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12 2 1.1 11 6.9 11.4 1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> horizontal composition (changes) </h1> <div class="navigation"> <span class="skipNav"><a href='#navEnd'>Skip the Navigation Links</a> | </span> <span style="display:inline-block; width: 0.3em;"></span> <a href="/nlab/show/diff/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/7898/#Item_6" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <p class="show_diff"> Showing changes from revision #11 to #12: <ins class="diffins">Added</ins> | <del class="diffdel">Removed</del> | <del class="diffmod">Chan</del><ins class="diffmod">ged</ins> </p> <div class='rightHandSide'> <div class='toc clickDown' tabindex='0'> <h3 id='context'>Context</h3> <h4 id='2category_theory'>2-Category theory</h4> <div class='hide'> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/2-category+theory'>2-category theory</a></strong></p> <p><strong>Definitions</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/2-category'>2-category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/strict+2-category'>strict 2-category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/bicategory'>bicategory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/enriched+bicategory'>enriched bicategory</a></p> </li> </ul> <p><strong>Transfors between 2-categories</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/2-functor'>2-functor</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/pseudofunctor'>pseudofunctor</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/lax+functor'>lax functor</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/equivalence+of+2-categories'>equivalence of 2-categories</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/strict+2-natural+transformation'>2-natural transformation</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/lax+natural+transformation'>lax natural transformation</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/icon'>icon</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/modification'>modification</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Yoneda+lemma+for+bicategories'>Yoneda lemma for bicategories</a></p> </li> </ul> <p><strong>Morphisms in 2-categories</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fully+faithful+morphism'>fully faithful morphism</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/faithful+morphism'>faithful morphism</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/conservative+morphism'>conservative morphism</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/pseudomonic+morphism'>pseudomonic morphism</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/discrete+morphism'>discrete morphism</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/eso+morphism'>eso morphism</a></p> </li> </ul> <p><strong>Structures in 2-categories</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/adjunction'>adjunction</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/mate'>mate</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/monad'>monad</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cartesian+object'>cartesian object</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fibration+in+a+2-category'>fibration in a 2-category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/codiscrete+cofibration'>codiscrete cofibration</a></p> </li> </ul> <p><strong>Limits in 2-categories</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/2-limit'>2-limit</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/2-pullback'>2-pullback</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/comma+object'>comma object</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/inserter'>inserter</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/inverter'>inverter</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/equifier'>equifier</a></p> </li> </ul> <p><strong>Structures on 2-categories</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/2-monad'>2-monad</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/lax-idempotent+2-monad'>lax-idempotent 2-monad</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/2-monad'>pseudomonad</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/pseudoalgebra+for+a+2-monad'>pseudoalgebra for a 2-monad</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/monoidal+bicategory'>monoidal 2-category</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/cartesian+bicategory'>cartesian bicategory</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Gray+tensor+product'>Gray tensor product</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/2-category+equipped+with+proarrows'>proarrow equipment</a></p> </li> </ul> </div> <h4 id='higher_category_theory'>Higher category theory</h4> <div class='hide'> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/higher+category+theory'>higher category theory</a></strong></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/category+theory'>category