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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"> <div class='rightHandSide'> <div class='toc clickDown' tabindex='0'> <h3 id='context'>Context</h3> <h4 id='monoidal_categories'>Monoidal categories</h4> <div class='hide'> <p><strong><a class='existingWikiWord' href='/nlab/show/monoidal+category'>monoidal categories</a></strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/enriched+monoidal+category'>enriched monoidal category</a>, <a class='existingWikiWord' href='/nlab/show/tensor+category'>tensor category</a></p> </li> <li> <p><a class='existingWikiWord' 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/rigid+monoidal+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/ribbon+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/monoidal+functor'>lax</a>, <a class='existingWikiWord' href='/nlab/show/oplax+monoidal+functor'>oplax</a>, <a class='existingWikiWord' href='/nlab/show/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+and+strictification+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+bicategory'>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+%28infinity%2C1%29-category'>monoidal (∞,1)-category</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/symmetric+monoidal+%28infinity%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> <h4 id='category_theory'>Category theory</h4> <div class='hide'> <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/end'>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%E2%80%93Ulmer+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> <h1 id='contents'>Contents</h1> <div class='maruku_toc'><ul><li><a href='#idea'>Idea</a></li><li><a href='#definition'>Definition</a><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><li><a href='#properties'>Properties</a><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></ul></li><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/morphism'>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 class='maruku-mathml' display='inline' id='mathml_771576dfa7e3cbeadd6560a680f7a33450ede0f2_1' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_771576dfa7e3cbeadd6560a680f7a33450ede0f2_2' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_771576dfa7e3cbeadd6560a680f7a33450ede0f2_3' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_771576dfa7e3cbeadd6560a680f7a33450ede0f2_4' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_771576dfa7e3cbeadd6560a680f7a33450ede0f2_5' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_771576dfa7e3cbeadd6560a680f7a33450ede0f2_6' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_771576dfa7e3cbeadd6560a680f7a33450ede0f2_7' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_771576dfa7e3cbeadd6560a680f7a33450ede0f2_8' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_771576dfa7e3cbeadd6560a680f7a33450ede0f2_9' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_771576dfa7e3cbeadd6560a680f7a33450ede0f2_10' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_771576dfa7e3cbeadd6560a680f7a33450ede0f2_11' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_771576dfa7e3cbeadd6560a680f7a33450ede0f2_12' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_771576dfa7e3cbeadd6560a680f7a33450ede0f2_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-tuple <math class='maruku-mathml' display='inline' id='mathml_771576dfa7e3cbeadd6560a680f7a33450ede0f2_14' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='block' id='mathml_771576dfa7e3cbeadd6560a680f7a33450ede0f2_15' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_771576dfa7e3cbeadd6560a680f7a33450ede0f2_16' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_771576dfa7e3cbeadd6560a680f7a33450ede0f2_17' xmlns='http://www.w3.org/1998/Math/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+monad'>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 class='maruku-mathml' display='inline' id='mathml_771576dfa7e3cbeadd6560a680f7a33450ede0f2_18' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_771576dfa7e3cbeadd6560a680f7a33450ede0f2_19' xmlns='http://www.w3.org/1998/Math/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/pullback'>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 class='maruku-mathml' display='inline' id='mathml_771576dfa7e3cbeadd6560a680f7a33450ede0f2_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>T</mi></mrow><annotation encoding='application/x-tex'>T</annotation></semantics></math>-span</strong> from <math class='maruku-mathml' display='inline' id='mathml_771576dfa7e3cbeadd6560a680f7a33450ede0f2_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> to <math class='maruku-mathml' display='inline' id='mathml_771576dfa7e3cbeadd6560a680f7a33450ede0f2_22' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_771576dfa7e3cbeadd6560a680f7a33450ede0f2_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi></mrow><annotation encoding='application/x-tex'>p</annotation></semantics></math> from <math class='maruku-mathml' display='inline' id='mathml_771576dfa7e3cbeadd6560a680f7a33450ede0f2_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>T</mi><mi>X</mi></mrow><annotation encoding='application/x-tex'>T X</annotation></semantics></math> to <math class='maruku-mathml' display='inline' id='mathml_771576dfa7e3cbeadd6560a680f7a33450ede0f2_25' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='block' id='mathml_771576dfa7e3cbeadd6560a680f7a33450ede0f2_26' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_771576dfa7e3cbeadd6560a680f7a33450ede0f2_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>T</mi></mrow><annotation encoding='application/x-tex'>T</annotation></semantics></math>-span is often written as <math class='maruku-mathml' display='inline' id='mathml_771576dfa7e3cbeadd6560a680f7a33450ede0f2_28' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_771576dfa7e3cbeadd6560a680f7a33450ede0f2_29' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>T</mi></mrow><annotation encoding='application/x-tex'>T</annotation></semantics></math> is the free monoid monad on <math class='maruku-mathml' display='inline' id='mathml_771576dfa7e3cbeadd6560a680f7a33450ede0f2_30' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Set</mi></mrow><annotation encoding='application/x-tex'>Set</annotation></semantics></math>, a <math class='maruku-mathml' display='inline' id='mathml_771576dfa7e3cbeadd6560a680f7a33450ede0f2_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>T</mi></mrow><annotation encoding='application/x-tex'>T</annotation></semantics></math>-span from <math class='maruku-mathml' display='inline' id='mathml_771576dfa7e3cbeadd6560a680f7a33450ede0f2_32' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_771576dfa7e3cbeadd6560a680f7a33450ede0f2_33' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>.</p> <ul> <li> <p><math class='maruku-mathml' display='inline' id='mathml_771576dfa7e3cbeadd6560a680f7a33450ede0f2_34' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_771576dfa7e3cbeadd6560a680f7a33450ede0f2_35' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>T</mi></mrow><annotation encoding='application/x-tex'>T</annotation></semantics></math>-spans <math class='maruku-mathml' display='inline' id='mathml_771576dfa7e3cbeadd6560a680f7a33450ede0f2_36' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_771576dfa7e3cbeadd6560a680f7a33450ede0f2_37' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>T</mi><mi>X</mi></mrow><annotation encoding='application/x-tex'>T X</annotation></semantics></math> to <math class='maruku-mathml' display='inline' id='mathml_771576dfa7e3cbeadd6560a680f7a33450ede0f2_38' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math>. To horizontally compose <math class='maruku-mathml' display='inline' id='mathml_771576dfa7e3cbeadd6560a680f7a33450ede0f2_39' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>T</mi></mrow><annotation encoding='application/x-tex'>T</annotation></semantics></math>-spans <math class='maruku-mathml' display='inline' id='mathml_771576dfa7e3cbeadd6560a680f7a33450ede0f2_40' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_771576dfa7e3cbeadd6560a680f7a33450ede0f2_41' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='block' id='mathml_771576dfa7e3cbeadd6560a680f7a33450ede0f2_42' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_771576dfa7e3cbeadd6560a680f7a33450ede0f2_43' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_771576dfa7e3cbeadd6560a680f7a33450ede0f2_44' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>T</mi></mrow><annotation encoding='application/x-tex'>T</annotation></semantics></math>-span from <math class='maruku-mathml' display='inline' id='mathml_771576dfa7e3cbeadd6560a680f7a33450ede0f2_45' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='block' id='mathml_771576dfa7e3cbeadd6560a680f7a33450ede0f2_46' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_771576dfa7e3cbeadd6560a680f7a33450ede0f2_47' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_771576dfa7e3cbeadd6560a680f7a33450ede0f2_48' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_771576dfa7e3cbeadd6560a680f7a33450ede0f2_49' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_771576dfa7e3cbeadd6560a680f7a33450ede0f2_50' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>T</mi></mrow><annotation encoding='application/x-tex'>T</annotation></semantics></math>-spans.