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style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title></title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="monoidal_categories">Monoidal categories</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/monoidal+categories">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/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="enriched_category_theory">Enriched category theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a></strong></p> <h2 id="background">Background</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a>, <a class="existingWikiWord" href="/nlab/show/closed+monoidal+category">closed monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cosmos">cosmos</a>, <a class="existingWikiWord" href="/nlab/show/multicategory">multicategory</a>, <a class="existingWikiWord" href="/nlab/show/bicategory">bicategory</a>, <a class="existingWikiWord" href="/nlab/show/double+category">double category</a>, <a class="existingWikiWord" href="/nlab/show/virtual+double+category">virtual double category</a></p> </li> </ul> <h2 id="basic_concepts">Basic concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+category">enriched category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+functor">enriched functor</a>, <a class="existingWikiWord" href="/nlab/show/profunctor">profunctor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+natural+transformation">enriched natural transformation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+adjoint+functor">enriched adjoint functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+product+category">enriched product category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+functor+category">enriched functor category</a></p> </li> </ul> <h2 id="universal_constructions">Universal constructions</h2> <ul> <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> </ul> <h2 id="extra_stuff_structure_property">Extra stuff, structure, property</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/copowering">copowering</a> (<a class="existingWikiWord" href="/nlab/show/tensoring">tensoring</a>)</p> <p><a class="existingWikiWord" href="/nlab/show/powering">powering</a> (<a class="existingWikiWord" href="/nlab/show/cotensoring">cotensoring</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+enriched+category">monoidal enriched category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cartesian+closed+enriched+category">cartesian closed enriched category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+cartesian+closed+enriched+category">locally cartesian closed enriched category</a></p> </li> </ul> </li> </ul> <h3 id="homotopical_enrichment">Homotopical enrichment</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+homotopical+category">enriched homotopical category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+model+category">enriched model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+homotopical+presheaves">model structure on homotopical presheaves</a></p> </li> </ul> </div></div> </div> </div> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#versus_monoidal_categories'>Versus monoidal categories</a></li> <li><a href='#day_convolution'>Day convolution</a></li> <li><a href='#versus_multicategories'>Versus multicategories</a></li> </ul> <li><a href='#notes'>Notes</a></li> <li><a href='#related_pages'>Related pages</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>A <strong>promonoidal category</strong> is like a <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> in whose structure (namely, <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> and <a class="existingWikiWord" href="/nlab/show/unit+object">unit object</a>) we have replaced <a class="existingWikiWord" href="/nlab/show/functors">functors</a> by <a class="existingWikiWord" href="/nlab/show/profunctors">profunctors</a>.</p> <h2 id="definition">Definition</h2> <p>A <strong>promonoidal category</strong> is a <a class="existingWikiWord" href="/nlab/show/pseudomonoid">pseudomonoid</a> in the <a class="existingWikiWord" href="/nlab/show/monoidal+bicategory">monoidal bicategory</a> <a class="existingWikiWord" href="/nlab/show/Prof">Prof</a>. This means that it is a <a class="existingWikiWord" href="/nlab/show/category">category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> together with</p> <ul> <li>A <a class="existingWikiWord" href="/nlab/show/profunctor">profunctor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>×</mo><mi>A</mi><mi>⇸</mi><mi>A</mi></mrow><annotation encoding="application/x-tex">P \colon A\times A &#8696; A</annotation></semantics></math>.</li> <li>A profunctor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi><mo lspace="verythinmathspace">:</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">J\colon 1</annotation></semantics></math> ⇸ <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>.