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1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> semicategory </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/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/4541/#Item_10" 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"> <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>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="category_theory">Category theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></strong></p> <h2 id="sidebar_concepts">Concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category">category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cat">Cat</a></p> </li> </ul> <h2 id="sidebar_universal_constructions">Universal constructions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+construction">universal construction</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/representable+functor">representable functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+functor">adjoint functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/limit">limit</a>/<a class="existingWikiWord" href="/nlab/show/colimit">colimit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/weighted+limit">weighted limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/end">end</a>/<a class="existingWikiWord" href="/nlab/show/coend">coend</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kan+extension">Kan extension</a></p> </li> </ul> </li> </ul> <h2 id="sidebar_theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+construction">Grothendieck construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+functor+theorem">adjoint functor theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monadicity+theorem">monadicity theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+lifting+theorem">adjoint lifting theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gabriel-Ulmer+duality">Gabriel-Ulmer duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freyd-Mitchell+embedding+theorem">Freyd-Mitchell embedding theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relation+between+type+theory+and+category+theory">relation between type theory and category theory</a></p> </li> </ul> <h2 id="sidebar_extensions">Extensions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf+and+topos+theory">sheaf and topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a></p> </li> </ul> <h2 id="sidebar_applications">Applications</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/applications+of+%28higher%29+category+theory">applications of (higher) category theory</a></li> </ul> <div> <p> <a href="/nlab/edit/category+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="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="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#RelationToCategories'>Relation to categories</a></li> <li><a href='#transitive_relations'>Transitive relations</a></li> <li><a href='#nerves_and_semisimplicial_sets'>Nerves and semi-simplicial sets</a></li> </ul> <li><a href='#examples'>Examples</a></li> <li><a href='#in_higher_category_theory'>In higher category theory</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>The notion of <em>semicategory</em> or <em>non-unital category</em> is like that of <em><a class="existingWikiWord" href="/nlab/show/category">category</a></em> but omitting the requirement of <a class="existingWikiWord" href="/nlab/show/identity">identity</a>-morphisms.</p> <p>This generalizes the notions of <a class="existingWikiWord" href="/nlab/show/semigroup">semigroup</a>, <a class="existingWikiWord" href="/nlab/show/semiring">semiring</a>, etc:</p> <ul> <li> <p>a <a class="existingWikiWord" href="/nlab/show/semigroup">semigroup</a> is (the <a class="existingWikiWord" href="/nlab/show/hom-set">hom-set</a> of) a semicategory with a single object;</p> </li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/semiring">semiring</a> is (the <a class="existingWikiWord" href="/nlab/show/hom-set">hom-set</a> of) a semicategory <a class="existingWikiWord" href="/nlab/show/enriched+category">enriched</a> in <a class="existingWikiWord" href="/nlab/show/Ab">Ab</a> with a single object.</p> </li> </ul> <p>Semicategories, like categories, appear as <a class="existingWikiWord" href="/nlab/show/semipresheaf">semipresheaves</a> on the category with two objects and two morphisms.