theory</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/homotopy+theory'>homotopy theory</a></li> </ul> <h2 id='basic_concepts'>Basic concepts</h2> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/k-morphism'>k-morphism</a>, <a class='existingWikiWord' href='/nlab/show/diff/coherence+law'>coherence</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/looping'>looping and delooping</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/stabilization'>looping and suspension</a></li> </ul> <h2 id='basic_theorems'>Basic theorems</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+hypothesis'>homotopy hypothesis</a>-theorem</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/delooping+hypothesis'>delooping hypothesis</a>-theorem</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/periodic+table'>periodic table</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/stabilization+hypothesis'>stabilization hypothesis</a>-theorem</p> </li> <li> <p><a class='existingWikiWord' href='/michaelshulman/show/diff/exactness+hypothesis' title='michaelshulman'>exactness hypothesis</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/holographic+principle+of+higher+category+theory'>holographic principle</a></p> </li> </ul> <h2 id='applications'>Applications</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/applications+of+%28higher%29+category+theory'>applications of (higher) category theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/higher+category+theory+and+physics'>higher category theory and physics</a></p> </li> </ul> <h2 id='models'>Models</h2> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/%28n%2Cr%29-category'>(n,r)-category</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Theta-space'>Theta-space</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/infinity-category'>∞-category</a>/<a class='existingWikiWord' href='/nlab/show/diff/infinity-category'>∞-category</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2Cn%29-category'>(∞,n)-category</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/n-fold+complete+Segal+space'>n-fold complete Segal space</a></li> </ul> </li> <li><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C2%29-category'>(∞,2)-category</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category'>(∞,1)-category</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/quasi-category'>quasi-category</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/algebraic+quasi-category'>algebraic quasi-category</a></li> </ul> </li> <li><a class='existingWikiWord' href='/nlab/show/diff/simplicially+enriched+category'>simplicially enriched category</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/complete+Segal+space'>complete Segal space</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/model+category'>model category</a></li> </ul> </li> <li><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C0%29-category'>(∞,0)-category</a>/<a class='existingWikiWord' href='/nlab/show/diff/infinity-groupoid'>∞-groupoid</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Kan+complex'>Kan complex</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/algebraic+Kan+complex'>algebraic Kan complex</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/simplicial+T-complex'>simplicial T-complex</a></li> </ul> </li> </ul> </li> <li><a class='existingWikiWord' href='/nlab/show/diff/%28%E2%88%9E%2CZ%29-category'>(∞,Z)-category</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/n-category'>n-category</a> = (n,n)-category <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/2-category'>2-category</a>, <a class='existingWikiWord' href='/nlab/show/diff/%282%2C1%29-category'>(2,1)-category</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/1-category'>1-category</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/0-category'>0-category</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/%28-1%29-category'>(-1)-category</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/%28-2%29-category'>(-2)-category</a></li> </ul> </li> <li><a class='existingWikiWord' href='/nlab/show/diff/n-poset'>n-poset</a> = <a class='existingWikiWord' href='/nlab/show/diff/n-poset'>(n-1,n)-category</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/partial+order'>poset</a> = <a class='existingWikiWord' href='/nlab/show/diff/%280%2C1%29-category'>(0,1)-category</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/2-poset'>2-poset</a> = <a class='existingWikiWord' href='/nlab/show/diff/2-poset'>(1,2)-category</a></li> </ul> </li> <li><a class='existingWikiWord' href='/nlab/show/diff/n-groupoid'>n-groupoid</a> = (n,0)-category <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/2-groupoid'>2-groupoid</a>, <a class='existingWikiWord' href='/nlab/show/diff/3-groupoid'>3-groupoid</a></li> </ul> </li> </ul> </li> <li><a class='existingWikiWord' href='/nlab/show/diff/vertical+categorification'>categorification</a>/<a