</li> <li><a class="existingWikiWord" href="/nlab/show/associator">Associativity</a> and <a class="existingWikiWord" href="/nlab/show/unitor">unit isomorphisms</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>⊙</mo><mo stretchy="false">(</mo><mi>P</mi><mo>×</mo><mn>1</mn><mo stretchy="false">)</mo><mo>≅</mo><mi>P</mi><mo>⊙</mo><mo stretchy="false">(</mo><mn>1</mn><mo>×</mo><mi>P</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P \odot (P\times 1) \cong P\odot (1\times P)</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>⊙</mo><mo stretchy="false">(</mo><mi>J</mi><mo>×</mo><mn>1</mn><mo stretchy="false">)</mo><mo>≅</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">P\odot (J\times 1) \cong 1</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>⊙</mo><mo stretchy="false">(</mo><mn>1</mn><mo>×</mo><mi>J</mi><mo stretchy="false">)</mo><mo>≅</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">P\odot (1\times J) \cong 1</annotation></semantics></math>.</li> <li>The usual <a class="existingWikiWord" href="/nlab/show/pentagon+identity">pentagon</a> and unit conditions hold, as in a <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a>.</li> </ul> <p>Recalling that a profunctor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> ⇸ <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> is defined to be a functor of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>B</mi> <mi>op</mi></msup><mo>×</mo><mi>A</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">B^{op}\times A \to Set</annotation></semantics></math>, we can make this more explicit. We can also generalize it by replacing <a class="existingWikiWord" href="/nlab/show/Set">Set</a> by a <a class="existingWikiWord" href="/nlab/show/Benabou+cosmos">Benabou cosmos</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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> by 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 category</a>; then a profunctor is 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+functor">enriched functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>B</mi> <mi>op</mi></msup><mo>×</mo><mi>A</mi><mo>→</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">B^{op}\times A \to V</annotation></semantics></math>.</p> <p>Thus, we obtain the following as an explicit definition of <strong>promonoidal <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>-category</strong>:</p> <p>We have the following data</p> <ol> <li> <p>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 category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>.</p> </li> <li> <p>A <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>3</mn></mrow><annotation encoding="application/x-tex">3</annotation></semantics></math>-ary <a class="existingWikiWord" href="/nlab/show/enriched+functor">enriched functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>:</mo><msup><mi>A</mi> <mo lspace="0em" rspace="thinmathspace">op</mo></msup><mo>⊗</mo><mi>A</mi><mo>⊗</mo><mi>A</mi><mo>→</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">P:A^\op \otimes A \otimes A\to V</annotation></semantics></math>. For notational clarity, we may write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(a,b,c)</annotation></semantics></math> as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>⋄</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(a,b \diamond c)</annotation></semantics></math>.</p> </li> <li> <p>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>-functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi><mo>:</mo><msup><mi>A</mi> <mi>op</mi></msup><mo>→</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">J:A^{op}\to V</annotation></semantics></math>.</p> </li> </ol> <p>and <a class="existingWikiWord" href="/nlab/show/enriched+natural+isomorphisms">enriched natural isomorphisms</a></p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>λ</mi> <mi>ab</mi></msub><mo>:</mo><msup><mo>∫</mo> <mi>x</mi></msup><mo stretchy="false">(</mo><mi>J</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>P</mi><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>a</mi><mo>⋄</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><mi>A</mi><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\lambda_{ab}:\int^x (J(x) \otimes P(b,a \diamond x))\to A(b,a)</annotation></semantics></math></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ρ</mi> <mi>ab</mi></msub><mo>:</mo><msup><mo>∫</mo> <mi>x</mi></msup><mo stretchy="false">(</mo><mi>J</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>P</mi><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>x</mi><mo>⋄</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><mi>A</mi><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\rho_{ab}: \int^x ( J(x)\otimes P(b,x \diamond a))\to A(b,a)</annotation></semantics></math></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>α</mi> <mi>abcd</mi></msub><mo>:</mo><msup><mo>∫</mo> <mi>x</mi></msup><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>a</mi><mo>⋄</mo><mi>b</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>P</mi><mo stretchy="false">(</mo><mi>d</mi><mo>,</mo><mi>x</mi><mo>⋄</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><msup><mo>∫</mo> <mi>x</mi></msup><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>b</mi><mo>⋄</mo><mi>c</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>P</mi><mo stretchy="false">(</mo><mi>d</mi><mo>,</mo><mi>a</mi><mo>⋄</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\alpha_{abcd}: \int^x (P(x,a\diamond b)\otimes P(d,x\diamond c)) \to \int^x(P(x,b\diamond