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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/17607/#Item_2" 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"> <h1 id='the_idea'>The idea</h1> <p>A <a class='existingWikiWord' href='/nlab/show/category'>category</a> consists of a collection of arrows each having a single object 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 multicategory is like a category, except that one allows multiple inputs and a single output.</p> <p>Thus a multicategory <math class='maruku-mathml' display='inline' id='mathml_9937a2ab11f5ffb86de41c7b08fd18affaed6d6a_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_9937a2ab11f5ffb86de41c7b08fd18affaed6d6a_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_9937a2ab11f5ffb86de41c7b08fd18affaed6d6a_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_9937a2ab11f5ffb86de41c7b08fd18affaed6d6a_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 (thus <math class='maruku-mathml' display='inline' id='mathml_9937a2ab11f5ffb86de41c7b08fd18affaed6d6a_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_9937a2ab11f5ffb86de41c7b08fd18affaed6d6a_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_9937a2ab11f5ffb86de41c7b08fd18affaed6d6a_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_9937a2ab11f5ffb86de41c7b08fd18affaed6d6a_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_9937a2ab11f5ffb86de41c7b08fd18affaed6d6a_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_9937a2ab11f5ffb86de41c7b08fd18affaed6d6a_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_9937a2ab11f5ffb86de41c7b08fd18affaed6d6a_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_9937a2ab11f5ffb86de41c7b08fd18affaed6d6a_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_9937a2ab11f5ffb86de41c7b08fd18affaed6d6a_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_9937a2ab11f5ffb86de41c7b08fd18affaed6d6a_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_9937a2ab11f5ffb86de41c7b08fd18affaed6d6a_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 associativity and identity axioms which the reader can probably figure out, but see for example <a class='existingWikiWord' href='/nlab/show/Tom+Leinster'>Tom Leinster</a>’s <a href='http://arxiv.org/abs/math.CT/0305049'>book</a>, page 35 ff., for details.</p> <p>Many people (especially non-category theorists) use <em>multicategory</em> to mean what we would call a <em>symmetric multicategory</em>, in which there is also an action of the symmetric group <math class='maruku-mathml' display='inline' id='mathml_9937a2ab11f5ffb86de41c7b08fd18affaed6d6a_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_9937a2ab11f5ffb86de41c7b08fd18affaed6d6a_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> and the composition is equivariant.</p> <h1 id='further_details_and_generalizations'>Further details and generalizations</h1> <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 monad</a>s. For ordinary categories, one uses the identity monad on <a class='existingWikiWord' href='/nlab/show/Set'>Set</a>; for ordinary multicategories, one uses the free monoid monad <math class='maruku-mathml' display='inline' id='mathml_9937a2ab11f5ffb86de41c7b08fd18affaed6d6a_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>. There is a very general notion of <math class='maruku-mathml' display='inline' id='mathml_9937a2ab11f5ffb86de41c7b08fd18affaed6d6a_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>T</mi></mrow><annotation encoding='application/x-tex'>T</annotation></semantics></math>-multicategory, where <math class='maruku-mathml' display='inline' id='mathml_9937a2ab11f5ffb86de41c7b08fd18affaed6d6a_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>T</mi></mrow><annotation encoding='application/x-tex'>T</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/cartesian+monad'>cartesian monad</a> on a category with pullbacks, which we outline as follows.</p> <ul> <li>First, a <math class='maruku-mathml' display='inline' id='mathml_9937a2ab11f5ffb86de41c7b08fd18affaed6d6a_21' 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_9937a2ab11f5ffb86de41c7b08fd18affaed6d6a_22' 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_9937a2ab11f5ffb86de41c7b08fd18affaed6d6a_23' 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_9937a2ab11f5ffb86de41c7b08fd18affaed6d6a_24' 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_9937a2ab11f5ffb86de41c7b08fd18affaed6d6a_25' 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_9937a2ab11f5ffb86de41c7b08fd18affaed6d6a_26' 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_9937a2ab11f5ffb86de41c7b08fd18affaed6d6a_27' 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_9937a2ab11f5ffb86de41c7b08fd18affaed6d6a_28' 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_9937a2ab11f5ffb86de41c7b08fd18affaed6d6a_29' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi><mo>:</mo><mi>X</mi><mo>⤏</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>p: X \dashrightarrow Y</annotation></semantics></math>.