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href="/nlab/show/string+diagram">string diagram</a>, <a class="existingWikiWord" href="/nlab/show/tensor+network">tensor network</a></p> </li> </ul> <p><strong>With braiding</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/braided+monoidal+category">braided monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/balanced+monoidal+category">balanced monoidal category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/twist">twist</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a></p> </li> </ul> <p><strong>With duals for objects</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+with+duals">category with duals</a> (list of them)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dualizable+object">dualizable object</a> (what they have)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/rigid+monoidal+category">rigid monoidal category</a>, a.k.a. <a class="existingWikiWord" href="/nlab/show/autonomous+category">autonomous category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pivotal+category">pivotal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spherical+category">spherical category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ribbon+category">ribbon category</a>, a.k.a. <a class="existingWikiWord" href="/nlab/show/tortile+category">tortile category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+closed+category">compact closed category</a></p> </li> </ul> <p><strong>With duals for morphisms</strong></p> <ul> <li> <p><span class="newWikiWord">monoidal dagger-category<a href="/nlab/new/monoidal+dagger-category">?</a></span></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+dagger-category">symmetric monoidal dagger-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dagger+compact+category">dagger compact category</a></p> </li> </ul> <p><strong>With traces</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/trace">trace</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/traced+monoidal+category">traced monoidal category</a></p> </li> </ul> <p><strong>Closed structure</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/closed+monoidal+category">closed monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cartesian+closed+category">cartesian closed category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/closed+category">closed category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/star-autonomous+category">star-autonomous category</a></p> </li> </ul> <p><strong>Special sorts of products</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cartesian+monoidal+category">cartesian monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/semicartesian+monoidal+category">semicartesian monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+category+with+diagonals">monoidal category with diagonals</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/multicategory">multicategory</a></p> </li> </ul> <p><strong>Semisimplicity</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/semisimple+category">semisimple category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fusion+category">fusion category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/modular+tensor+category">modular tensor category</a></p> </li> </ul> <p><strong>Morphisms</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+functor">monoidal functor</a></p> <p>(<a class="existingWikiWord" href="/nlab/show/lax+monoidal+functor">lax</a>, <a class="existingWikiWord" href="/nlab/show/oplax+monoidal+functor">oplax</a>, <a class="existingWikiWord" href="/nlab/show/strong+monoidal+functor">strong</a> <a class="existingWikiWord" href="/nlab/show/bilax+monoidal+functor">bilax</a>, <a class="existingWikiWord" href="/nlab/show/Frobenius+monoidal+functor">Frobenius</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/braided+monoidal+functor">braided monoidal functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+functor">symmetric monoidal functor</a></p> </li> </ul> <p><strong>Internal monoids</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monoid+in+a+monoidal+category">monoid in a monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/commutative+monoid+in+a+symmetric+monoidal+category">commutative monoid in a symmetric monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/module+over+a+monoid">module over a monoid</a></p> </li> </ul> <p><strong id="_examples">Examples</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/closed+monoidal+structure+on+presheaves">closed monoidal structure on presheaves</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Day+convolution">Day convolution</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/coherence+theorem+for+monoidal+categories">coherence theorem for monoidal categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal Dold-Kan correspondence</a></p> </li> </ul> <p><strong>In higher category theory</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+2-category">monoidal 2-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/braided+monoidal+2-category">braided monoidal 2-category</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+bicategory">monoidal