</p> </li> <li id="Richter19"> <p><a class="existingWikiWord" href="/nlab/show/Birgit+Richter">Birgit Richter</a>, Section 9.5 of: <em>From categories to homotopy theory</em>, Cambridge Studies in Advanced Mathematics 188, Cambridge University Press 2020 (<a href="https://doi.org/10.1017/9781108855891">doi:10.1017/9781108855891</a>, <a href="https://www.math.uni-hamburg.de/home/richter/catbook.html">book webpage</a>, <a href="https://www.math.uni-hamburg.de/home/richter/bookdraft.pdf">pdf</a>)</p> </li> <li id="JohnsonYau20"> <p><a class="existingWikiWord" href="/nlab/show/Niles+Johnson">Niles Johnson</a>, <a class="existingWikiWord" href="/nlab/show/Donald+Yau">Donald Yau</a>, Section 2.3 of: <em>2-Dimensional Categories</em>, Oxford University Press 2021 (<a href="http://arxiv.org/abs/2002.06055">arXiv:2002.06055</a>, <a href="https://oxford.universitypressscholarship.com/view/10.1093/oso/9780198871378.001.0001/oso-9780198871378">doi:10.1093/oso/9780198871378.001.0001</a>)</p> </li> </ul> <p>See also</p> <ul> <li>Wikipedia, <em><a href="https://en.wikipedia.org/wiki/Strict_2-category">Strict 2-category</a></em></li> </ul> <p>The special case of <a class="existingWikiWord" href="/nlab/show/strict+%282%2C1%29-categories">strict (2,1)-categories</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Peter+H.+H.+Fantham">Peter H. H. Fantham</a>, <a class="existingWikiWord" href="/nlab/show/Eric+J.+Moore">Eric J. Moore</a>, <em>Groupoid Enriched Categories and Homotopy Theory</em>, Canadian Journal of Mathematics <strong>35</strong> 3 (1983) 385-416 (<a href="https://doi.org/10.4153/CJM-1983-022-8">doi:10.4153/CJM-1983-022-8</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on February 21, 2024 at 08:01:04. 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