</p> <h2 id="definition">Definition</h2> <div class="num_defn" id="SemiCategory"> <h6 id="definition_2">Definition</h6> <p>A (small) <strong>semicategory</strong> or <strong>non-unital category</strong> <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">\mathcal{C}</annotation></semantics></math> consists of</p> <ul> <li> <p>a <a class="existingWikiWord" href="/nlab/show/set">set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\mathcal{C}_0</annotation></semantics></math> of <em><a class="existingWikiWord" href="/nlab/show/objects">objects</a></em>;</p> </li> <li> <p>a set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\mathcal{C}_1</annotation></semantics></math> of <em><a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a></em> (or <em>arrows</em>);</p> </li> <li> <p>two <a class="existingWikiWord" href="/nlab/show/functions">functions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo>,</mo><mi>t</mi><mo>:</mo><msub><mi>𝒞</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>𝒞</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">s, t : \mathcal{C}_1 \to \mathcal{C}_0</annotation></semantics></math> called <em>source</em> (or <em>domain</em>) and <em>target</em> (or <em>codomain</em>);</p> <ul> <li>one writes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>x</mi><mo>→</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">f : x \to y</annotation></semantics></math> if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>=</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">s(f) = x</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>=</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">t(f) = y</annotation></semantics></math>;</li> </ul> </li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/function">function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∘</mo><mo lspace="verythinmathspace">:</mo><msub><mi>𝒞</mi> <mn>1</mn></msub><msub><mo>×</mo> <mrow><mi>t</mi><mo>,</mo><mi>s</mi></mrow></msub><msub><mi>𝒞</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>𝒞</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\circ \colon \mathcal{C}_1 \times_{t,s} \mathcal{C}_1 \to \mathcal{C}_1</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/composition">composition</a>) from the set of pairs of morphisms such that the target of the first is the source of the second;</p> </li> </ul> <p>such that the following properties are satisfied:</p> <ul> <li> <p>source and target are respected by composition: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo stretchy="false">(</mo><mi>g</mi><mo>∘</mo><mi>f</mi><mo stretchy="false">)</mo><mo>=</mo><mi>s</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">s(g \circ f) = s(f)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi><mo stretchy="false">(</mo><mi>g</mi><mo>∘</mo><mi>f</mi><mo stretchy="false">)</mo><mo>=</mo><mi>t</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">t(g\circ f) = t(g)</annotation></semantics></math>;</p> </li> <li> <p>composition is <em><a class="existingWikiWord" href="/nlab/show/associativity">associative</a></em>: <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><mo>=</mo><mi>h</mi><mo>∘</mo><mo stretchy="false">(</mo><mi>g</mi><mo>∘</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(h \circ g)\circ f = h\circ (g \circ f)</annotation></semantics></math> whenever <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>=</mo><mi>s</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">t(f) = s(g)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>=</mo><mi>s</mi><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">t(g) = s(h)</annotation></semantics></math>.</p> </li> </ul> </div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>If one added to this definition the existence of a function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo lspace="verythinmathspace">:</mo><msub><mi>C</mi> <mn>0</mn></msub><mo>→</mo><msub><mi>C</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">i \colon C_0 \to C_1</annotation></semantics></math> such that for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><msub><mi>C</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">c \in C_0</annotation></semantics></math> the morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">i(c)</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/identity">identity</a> on <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> under the given <a class="existingWikiWord" href="/nlab/show/composition">composition</a>, then one has the defintion of a <a class="existingWikiWord" href="/nlab/show/category">category</a>.</p> <p>However, having <a class="existingWikiWord" href="/nlab/show/identities">identities</a> is just an extra <a class="existingWikiWord" href="/nlab/show/property">property</a> on a semi-category, not extra <a class="existingWikiWord" href="/nlab/show/structure">structure</a>. For more on this see below at <em><a href="#RelationToCategories">Relation to categories</a></em>.</p> </div> <div class="num_remark"> <h6 id="remark_2">Remark</h6> <p>One often writes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>hom</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">hom(x,y)</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>hom</mi> <mi>C</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">hom_C(x,y)</annotation></semantics></math>, or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C(x,y)</annotation></semantics></math> for the collection of morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>x</mi><mo>→</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">f : x \to y</annotation></semantics></math>; the latter two have the advantage of making clear which category is being discussed. People also often write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">x \in C</annotation></semantics></math> instead of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><msub><mi>C</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">x \in C_0</annotation></semantics></math> as a short way to indicate that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> is an object of <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>. Also, some people write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ob</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ob(C)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Mor</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Mor(C)</annotation></semantics></math> instead of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">C_0</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">C_1</annotation></semantics></math>.</p> </div> <div class="num_defn" id="SemiFunctor"> <h6 id="definition_3">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo>,</mo><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}, \mathcal{D}</annotation></semantics></math> two semicategories, a <a class="existingWikiWord" href="/nlab/show/semi-functor">semi-functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo lspace="verythinmathspace">:</mo><mi>𝒞</mi><mo>→</mo><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">F \colon \mathcal{C} \to \mathcal{D}</annotation></semantics></math> is a pair of <a class="existingWikiWord" href="/nlab/show/functions">functions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mn>0</mn></msub><mo lspace="verythinmathspace">:</mo><msub><mi>𝒞</mi> <mn>0</mn></msub><mo>→</mo><msub><mi>𝒟</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">F_0 \colon \mathcal{C}_0 \to \mathcal{D}_0</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mn>1</mn></msub><mo lspace="verythinmathspace">:</mo><msub><mi>𝒞</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>𝒟</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">F_1 \colon \mathcal{C}_1 \to \mathcal{D}_1</annotation></semantics></math> that respects all the given <a class="existingWikiWord" href="/nlab/show/structure">structure</a> in the obvious way.</p> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SemiCat</mi></mrow><annotation encoding="application/x-tex">SemiCat</annotation></semantics></math> for the (<a class="existingWikiWord" href="/nlab/show/large+category">large</a>) <a class="existingWikiWord" href="/nlab/show/category">category</a> whose objects are semicategories, and whose morphisms are semifunctors.</p> </div> <h2 id="properties">Properties</h2> <h3 id="RelationToCategories">Relation to categories</h3> <p>We discuss the relation of semicategories to <a class="existingWikiWord" href="/nlab/show/categories">categories</a>. (See for instance the beginning of (<a href="#Harpaz">Harpaz</a>) for a quick review of basics, with an eye towards their generalization to the relation between <a class="existingWikiWord" href="/nlab/show/complete+Segal+spaces">complete Segal spaces</a> and <a class="existingWikiWord" href="/nlab/show/complete+semi-Segal+spaces">complete semi-Segal spaces</a>.)</p> <div class="num_defn" id="ForgetfulFromCatToSemicat"> <h6 id="definition_4">Definition</h6> <p>There is an evident <a class="existingWikiWord" href="/nlab/show/forgetful+functor">forgetful functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo lspace="verythinmathspace">:</mo><mi>Cat</mi><mo>→</mo><mi>SemiCat</mi></mrow><annotation encoding="application/x-tex"> U \colon Cat \to SemiCat </annotation></semantics></math></div> <p>from the <a class="existingWikiWord" href="/nlab/show/category">category</a> <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a> of <a class="existingWikiWord" href="/nlab/show/categories">categories</a> to that of semicategories, def. <a class="maruku-ref" href="#SemiFunctor"></a>, given simply by forgetting the identity-assigning map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo lspace="verythinmathspace">:</mo><msub><mi>𝒞</mi> <mn>0</mn></msub><mo>→</mo><msub><mi>𝒞</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">i \colon \mathcal{C}_0 \to \mathcal{C}_1</annotation></semantics></math> in a category.</p> </div> <div class="num_defn" id="SetNeutralElementsInEndoSemiMonoids"> <h6 id="definition_5">Definition</h6> <p>For <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">\mathcal{C}</annotation></semantics></math> a semi-category, def. <a class="maruku-ref" href="#SemiCategory"></a>, write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Id</mi><mo stretchy="false">(</mo><msub><mi>𝒞</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>↪</mo><msub><mi>𝒞</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex"> Id(\mathcal{C}_1) \hookrightarrow \mathcal{C}_1 </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/subset">subset</a> on those morphisms which are <a class="existingWikiWord" href="/nlab/show/endomorphisms">endomorphisms</a> on some object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><msub><mi>𝒞</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">x \in \mathcal{C}_0</annotation></semantics></math> and such that they are neutral elements with respect to composition in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>.