bicategory</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/k-tuply+monoidal+n-category">k-tuply monoidal n-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/little+cubes+operad">little cubes operad</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+%28%E2%88%9E%2C1%29-category">monoidal (∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28%E2%88%9E%2C1%29-category">symmetric monoidal (∞,1)-category</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+double+category">compact double category</a></p> </li> </ul> </div></div> <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> </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> <ul> <li><a href='#in_components'>In components</a></li> <li><a href='#in_terms_of_cartesian_monads'>In terms of cartesian monads</a></li> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#relation_to_operads'>Relation to operads</a></li> <li><a href='#relation_to_monoidal_categories'>Relation to monoidal categories</a></li> <li><a href='#exponentiability'>Exponentiability</a></li> </ul> <li><a href='#examples_and_special_cases'>Examples and special cases</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>Recall that a <a class="existingWikiWord" href="/nlab/show/category">category</a> consists of a collection of <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> each having a single <a class="existingWikiWord" href="/nlab/show/object">object</a> as source or input, and a single object as target or output, together with laws for composition and identity obeying associativity and identity axioms. A <em>multicategory</em> is like a category, except that one allows multiple inputs and a single output.</p> <p>Another term for <em>multicategory</em> is <em>coloured <a class="existingWikiWord" href="/nlab/show/operad">operad</a></em>.</p> <h2 id="definition">Definition</h2> <h3 id="in_components">In components</h3> <p>A <strong>multicategory</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">C</annotation></semantics></math> consists of</p> <ul> <li>A collection of <em>objects</em>, <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>.</li> <li>A collection of <em>multimorphisms</em>, <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>.</li> <li>A source map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo>:</mo><msub><mi>C</mi> <mn>1</mn></msub><mo>→</mo><mo stretchy="false">(</mo><msub><mi>C</mi> <mn>0</mn></msub><mo stretchy="false">)</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">s: C_1 \to (C_0)*</annotation></semantics></math> to the collection of finite, possibly empty <a class="existingWikiWord" href="/nlab/show/list">list</a>s of objects (where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>C</mi> <mn>0</mn></msub><mo stretchy="false">)</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">(C_0)*</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/free+monoid">free monoid</a> generated by <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 a target map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><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">t: C_1 \to C_0</annotation></semantics></math>. We write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><msub><mi>c</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>c</mi> <mi>n</mi></msub><mo>→</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">f: c_1, \ldots, c_n \to c</annotation></semantics></math> to indicate the source and target of a multimorphism <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>.</li> <li>Identity and composition laws. The identity law is a map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mn>1</mn> <mo lspace="verythinmathspace" rspace="0em">−</mo></msub><mo>:</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">1_{-}: C_0 \to C_1</annotation></semantics></math> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mn>1</mn> <mi>c</mi></msub><mo>:</mo><mi>c</mi><mo>→</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">1_c: c \to c</annotation></semantics></math>. The composition law assigns, to each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><msub><mi>c</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>c</mi> <mi>n</mi></msub><mo>→</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">f: c_1, \ldots, c_n \to c</annotation></semantics></math> together with an <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>-tuple <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><msub><mi>f</mi> <mi>i</mi></msub><mo>:</mo><msub><mover><mi>c</mi><mo stretchy="false">→</mo></mover> <mi>i</mi></msub><mo>→</mo><msub><mi>c</mi> <mi>i</mi></msub><mo>:</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mi>…</mi><mo>,</mo><mi>n</mi><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle f_i: \vec{c}_i \to c_i: i = 1, \ldots, n \rangle</annotation></semantics></math>, a composite<div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∘</mo><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>f</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>:</mo><msub><mover><mi>c</mi><mo stretchy="false">→</mo></mover> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mover><mi>c</mi><mo stretchy="false">→</mo></mover> <mi>n</mi></msub><mo>→</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">f \circ (f_1, \ldots, f_n): \vec{c}_1, \ldots, \vec{c}_n \to c</annotation></semantics></math></div> <p>where the source is obtained by concatenating lists in the evident way.