class='existingWikiWord' href='/nlab/show/diff/decategorification'>decategorification</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/geometric+definition+of+higher+categories'>geometric definition of higher category</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Kan+complex'>Kan complex</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/quasi-category'>quasi-category</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/simplicial+model+for+weak+omega-categories'>simplicial model for weak ∞-categories</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/complicial+set'>complicial set</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/weak+complicial+set'>weak complicial set</a></li> </ul> </li> </ul> </li> <li><a class='existingWikiWord' href='/nlab/show/diff/algebraic+definition+of+higher+categories'>algebraic definition of higher category</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/bicategory'>bicategory</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/bigroupoid'>bigroupoid</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/tricategory'>tricategory</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/tetracategory'>tetracategory</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/strict+omega-category'>strict ∞-category</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/Batanin+omega-category'>Batanin ∞-category</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/Trimble+n-category'>Trimble ∞-category</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/Grothendieck-Maltsiniotis+infinity-category'>Grothendieck-Maltsiniotis ∞-categories</a></li> </ul> </li> <li><a class='existingWikiWord' href='/nlab/show/diff/stable+homotopy+theory'>stable homotopy theory</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/symmetric+monoidal+category'>symmetric monoidal category</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/symmetric+monoidal+%28infinity%2C1%29-category'>symmetric monoidal (∞,1)-category</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/stable+%28infinity%2C1%29-category'>stable (∞,1)-category</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/dg-category'>dg-category</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/A-infinity-category'>A-∞ category</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/triangulated+category'>triangulated category</a></li> </ul> </li> </ul> </li> </ul> <h2 id='morphisms'>Morphisms</h2> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/k-morphism'>k-morphism</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/2-morphism'>2-morphism</a></li> </ul> </li> <li><a class='existingWikiWord' href='/nlab/show/diff/transfor'>transfor</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/natural+transformation'>natural transformation</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/modification'>modification</a></li> </ul> </li> </ul> <h2 id='functors'>Functors</h2> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/functor'>functor</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/2-functor'>2-functor</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/pseudofunctor'>pseudofunctor</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/lax+functor'>lax functor</a></li> </ul> </li> <li><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-functor'>(∞,1)-functor</a></li> </ul> <h2 id='universal_constructions'>Universal constructions</h2> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/2-limit'>2-limit</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/adjoint+%28infinity%2C1%29-functor'>(∞,1)-adjunction</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-Kan+extension'>(∞,1)-Kan extension</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/%28%E2%88%9E%2C1%29-limit'>(∞,1)-limit</a></li> </ul> </li> <li><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-Grothendieck+construction'>(∞,1)-Grothendieck construction</a></li> </ul> <h2 id='extra_properties_and_structure'>Extra properties and structure</h2> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/cosmic+cube'>cosmic cube</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/k-tuply+monoidal+n-category'>k-tuply monoidal n-category</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/strict+omega-category'>strict ∞-category</a>, <a class='existingWikiWord' href='/nlab/show/diff/strict+omega-groupoid'>strict ∞-groupoid</a></li> </ul> </li> <li><a class='existingWikiWord' href='/nlab/show/diff/stable+%28infinity%2C1%29-category'>stable (∞,1)-category</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-topos'>(∞,1)-topos</a></li> </ul> <h2 id='1categorical_presentations'>1-categorical presentations</h2> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/homotopical+category'>homotopical category</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/model+category'>model category theory</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/enriched+category+theory'>enriched category