</p> </li> </ul> <p>When <math class='maruku-mathml' display='inline' id='mathml_771576dfa7e3cbeadd6560a680f7a33450ede0f2_51' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_771576dfa7e3cbeadd6560a680f7a33450ede0f2_52' xmlns='http://www.w3.org/1998/Math/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/planar+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 class='maruku-mathml' display='inline' id='mathml_771576dfa7e3cbeadd6560a680f7a33450ede0f2_53' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi></mrow><annotation encoding='application/x-tex'>V</annotation></semantics></math> is equivalent to a <math class='maruku-mathml' display='inline' id='mathml_771576dfa7e3cbeadd6560a680f7a33450ede0f2_54' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_771576dfa7e3cbeadd6560a680f7a33450ede0f2_55' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>T</mi></mrow><annotation encoding='application/x-tex'>T</annotation></semantics></math>, there is a corresponding notion of <math class='maruku-mathml' display='inline' id='mathml_771576dfa7e3cbeadd6560a680f7a33450ede0f2_56' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>T</mi></mrow><annotation encoding='application/x-tex'>T</annotation></semantics></math>-operad, namely a <math class='maruku-mathml' display='inline' id='mathml_771576dfa7e3cbeadd6560a680f7a33450ede0f2_57' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>T</mi></mrow><annotation encoding='application/x-tex'>T</annotation></semantics></math>-multicategory whose underlying <math class='maruku-mathml' display='inline' id='mathml_771576dfa7e3cbeadd6560a680f7a33450ede0f2_58' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>T</mi></mrow><annotation encoding='application/x-tex'>T</annotation></semantics></math>-span has the form <math class='maruku-mathml' display='inline' id='mathml_771576dfa7e3cbeadd6560a680f7a33450ede0f2_59' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_771576dfa7e3cbeadd6560a680f7a33450ede0f2_60' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_771576dfa7e3cbeadd6560a680f7a33450ede0f2_61' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>T</mi></mrow><annotation encoding='application/x-tex'>T</annotation></semantics></math>-operad, where <math class='maruku-mathml' display='inline' id='mathml_771576dfa7e3cbeadd6560a680f7a33450ede0f2_62' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>T</mi></mrow><annotation encoding='application/x-tex'>T</annotation></semantics></math> is the free (strict) <math class='maruku-mathml' display='inline' id='mathml_771576dfa7e3cbeadd6560a680f7a33450ede0f2_63' xmlns='http://www.w3.org/1998/Math/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+multicategory'>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+category'>monoidal categories</a> to <a class='existingWikiWord' href='/nlab/show/multicategory'>multicategories</a>, given by forming <a class='existingWikiWord' href='/nlab/show/representable+multicategory'>represented multicategories</a>. Conversely, to any multicategory <math class='maruku-mathml' display='inline' id='mathml_771576dfa7e3cbeadd6560a680f7a33450ede0f2_64' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> there is an associated (strict) monoidal category <math class='maruku-mathml' display='inline' id='mathml_771576dfa7e3cbeadd6560a680f7a33450ede0f2_65' xmlns='http://www.w3.org/1998/Math/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/list'>lists</a> of objects (respectively, arrows) of <math class='maruku-mathml' display='inline' id='mathml_771576dfa7e3cbeadd6560a680f7a33450ede0f2_66' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math>, and where the tensor product in <math class='maruku-mathml' display='inline' id='mathml_771576dfa7e3cbeadd6560a680f7a33450ede0f2_67' xmlns='http://www.w3.org/1998/Math/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> <h2 id='examples_and_special_cases'>Examples and special cases</h2> <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/polycategory'>polycategory</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/generalized+multicategory'>generalized multicategory</a>, <a class='existingWikiWord' href='/nlab/show/operad'>operad</a></p> </li> </ul> <h2 id='references'>References</h2> <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> <p> </p> <p> </p> </div>  <div class="revisedby"> <p> Revision on June 23, 2016 at 14:35:58 by <a href="/nlab/author/David+Tanzer" style="color: #005c19">David Tanzer</a> See the <a href="/nlab/history/multicategory" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="https://nforum.ncatlab.org/discussion/17607/#Item_2">Discuss</a><span class="backintime"><a href="/nlab/revision/multicategory/29" accesskey="F" class="navlinkbackintime" id="to_next_revision" rel="nofollow">Next revision</a> (7 more)</span><span class="backintime"><a href="/nlab/revision/multicategory/27" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a> (27 more)</span><a href="/nlab/show/multicategory" class="navlink" id="to_current_revision">Current version of page</a><a href="/nlab/revision/diff/multicategory/28" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/multicategory" accesskey="S" class="navlink" id="history" rel="nofollow">History (34 revisions)</a><a href="/nlab/rollback/multicategory?rev=28" class="navlink" id="rollback" rel="nofollow">Rollback</a> <a href="/nlab/revision/multicategory/28/cite" style="color: black">Cite</a> <a href="/nlab/source/multicategory/28" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>