</p> </li> </ul> <p>These operations are subject to <a class="existingWikiWord" href="/nlab/show/associativity">associativity</a> and <a class="existingWikiWord" href="/nlab/show/identity">identity</a> axioms which the reader can probably figure out, but see for example (<a href="#Leinster">Leinster, page 35 ff.</a>), for details.</p> <p>Many people (especially non-category theorists) use the word <em>multicategory</em> or the word <em>colored <a class="existingWikiWord" href="/nlab/show/operad">operad</a></em> to mean what we would call a <em><a class="existingWikiWord" href="/nlab/show/symmetric+multicategory">symmetric multicategory</a></em> / <em><a class="existingWikiWord" href="/nlab/show/symmetric+operad">symmetric operad</a></em>. These are multicategories equipped with an <a class="existingWikiWord" href="/nlab/show/action">action</a> of the <a class="existingWikiWord" href="/nlab/show/symmetric+group">symmetric group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">S_n</annotation></semantics></math> on the multimorphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>c</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>c</mi> <mi>n</mi></msub><mo>→</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">c_1, \ldots, c_n \to c</annotation></semantics></math> such that the composition is equivariant with respect to these actions.</p> <h3 id="in_terms_of_cartesian_monads">In terms of cartesian monads</h3> <p>An efficient abstract method for defining multicategories and related structures is through the formalism of <a class="existingWikiWord" href="/nlab/show/cartesian+monads">cartesian monads</a>. For ordinary categories, one uses the identity monad on <a class="existingWikiWord" href="/nlab/show/Set">Set</a>; for ordinary multicategories, one uses the <a class="existingWikiWord" href="/nlab/show/free+monoid">free monoid</a> monad <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>*</mo><mo>:</mo><mi>Set</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">(-)*: Set \to Set</annotation></semantics></math>. This is a special case of the yet more general notion of <a class="existingWikiWord" href="/nlab/show/generalized+multicategory">generalized multicategory</a>.</p> <p>We summarize here how the theory applies to the case of a <a class="existingWikiWord" href="/nlab/show/cartesian+monad">cartesian monad</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> on a category with <a class="existingWikiWord" href="/nlab/show/pullbacks">pullbacks</a>; see <a class="existingWikiWord" href="/nlab/show/generalized+multicategory">generalized multicategory</a> for the fully general context.</p> <ul> <li>First, a <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>-span</strong> from <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> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/span">span</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">T X</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math>, that is, a diagram<div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>T</mi><mi>X</mi><mover><mo>←</mo><mrow><msub><mi>p</mi> <mn>1</mn></msub></mrow></mover><mi>P</mi><mover><mo>→</mo><mrow><msub><mi>p</mi> <mn>2</mn></msub></mrow></mover><mi>Y</mi></mrow><annotation encoding="application/x-tex">T X \stackrel{p_1}{\leftarrow} P \stackrel{p_2}{\to} Y</annotation></semantics></math></div> <p>A <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>-span is often written as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><mi>X</mi><mi>⇸</mi><mi>Y</mi></mrow><annotation encoding="application/x-tex">p: X &#8696; Y</annotation></semantics></math>.</p> </li> </ul> <p>When <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> is the free monoid monad on <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>, a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>-span from <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> to itself is called a <em>multigraph</em> on <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>.</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>-spans are the 1-cells of a <a class="existingWikiWord" href="/nlab/show/bicategory">bicategory</a>. A 2-cell between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>-spans <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi><mo>,</mo><mi>f</mi><mo>:</mo><mi>X</mi><mi>⇸</mi><mi>Y</mi></mrow><annotation encoding="application/x-tex">e, f: X &#8696; Y</annotation></semantics></math> is a 2-cell between ordinary spans from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">T X</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math>. To horizontally compose <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>-spans <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi><mo>:</mo><mi>X</mi><mi>⇸</mi><mi>Y</mi></mrow><annotation encoding="application/x-tex">e: X &#8696; Y</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>Y</mi><mi>⇸</mi><mi>Z</mi></mrow><annotation