theory</a></li> </ul> </div> </div> </div> <h1 id='contents'>Contents</h1> <div class='maruku_toc'><ul><li><a href='#definition'>Definition</a></li><li><a href='#properties'>Properties</a></li><li><a href='#examples'>Examples</a></li><li><a href='#related_topics'>Related topics</a></li><ins class='diffins'><li><a href='#references'>References</a></li></ins></ul></div> <h2 id='definition'>Definition</h2> <p>In a <a class='existingWikiWord' href='/nlab/show/diff/2-category'>2-category</a>, the composition of <a class='existingWikiWord' href='/nlab/show/diff/2-morphism'><span><del class='diffmod'> 2-morphism</del><ins class='diffmod'> 2-morphisms</ins></span></a><span><del class='diffmod'> s</del><ins class='diffmod'> </ins> along</span><a class='existingWikiWord' href='/nlab/show/diff/object'>object</a>s is called <em>horizontal composition</em> .</p> <div style='text-align: center;'><div style='text-align: center;'><div> <svg height='84' se:nonce='57183' xmlns:se='http://svg-edit.googlecode.com' width='340' xmlns='http://www.w3.org/2000/svg'> <g> <title>Layer 1</title> <foreignObject font-size='16' height='22' id='svg_57183_1' width='1em' x='5' y='33'> <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'> <semantics> <mrow> <mi>A</mi> </mrow> <annotation encoding='application/x-tex'>A</annotation> </semantics> </math> </foreignObject> <foreignObject font-size='16' height='22' id='svg_57183_1' width='2em' x='45' y='33'> <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'> <semantics> <mrow> <mo lspace='verythinmathspace' rspace='verythinmathspace'>⇓</mo> <mi>α</mi> </mrow> <annotation encoding='application/x-tex'>\Downarrow\mathrlap{\alpha}</annotation> </semantics> </math> </foreignObject> <foreignObject font-size='16' height='22' id='svg_57183_1' width='1em' x='92' y='33'> <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'> <semantics> <mrow> <mi>B</mi> </mrow> <annotation encoding='application/x-tex'>B</annotation> </semantics> </math> </foreignObject> <foreignObject font-size='16' height='22' id='svg_57183_1' width='2em' x='130' y='33'> <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'> <semantics> <mrow> <mo lspace='verythinmathspace' rspace='verythinmathspace'>⇓</mo> <mi>β</mi> </mrow> <annotation encoding='application/x-tex'>\Downarrow\mathrlap{\beta}</annotation> </semantics> </math> </foreignObject> <foreignObject font-size='16' height='22' id='svg_57183_1' width='1em' x='180' y='33'> <math display='inline' 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rspace='verythinmathspace'>○</mo> <mi>α</mi> </mrow> </mrow> <annotation encoding='application/x-tex'>\Downarrow\beta\circ\alpha</annotation> </semantics> </math> </foreignObject> <foreignObject font-size='16' height='22' id='svg_57183_1' width='1em' x='327' y='33'> <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'> <semantics> <mrow> <mi>C</mi> </mrow> <annotation encoding='application/x-tex'>C</annotation> </semantics> </math> </foreignObject> <path d='m19.5,33.75c22.66666,-15 49.069603,-15 68,0' fill='none' id='svg_57183_2' marker-end='url(#se_marker_end_svg_57183_2)' stroke='#000000' /> <path d='m19.5,52.5c22.66666,14.333344 42.133331,15.666656 68,1' fill='none' id='svg_57183_3' marker-end='url(#se_marker_end_svg_57183_3)' stroke='#000000' /> <path d='m102.5,33.75c24.666656,-15 53.399261,-15 74,0' fill='none' id='svg_57183_4' marker-end='url(#se_marker_end_svg_57183_4)' stroke='#000000' /> <path d='m239.5,33.75c27.999969,-15 60.615387,-15 84,0' fill='none' 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</math> </foreignObject> <foreignObject font-size='16' height='22' id='svg_57183_17' width='20' x='45' y='63'> <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'> <semantics> <mrow> <msub> <mi>G</mi> <mn>1</mn> </msub> </mrow> <annotation encoding='application/x-tex'>G_1</annotation> </semantics> </math> </foreignObject> <foreignObject font-size='16' height='22' id='svg_57183_25' width='20' x='133' y='63'> <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'> <semantics> <mrow> <msub> <mi>G</mi> <mn>2</mn> </msub> </mrow> <annotation encoding='application/x-tex'>G_2</annotation> </semantics> </math> </foreignObject> <foreignObject font-size='16' height='22' id='svg_57183_33' width='5em' x='250' y='0'> <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'> <semantics> <mrow> <msub> <mi>F</mi> <mn>2</mn> </msub> <mo lspace='verythinmathspace' rspace='verythinmathspace'>○</mo> <msub> <mi>F</mi> <mn>1</mn> </msub> </mrow> <annotation encoding='application/x-tex'>F_2\circ F_1</annotation> </semantics> </math> </foreignObject> <foreignObject font-size='16' height='22' id='svg_57183_34' width='5em' x='250' y='63'> <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'> <semantics> <mrow> <msub> <mi>G</mi> <mn>2</mn> </msub> <mo lspace='verythinmathspace' rspace='verythinmathspace'>○</mo> <msub> <mi>G</mi> <mn>1</mn> </msub> </mrow> <annotation encoding='application/x-tex'>G_2\circ G_1</annotation> </semantics> </math> </foreignObject> </g> <defs> <marker id='se_marker_end_svg_57183_2' markerHeight='5' markerUnits='strokeWidth' markerWidth='5' orient='auto' refX='50' refY='50' viewBox='0 0 100 100'> <path d='m100,50l-100,40l30,-40l-30,-40l100,40z' fill='#000000' id='svg_57183_10' stroke='#000000' stroke-width='10' /> </marker> <marker id='se_marker_end_svg_57183_3' markerHeight='5' markerUnits='strokeWidth' markerWidth='5' orient='auto' refX='50' refY='50' viewBox='0 0 100 100'> <path d='m100,50l-100,40l30,-40l-30,-40l100,40z' fill='#000000' id='svg_57183_11' stroke='#000000' stroke-width='10' /> </marker> <marker id='se_marker_end_svg_57183_4' markerHeight='5' markerUnits='strokeWidth' markerWidth='5' orient='auto' refX='50' refY='50' viewBox='0 0 100 100'> <path d='m100,50l-100,40l30,-40l-30,-40l100,40z' fill='#000000' id='svg_57183_12' stroke='#000000' stroke-width='10' /> </marker> <marker id='se_marker_end_svg_57183_5' markerHeight='5' markerUnits='strokeWidth' markerWidth='5' orient='auto' refX='50' refY='50' viewBox='0 0 100 100'> <path d='m100,50l-100,40l30,-40l-30,-40l100,40z' fill='#000000' id='svg_57183_13' stroke='#000000' stroke-width='10' /> </marker> <marker id='se_marker_end_svg_57183_6' markerHeight='5' markerUnits='strokeWidth' markerWidth='5' orient='auto' refX='50' refY='50' viewBox='0 0 100 100'> <path d='m100,50l-100,40l30,-40l-30,-40l100,40z' fill='#000000' id='svg_57183_14' stroke='#000000' stroke-width='10' /> </marker> <marker id='se_marker_end_svg_57183_7' markerHeight='5' markerUnits='strokeWidth' markerWidth='5' orient='auto' refX='50' refY='50' viewBox='0 0 100 100'> <path d='m100,50l-100,40l30,-40l-30,-40l100,40z' fill='#000000' id='svg_57183_15' stroke='#000000' stroke-width='10' /> </marker> </defs> </svg></div></div></div> <p>This is in contrast to the <a class='existingWikiWord' href='/nlab/show/diff/vertical+composition'>vertical composition</a> of 2-morphisms, which is their composition along 1-<a class='existingWikiWord' href='/nlab/show/diff/morphism'>morphism</a>s.</p> <h2 id='properties'>Properties</h2> <p>Horizontal and vertical composition are subject to the compatibility condition called the <a class='existingWikiWord' href='/nlab/show/diff/exchange+law'>interchange law</a>, which in this case and the higher categorical ones means that if we fix three objects <math class='maruku-mathml' display='inline' id='mathml_1ce79634527f4f5ebf99cbbf03c346e8f17c4e57_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math>, <math class='maruku-mathml' display='inline' id='mathml_1ce79634527f4f5ebf99cbbf03c346e8f17c4e57_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math>, and <math class='maruku-mathml' display='inline' id='mathml_1ce79634527f4f5ebf99cbbf03c346e8f17c4e57_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> then the composition of 1-cells between them defines a functor.</p> <h2 id='examples'>Examples</h2> <ul> <li>In <a class='existingWikiWord' href='/nlab/show/diff/Cat'>Cat</a>, horizontal composition is the <a class='existingWikiWord' href='/nlab/show/diff/Godement+product'>Godement product</a> of <a class='existingWikiWord' href='/nlab/show/diff/natural+transformation'>natural transformation</a>s.</li> </ul> <h2 id='related_topics'>Related topics</h2> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/whiskering'>whiskering</a></li> </ul><ins class='diffins'> </ins><ins class='diffins'><h2 id='references'>References</h2></ins><ins class='diffins'> </ins><ins class='diffins'><p>Horizontal composition of <a class='existingWikiWord' href='/nlab/show/diff/natural+transformation'>natural transformations</a> is first describes in</p></ins><ins class='diffins'> </ins><ins class='diffins'><ul> <li id='Godement58'><a class='existingWikiWord' href='/nlab/show/diff/Roger+Godement'>Roger Godement</a>, Appendix (pp. 269) of: <em>Topologie algébrique et theorie des faisceaux</em>, Actualités Sci. Ind. <strong>1252</strong>, Hermann, Paris (1958) [[webpage](https://www.editions-hermann.fr/livre/topologie-algebrique-et-theorie-des-faisceaux-roger-godement), <a class='existingWikiWord' href='/nlab/files/Godement-TopologieAlgebrique.pdf' title='pdf'>pdf</a>]</li> </ul></ins><ins class='diffins'> </ins><ins class='diffins'><p>for textbook accounts see most of those listed at <em><a class='existingWikiWord' href='/nlab/show/diff/category'>category</a></em>, such as</p></ins><ins class='diffins'> </ins><ins class='diffins'><ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Saunders+Mac+Lane'>Saunders MacLane</a>, §II.5, p. 43 of: <em><a class='existingWikiWord' href='/nlab/show/diff/Categories+for+the+Working+Mathematician'>Categories for the Working Mathematician</a></em>, Graduate Texts in Mathematics <strong>5</strong> Springer (1971, second ed. 1997) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]</li> </ul></ins> </div> <div class="revisedby"> <p> Last revised on August 18, 2023 at 15:20:12. 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