</p> </li> </ul> <div class='query'> <p>Dang… how do you make an arrow with a vertical slash in the middle? – Todd</p> <p>I don’t think that has been implemented yet. <a class='existingWikiWord' href='/ericforgy/published/itex2MML' title='ericforgy'>Here</a> is a list of all available arrows. – Eric</p> </div> <p>When <math class='maruku-mathml' display='inline' id='mathml_9937a2ab11f5ffb86de41c7b08fd18affaed6d6a_30' 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_9937a2ab11f5ffb86de41c7b08fd18affaed6d6a_31' 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_9937a2ab11f5ffb86de41c7b08fd18affaed6d6a_32' 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_9937a2ab11f5ffb86de41c7b08fd18affaed6d6a_33' 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_9937a2ab11f5ffb86de41c7b08fd18affaed6d6a_34' 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_9937a2ab11f5ffb86de41c7b08fd18affaed6d6a_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 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_9937a2ab11f5ffb86de41c7b08fd18affaed6d6a_36' 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_9937a2ab11f5ffb86de41c7b08fd18affaed6d6a_37' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>e</mi><mo>,</mo><mi>f</mi><mo>:</mo><mi>X</mi><mo>⤏</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>e, f: X \dashrightarrow Y</annotation></semantics></math> is a 2-cell between ordinary spans from <math class='maruku-mathml' display='inline' id='mathml_9937a2ab11f5ffb86de41c7b08fd18affaed6d6a_38' 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_9937a2ab11f5ffb86de41c7b08fd18affaed6d6a_39' 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_9937a2ab11f5ffb86de41c7b08fd18affaed6d6a_40' 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_9937a2ab11f5ffb86de41c7b08fd18affaed6d6a_41' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>e</mi><mo>:</mo><mi>X</mi><mo>⤏</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>e: X \dashrightarrow Y</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_9937a2ab11f5ffb86de41c7b08fd18affaed6d6a_42' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>:</mo><mi>Y</mi><mo>⤏</mo><mi>Z</mi></mrow><annotation encoding='application/x-tex'>f: Y \dashrightarrow Z</annotation></semantics></math>, take the ordinary span composite of</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_9937a2ab11f5ffb86de41c7b08fd18affaed6d6a_43' 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_9937a2ab11f5ffb86de41c7b08fd18affaed6d6a_44' 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_9937a2ab11f5ffb86de41c7b08fd18affaed6d6a_45' 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_9937a2ab11f5ffb86de41c7b08fd18affaed6d6a_46' 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_9937a2ab11f5ffb86de41c7b08fd18affaed6d6a_47' 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_9937a2ab11f5ffb86de41c7b08fd18affaed6d6a_48' 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_9937a2ab11f5ffb86de41c7b08fd18affaed6d6a_49' 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 <math class='maruku-mathml' display='inline' id='mathml_9937a2ab11f5ffb86de41c7b08fd18affaed6d6a_50' 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 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_9937a2ab11f5ffb86de41c7b08fd18affaed6d6a_51' 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>The full details are carefully treated in Tom Leinster’s book, <em>loc. cit.</em>, who gives many illuminating examples.</p> <h1 id='connection_with_operads'>Connection with operads</h1> <p>A nonpermutative (or Stasheff) <a class='existingWikiWord' href='/nlab/show/operad'>operad</a> may be defined as an ordinary multicategory with exactly one object. For each cartesian monad <math class='maruku-mathml' display='inline' id='mathml_9937a2ab11f5ffb86de41c7b08fd18affaed6d6a_52' 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_9937a2ab11f5ffb86de41c7b08fd18affaed6d6a_53' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>T</mi></mrow><annotation encoding='application/x-tex'>T</annotation></semantics></math>-operad: a <math class='maruku-mathml' display='inline' id='mathml_9937a2ab11f5ffb86de41c7b08fd18affaed6d6a_54' 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_9937a2ab11f5ffb86de41c7b08fd18affaed6d6a_55' 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_9937a2ab11f5ffb86de41c7b08fd18affaed6d6a_56' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>1</mn><mo>⤏</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>1 \dashrightarrow 1</annotation></semantics></math>.</p> <p>For example, in Batanin’s approach to (weak) <math class='maruku-mathml' display='inline' id='mathml_9937a2ab11f5ffb86de41c7b08fd18affaed6d6a_57' 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_9937a2ab11f5ffb86de41c7b08fd18affaed6d6a_58' 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_9937a2ab11f5ffb86de41c7b08fd18affaed6d6a_59' 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_9937a2ab11f5ffb86de41c7b08fd18affaed6d6a_60' 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/omega-category'>category</a> monad on the category of <a class='existingWikiWord' href='/nlab/show/globular+set'>globular set</a>s.</p> <p>With a little care, ordinary (permutative) operads may also be treated within this framework. See Tom Leinster’s book for details.</p> </div>  <div class="revisedby"> <p> Revision on January 3, 2009 at 16:32:51 by <a href="/nlab/author/Mike+Shulman" style="color: #005c19">Mike Shulman</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/6" accesskey="F" class="navlinkbackintime" id="to_next_revision" rel="nofollow">Next revision</a> (30 more)</span><span class="backintime"><a href="/nlab/revision/multicategory/4" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a> (4 more)</span><a href="/nlab/show/multicategory" class="navlink" id="to_current_revision">Current version of page</a><a href="/nlab/revision/diff/multicategory/5" 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=5" class="navlink" id="rollback" rel="nofollow">Rollback</a> <a href="/nlab/revision/multicategory/5/cite" style="color: black">Cite</a> <a href="/nlab/source/multicategory/5" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>