encoding="application/x-tex">f: Y &#8696; Z</annotation></semantics></math>, take the ordinary span composite of</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>T</mi><mi>X</mi><mover><mo>←</mo><mrow><mi>m</mi><mi>X</mi></mrow></mover><msup><mi>T</mi> <mn>2</mn></msup><mi>X</mi><mover><mo>←</mo><mrow><mi>T</mi><msub><mi>e</mi> <mn>1</mn></msub></mrow></mover><mi>T</mi><mi>E</mi><mover><mo>→</mo><mrow><mi>T</mi><msub><mi>e</mi> <mn>2</mn></msub></mrow></mover><mi>T</mi><mi>Y</mi><mo stretchy="false">)</mo><mo>∘</mo><mo stretchy="false">(</mo><mi>T</mi><mi>Y</mi><mover><mo>←</mo><mrow><msub><mi>f</mi> <mn>1</mn></msub></mrow></mover><mi>F</mi><mover><mo>→</mo><mrow><msub><mi>f</mi> <mn>2</mn></msub></mrow></mover><mi>Z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(T X \stackrel{m X}{\leftarrow} T^2 X \stackrel{T e_1}{\leftarrow} T E \stackrel{T e_2}{\to} T Y) \circ (T Y \stackrel{f_1}{\leftarrow} F \stackrel{f_2}{\to} Z)</annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi><mo>:</mo><msup><mi>T</mi> <mn>2</mn></msup><mo>→</mo><mi>T</mi></mrow><annotation encoding="application/x-tex">m: T^2 \to T</annotation></semantics></math> is the monad multiplication. The identity <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>-span from <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> to itself is the span</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>T</mi><mi>X</mi><mover><mo>←</mo><mrow><mi>u</mi><mi>X</mi></mrow></mover><mi>X</mi><mover><mo>→</mo><mrow><msub><mn>1</mn> <mi>X</mi></msub></mrow></mover><mi>X</mi></mrow><annotation encoding="application/x-tex">T X \stackrel{u X}{\leftarrow} X \stackrel{1_X}{\to} X</annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mo>:</mo><mi>I</mi><mo>→</mo><mi>T</mi></mrow><annotation encoding="application/x-tex">u: I \to T</annotation></semantics></math> is the monad unit. The verification of the bicategory axioms uses the cartesianness of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> in concert with the corresponding axioms on the bicategory of spans.</p> </li> <li> <p>A <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>-multicategory</em> is defined to be a <a class="existingWikiWord" href="/nlab/show/monad">monad</a> in the bicategory of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>-spans.</p> </li> </ul> <p>When <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> is the free monoid monad on sets, then a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>-multicategory is a multicategory as defined above. For more examples and generalizations, see <a class="existingWikiWord" href="/nlab/show/generalized+multicategory">generalized multicategory</a>.</p> <h2 id="properties">Properties</h2> <h3 id="relation_to_operads">Relation to operads</h3> <p>A <a class="existingWikiWord" href="/nlab/show/nonpermutative+operad">nonpermutative</a> (or Stasheff-) <a class="existingWikiWord" href="/nlab/show/operad">operad</a> in <a class="existingWikiWord" href="/nlab/show/Set">Set</a> may be defined as an ordinary multicategory with exactly one object. Likewise, a <a class="existingWikiWord" href="/nlab/show/symmetric+operad">symmetric operad</a> in any <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> is equivalent to a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/enriched+category">enriched</a> multicategory with one object.</p> <p>More generally, the notion of <em>multi-colored <a class="existingWikiWord" href="/nlab/show/planar+operad">planar operad</a></em> is equivalent to that of multicategory, and the notion of <em>multi-colored <a class="existingWikiWord" href="/nlab/show/symmetric+operad">symmetric operad</a></em> is equivalent to that of <a class="existingWikiWord" href="/nlab/show/symmetric+multicategory">symmetric multicategory</a>.</p> <p>Fully generally, for each cartesian monad <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>, there is a corresponding notion of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>-operad, namely a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>-multicategory whose underlying <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>-span has the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mi>⇸</mi><mn>1</mn></mrow><annotation encoding="application/x-tex">1 &#8696; 1</annotation></semantics></math>.</p> <p>For example, in Batanin’s approach to (weak) <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>-<a class="existingWikiWord" href="/nlab/show/infinity-category">categories</a>, a <a class="existingWikiWord" href="/nlab/show/globular+operad">globular operad</a> is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>-operad, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> is the free (strict) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/strict+omega-category">category</a> monad on the category of <a class="existingWikiWord" href="/nlab/show/globular+set">globular set</a>s.</p> <p>Ordinary (permutative/symmetric) operads, and their generalization to <a class="existingWikiWord" href="/nlab/show/symmetric+multicategory">symmetric multicategories</a>, can also be treated in the framework of <a class="existingWikiWord" href="/nlab/show/generalized+multicategories">generalized multicategories</a>, but they require a framework more general than that of cartesian monads.