</p> </div> <div class="num_prop" id="CategoriesAreSemicategoriesWithUnits"> <h6 id="proposition">Proposition</h6> <p>A semicategory is the semicategory underlying a category, hence is in the image of the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> of def. <a class="maruku-ref" href="#ForgetfulFromCatToSemicat"></a>, precisely if every object has a neutral <a class="existingWikiWord" href="/nlab/show/endomorphism">endomorphism</a>, hence precisely if the composite diagonal function in</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>Id</mi><mo stretchy="false">(</mo><msub><mi>𝒞</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mtd> <mtd><mo>↪</mo></mtd> <mtd><msub><mi>𝒞</mi> <mn>1</mn></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mo>≃</mo></mpadded></msub><mo>↘</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>s</mi></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msub><mi>𝒞</mi> <mn>0</mn></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ Id(\mathcal{C}_1) &\hookrightarrow& \mathcal{C}_1 \\ & {}_{\mathllap{\simeq}}\searrow & \downarrow^{\mathrlap{s}} \\ && \mathcal{C}_0 } </annotation></semantics></math></div> <p>is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>, where the horizontal function is that of def. <a class="maruku-ref" href="#SetNeutralElementsInEndoSemiMonoids"></a>.</p> <p>Moreover, if a semicategory lifts to a category, it does so in a unique way: the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo lspace="verythinmathspace">:</mo><mi>Cat</mi><mo>→</mo><mi>SemiCat</mi></mrow><annotation encoding="application/x-tex">U \colon Cat \to SemiCat</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/injection">injection</a> on <a class="existingWikiWord" href="/nlab/show/isomorphism+classes">isomorphism classes</a>.</p> </div> <div class="num_remark"> <h6 id="remark_3">Remark</h6> <p>Equivalently one could use the target map instead of the source map in the formulation of prop. <a class="maruku-ref" href="#CategoriesAreSemicategoriesWithUnits"></a>.</p> </div> <div class="num_remark"> <h6 id="remark_4">Remark</h6> <p>The diagram appearing in prop. <a class="maruku-ref" href="#CategoriesAreSemicategoriesWithUnits"></a> is a simple version of the <a class="existingWikiWord" href="/nlab/show/univalence">univalence</a> condition appearing in definition of a <a class="existingWikiWord" href="/nlab/show/complete+semi-Segal+space">complete semi-Segal space</a>, a <a class="existingWikiWord" href="/nlab/show/category+object+in+an+%28infinity%2C1%29-category">semi-category object in an (infinity,1)-category</a>. See there for more on this.</p> </div> <div class="num_prop" id="idempotents"> <h6 id="proposition_2">Proposition</h6> <p>The functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> of def. <a class="maruku-ref" href="#ForgetfulFromCatToSemicat"></a> has a <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a>, which <a class="existingWikiWord" href="/nlab/show/free+functor">freely</a> adjoins identity morphisms to a semicategory in the obvious way. It also has a <a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjoint</a>, which sends a semicategory <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> to the category whose objects are the <a class="existingWikiWord" href="/nlab/show/idempotents">idempotents</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> and whose morphisms are the morphisms of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> that commute suitably with them, as described at <a class="existingWikiWord" href="/nlab/show/Karoubi+envelope">Karoubi envelope</a>. Indeed, the <a class="existingWikiWord" href="/nlab/show/monad">monad</a> on <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a> generated by this latter <a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a> is exactly the monad for <em><a class="existingWikiWord" href="/nlab/show/idempotent+completion">idempotent completion</a></em>, also called <a class="existingWikiWord" href="/nlab/show/Cauchy+completion">Cauchy completion</a>. (Note, however, that this is not a <a class="existingWikiWord" href="/nlab/show/2-monad">2-monad</a>, because the right adjoint of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> is not a <a class="existingWikiWord" href="/nlab/show/2-functor">2-functor</a>.)</p> </div> <h3 id="transitive_relations">Transitive relations</h3> <p>A <a class="existingWikiWord" href="/nlab/show/transitive+relation">transitive relation</a> is a semicategory <a class="existingWikiWord" href="/nlab/show/enriched+category">enriched</a> on <a class="existingWikiWord" href="/nlab/show/truth+values">truth values</a>, or a semicategory <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> where there is at most one morphism from every 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> to 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> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math></p> <h3 id="nerves_and_semisimplicial_sets">Nerves and semi-simplicial sets</h3> <p>The <a class="existingWikiWord" href="/nlab/show/nerve">nerve</a> of a semicategory is a <a class="existingWikiWord" href="/nlab/show/semi-simplicial+set">semi-simplicial set</a> which satisfies the <a class="existingWikiWord" href="/nlab/show/Segal+conditions">Segal conditions</a>.