</p> <h3 id="relation_to_monoidal_categories">Relation to monoidal categories</h3> <p>There is a <a class="existingWikiWord" href="/nlab/show/faithful+functor">faithful functor</a> from <a class="existingWikiWord" href="/nlab/show/monoidal+categories">monoidal categories</a> to <a class="existingWikiWord" href="/nlab/show/multicategories">multicategories</a>, given by forming <a class="existingWikiWord" href="/nlab/show/representable+multicategories">represented multicategories</a>. Conversely, to any multicategory <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> there is an associated (strict) monoidal category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(C)</annotation></semantics></math>, whose objects (respectively, arrows) are <a class="existingWikiWord" href="/nlab/show/lists">lists</a> of objects (respectively, arrows) 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>, and where the tensor product in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(C)</annotation></semantics></math> is given by concatenation.</p> <h3 id="exponentiability">Exponentiability</h3> <p>A multicategory is <a class="existingWikiWord" href="/nlab/show/exponentiable+object">exponentiable</a> if and only if it is <a class="existingWikiWord" href="/nlab/show/promonoidal+category">promonoidal</a> (Proposition 2.8 of <a href="#Pisani14">Pisani 2014</a>). In particular, <a class="existingWikiWord" href="/nlab/show/representable+multicategories">representable multicategories</a> and <a class="existingWikiWord" href="/nlab/show/sequential+multicategories">sequential multicategories</a> are exponentiable. This gives an abstract construction of the <a class="existingWikiWord" href="/nlab/show/Day+convolution">Day convolution</a> tensor product on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>C</mi><mo>,</mo><mi>D</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[C, D]</annotation></semantics></math> for any <a class="existingWikiWord" href="/nlab/show/promonoidal+category">promonoidal category</a> <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 any <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>: it is precisely the (representable) multicategory structure on the functor multicategory <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>D</mi> <mi>C</mi></msup></mrow><annotation encoding="application/x-tex">D^C</annotation></semantics></math>.</p> <h2 id="examples_and_special_cases">Examples and special cases</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/permutative+categories">permutative categories</a> with multi-linear functors between them form a multicategory <a class="existingWikiWord" href="/nlab/show/PermCat">PermCat</a></li> </ul> <p>See also the examples at <em><a class="existingWikiWord" href="/nlab/show/operad">operad</a></em>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/multimorphism">multimorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/multifunctor">multifunctor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/polycategory">polycategory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/properad">properad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fibration+of+multicategories">fibration of multicategories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+multicategory">symmetric multicategory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cartesian+multicategory">cartesian multicategory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sequential+multicategory">sequential multicategory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+multicategory">generalized multicategory</a>, <a class="existingWikiWord" href="/nlab/show/operad">operad</a></p> </li> </ul> <h2 id="references">References</h2> <p>Multicategories were introduced in:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Joachim+Lambek">Joachim Lambek</a>, <em>Deductive systems and categories II. Standard constructions and closed categories</em>, Category Theory, Homology Theory and their Applications I: Proceedings of the Conference held at the Seattle Research Center of the Battelle Memorial Institute, June 24–July 19, 1968 Volume One. Berlin, Heidelberg: Springer Berlin Heidelberg, 2006.</li> </ul> <p>and developed further in:</p> <ul> <li>Joachim Lambek, <em>Multicategories revisited</em>, Contemp. Math 92 (1989): 217-239.</li> </ul> <p>See also:</p> <ul id="Leinster"> <li><a class="existingWikiWord" href="/nlab/show/Tom+Leinster">Tom Leinster</a>, <em>Higher operads, higher categories</em>, London Math. Soc. Lec. Note Series <strong>298</strong>, <a href="http://arxiv.org/abs/math.CT/0305049">math.CT/0305049</a></li> </ul> <ul> <li id="Pisani14"><a class="existingWikiWord" href="/nlab/show/Claudio+Pisani">Claudio Pisani</a>, <em>Sequential multicategories</em>, Theory and Applications of Categories 29.19 (2014), <a href="https://arxiv.org/abs/1402.0253">arXiv:1402.0253</a></li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on September 27, 2024 at 08:38:48. 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