</p> <h2 id="examples">Examples</h2> <p>Start with the category of metric spaces and short maps. An occasionally useful semicategory can be formed from it by considering the nonempty spaces and strictly contractive functions.</p> <p>This is a semicategory, since:</p> <ul> <li>the composition of two strictly contractive functions is strictly contractive</li> <li>identity maps are not contractive (they are trivial isometries)</li> </ul> <p>The interest in this semicategory arises from the fact that all morphisms <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>A</mi></mrow><annotation encoding="application/x-tex">f : A \to A</annotation></semantics></math> have unique fixed points, by Banach’s fixed point theorem.</p> <h2 id="in_higher_category_theory">In higher category theory</h2> <p>The concept of semicategory has more or less evident analogs and generalizations in <a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a>.</p> <p>For models of higher categories by <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a>s, i.e. presheaves on the <a class="existingWikiWord" href="/nlab/show/simplex+category">simplex category</a> (such as <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a>es, <a class="existingWikiWord" href="/nlab/show/quasi-categories">quasi-categories</a>, <a class="existingWikiWord" href="/nlab/show/weak+complicial+set">weak complicial set</a>s) the corresponding semi-category notion is obtained by discarding the degeneracy maps (which are the identity-assigning maps in the simplicial framework), i.e. by considering just presheaves on the subcategory <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mo>+</mo></msub><mo>⊂</mo><mi>Δ</mi></mrow><annotation encoding="application/x-tex">\Delta_+ \subset \Delta</annotation></semantics></math> on injective morphisms (see the discuss of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mo>+</mo></msub></mrow><annotation encoding="application/x-tex">\Delta_+</annotation></semantics></math> at <a class="existingWikiWord" href="/nlab/show/Reedy+model+structure">Reedy model structure</a> for more details).</p> <p>Accordingly, there is the semi-category analog of a <a class="existingWikiWord" href="/nlab/show/Segal+space">Segal space</a>, called a <em><a class="existingWikiWord" href="/nlab/show/semi-Segal+space">semi-Segal space</a></em>.</p> <p><a class="existingWikiWord" href="/nlab/show/Simpson%27s+conjecture">Simpson's conjecture</a> says that every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-category has a model where all <a class="existingWikiWord" href="/nlab/show/composition">composition</a> operations are strict and only the <a class="existingWikiWord" href="/nlab/show/unit+law">unit law</a>s hold up to <a class="existingWikiWord" href="/nlab/show/coherent">coherent</a> homotopies. This would mean that the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-semicategory underlying any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-category can always be chosen to be strict.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><span class="newWikiWord">dagger semicategory<a href="/nlab/new/dagger+semicategory">?</a></span></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/semipresheaf">semipresheaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/regular+semicategory">regular semicategory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/non-unital+ring">non-unital ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/semi-simplicial+set">semi-simplicial set</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/semi-Segal+space">semi-Segal space</a></p> </li> </ul> <div> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/algebra">algebraic</a> <a class="existingWikiWord" href="/nlab/show/mathematical+structure">structure</a></th><th><a class="existingWikiWord" href="/nlab/show/oidification">oidification</a></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/magma">magma</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/magmoid">magmoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/pointed+set">pointed</a> <a class="existingWikiWord" href="/nlab/show/magma">magma</a> with an <a class="existingWikiWord" href="/nlab/show/endofunction">endofunction</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/setoid">setoid</a>/<a class="existingWikiWord" href="/nlab/show/Bishop+set">Bishop set</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/unital+magma">unital magma</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/unital+magmoid">unital magmoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quasigroup">quasigroup</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quasigroupoid">quasigroupoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/loop+%28algebra%29">loop</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/loopoid">loopoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/semigroup">semigroup</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/semicategory">semicategory</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/monoid">monoid</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/category">category</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/anti-involution">anti-involutive</a> <a class="existingWikiWord" href="/nlab/show/monoid">monoid</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dagger+category">dagger category</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/associative+quasigroup">associative quasigroup</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/associative+quasigroupoid">associative quasigroupoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/group">group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/flexible+magma">flexible magma</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/flexible+magmoid">flexible magmoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/alternative+magma">alternative magma</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/alternative+magmoid">alternative magmoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/absorption+monoid">absorption monoid</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/absorption+category">absorption category</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cancellative+monoid">cancellative monoid</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cancellative+category">cancellative category</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/rig">rig</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/CMon">CMon</a>-<a class="existingWikiWord" href="/nlab/show/enriched+category">enriched category</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/nonunital+ring">nonunital ring</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Ab">Ab</a>-<a class="existingWikiWord" href="/nlab/show/enriched+magmoid">enriched</a> <a class="existingWikiWord" href="/nlab/show/semicategory">semicategory</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/nonassociative+ring">nonassociative ring</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Ab">Ab</a>-<a class="existingWikiWord" href="/nlab/show/enriched+magmoid">enriched</a> <a class="existingWikiWord" href="/nlab/show/unital+magmoid">unital magmoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/ring">ring</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/ringoid">ringoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/nonassociative+algebra">nonassociative algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/linear+magmoid">linear magmoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/nonassociative+algebra">nonassociative unital algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/unital+magmoid">unital</a> <a class="existingWikiWord" href="/nlab/show/linear+magmoid">linear magmoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/nonunital+algebra">nonunital algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/linear+magmoid">linear</a> <a class="existingWikiWord" href="/nlab/show/semicategory">semicategory</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/associative+unital+algebra">associative unital algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/linear+category">linear category</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/C-star+algebra">C-star algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/C-star+category">C-star category</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/differential+algebra">differential algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/differential+algebroid">differential algebroid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/flexible+algebra">flexible algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/flexible+magmoid">flexible</a> <a class="existingWikiWord" href="/nlab/show/linear+magmoid">linear magmoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/alternative+algebra">alternative algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/alternative+magmoid">alternative</a> <a class="existingWikiWord" href="/nlab/show/linear+magmoid">linear magmoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Lie+algebroid">Lie algebroid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/monoidal+poset">monoidal poset</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/2-poset">2-poset</a></td></tr> <tr><td style="text-align: left;"><span class="newWikiWord">strict monoidal groupoid<a href="/nlab/new/strict+monoidal+groupoid">?</a></span></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/strict+%282%2C1%29-category">strict (2,1)-category</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/strict+2-group">strict 2-group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/strict+2-groupoid">strict 2-groupoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/strict+monoidal+category">strict monoidal category</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/strict+2-category">strict 2-category</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/monoidal+groupoid">monoidal groupoid</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/%282%2C1%29-category">(2,1)-category</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/2-group">2-group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/2-groupoid">2-groupoid</a>/<a class="existingWikiWord" href="/nlab/show/bigroupoid">bigroupoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/2-category">2-category</a>/<a class="existingWikiWord" href="/nlab/show/bicategory">bicategory</a></td></tr> </tbody></table> </div> <h2 id="references">References</h2> <p>Semicategories were introduced in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Barry+Mitchell">Barry Mitchell</a>, <em>The dominion of Isbell</em>, Transactions of the American Mathematical Society <strong>167</strong> (1972) 319-331 [<a href="https://doi.org/10.2307/1996142">doi:10.2307/1996142</a>]</li> </ul> <p>Enriched semicategory theory is developed in</p> <ul> <li id="MBB02">M.-A. Moens, U. Bernani-Canani, <a class="existingWikiWord" href="/nlab/show/Francis+Borceux">F. Borceux</a>, <em>On regular presheaves and regular semi-categories</em> , Cah. Top. Géom. Diff. Cat. <strong>XLIII</strong> no.3 (2002) pp.163-190. (<a href="http://www.numdam.org/item?id=CTGDC_2002__43_3_163_0">numdam</a>)</li> </ul> <p>This is turned one notch further in</p> <ul> <li id="Stubbe05"><a class="existingWikiWord" href="/nlab/show/Isar+Stubbe">Isar Stubbe</a>, <em>Categorical structures enriched in a quantaloid : regular presheaves, regular semicategories</em> , Cah. Top. Géom. Diff. Cat. <strong>XLVI</strong> no.2 (2005) pp.99-121. (<a href="http://www.numdam.org/item/CTGDC_2005__46_2_99_0">numdam</a>)</li> </ul> <p>Semicategories and semigroups are mentioned in section 2 in</p> <ul> <li>W. Dale Garraway, <em>Sheaves for an involutive quantaloid</em>, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 46 no. 4 (2005), p. 243-274 (<a href="http://www.numdam.org/item?id=CTGDC_2005__46_4_243_0">numdam</a>)</li> </ul> <p>Semicategories with an eye towards their generalization to <a class="existingWikiWord" href="/nlab/show/semi-Segal+spaces">semi-Segal spaces</a> are briefly discussed at the beginning of</p> <ul> <li id="Harpaz"><a class="existingWikiWord" href="/nlab/show/Yonatan+Harpaz">Yonatan Harpaz</a>, <em>Quasi-unital <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Categories</em> (<a href="http://arxiv.org/abs/1210.0212">arXiv:1210.0212</a>)</li> </ul> <p>Structures obtained by further relaxing also the <a class="existingWikiWord" href="/nlab/show/associativity">associativity</a> law are discussed in</p> <ul> <li>Salvatore Tringali, <em>Plots and Their Applications - Part I: Foundations</em> (<a href="http://arxiv.org/abs/1311.3524">arXiv:1311.3524</a>)</li> </ul> <p>Topologically enriched semicategories are used for studying some aspects of concurrency theory in computer science. It is necessary to work with semicategories to have functorial definitions of the branching and merging homologies of a concurrent system. A starting point for reading the theory can be the paper</p> <ul> <li id="Gaucher"><a class="existingWikiWord" href="/nlab/show/Philippe+Gaucher">Philippe Gaucher</a>, <em>Flows revisited: the model category structure and its left determinedness</em>, Cahiers de Topologie et Géométrie Différentielle Catégoriques, vol LXI-2 (2020) (<a href="http://cahierstgdc.com/wp-content/uploads/2020/04/GAUCHER-LXI-2.pdf">published</a>, <a href="https://arxiv.org/abs/1806.08197">arXiv:1806.08197</a>)</li> </ul> <p>Semi-categories, semi-adjunctions and semi-cartesian closed categories have been used to study the <a class="existingWikiWord" href="/nlab/show/lambda+calculus">lambda calculus</a> since</p> <ul> <li>Susumu Hayashi, <em>Adjunction of semifunctors: Categorical structures in nonextensional lambda calculus</em>, Theoretical Computer Science Volume 41, 1985, Pages 95-104 <a href="https://doi.org/10.1016/0304-3975%2885%2990062-3">doi:10.1016/0304-3975(85)90062-3</a></li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on June 5, 2023 at 10:38:38. See the <a href="/nlab/history/semicategory" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/semicategory" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/4541/#Item_10">Discuss</a><span class="backintime"><a href="/nlab/revision/semicategory/32" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/semicategory" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/semicategory" accesskey="S" class="navlink" id="history" rel="nofollow">History (32 revisions)</a> <a href="/nlab/show/semicategory/cite" style="color: black">Cite</a> <a href="/nlab/print/semicategory" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/semicategory" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>