c)\otimes P(d,a\diamond x))</annotation></semantics></math></p> </li> </ol> <p>satisfying the pentagon and unit axioms for promonoidal categories. Explicitly, writting <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>P</mi> <mrow><mi>B</mi><mo>,</mo><mi>C</mi></mrow> <mi>A</mi></msubsup></mrow><annotation encoding="application/x-tex">P^{A}_{B,C}</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>;</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(A,B;C)</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mstyle mathvariant="sans-serif"><mi>h</mi></mstyle> <mi>B</mi> <mi>A</mi></msubsup></mrow><annotation encoding="application/x-tex">\mathsf{h}^{A}_{B}</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi mathvariant="normal">Hom</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{Hom}_{A}(A,B)</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>J</mi> <mi>X</mi></msup></mrow><annotation encoding="application/x-tex">J^X</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">J(X)</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⋄</mo></mrow><annotation encoding="application/x-tex">\diamond</annotation></semantics></math> for composition of profunctors, we require the following conditions to hold:</p> <ol> <li> <p><strong>The triangle identity for promonoidal categories.</strong> For each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>,</mo><mi>B</mi><mo>,</mo><mi>C</mi><mo>∈</mo><mi mathvariant="normal">Obj</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A,B,C\in\mathrm{Obj}(A)</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a></p> <p><img src="/nlab/files/pro-triangle-corrected.svg" /></p> <p><a class="existingWikiWord" href="/nlab/show/commuting+diagram">commutes</a>;</p> </li> <li> <p><strong>The pentagon identity for promonoidal categories.</strong> For each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>,</mo><mi>B</mi><mo>,</mo><mi>C</mi><mo>,</mo><mi>D</mi><mo>,</mo><mi>E</mi><mo>∈</mo><mi mathvariant="normal">Obj</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A,B,C,D,E\in\mathrm{Obj}(A)</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a></p> <p><img src="/nlab/files/pro-pentagon.svg" /></p> <p><a class="existingWikiWord" href="/nlab/show/commuting+diagram">commutes</a>.</p> </li> </ol> <h2 id="properties">Properties</h2> <h3 id="versus_monoidal_categories">Versus monoidal categories</h3> <p>Since any functor induces a representable profunctor, any monoidal category can be regarded as a promonoidal category. A given promonoidal category arises in this way if and only if the profunctors <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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math> are representable.</p> <h3 id="day_convolution">Day convolution</h3> <p>A promonoidal structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> suffices to induce a monoidal structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>V</mi> <mrow><msup><mi>A</mi> <mi>op</mi></msup></mrow></msup></mrow><annotation encoding="application/x-tex">V^{A^{op}}</annotation></semantics></math> by <a class="existingWikiWord" href="/nlab/show/Day+convolution">Day convolution</a>. In fact, given a small <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>-category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>, there is an equivalence of categories between</p> <ol> <li> <p>the category of pro-monoidal structures on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>, with strong pro-monoidal functors between them, and</p> </li> <li> <p>the category of biclosed monoidal structures on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>V</mi> <mrow><msup><mi>A</mi> <mi>op</mi></msup></mrow></msup></mrow><annotation encoding="application/x-tex">V^{A^{op}}</annotation></semantics></math>, with strong monoidal functors between them.</p> </li> </ol> <h3 id="versus_multicategories">Versus multicategories</h3> <p>A promonoidal structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> can be identified with a particular sort of <a class="existingWikiWord" href="/nlab/show/multicategory">multicategory</a> structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>A</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">A^{op}</annotation></semantics></math>, i.e. with a co-multicategory structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>. The set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(x, y, z)</annotation></semantics></math> is regarded as the set of co-multimorphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>→</mo><mo stretchy="false">(</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x \to (y,z)</annotation></semantics></math>.</p> <p>More generally, we define a co-multicategory <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>A</mi><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">\bar A</annotation></semantics></math> as follows. The objects of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>A</mi><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">\bar A</annotation></semantics></math> are the objects of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>. The co-multimorphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>→</mo><msub><mi>a</mi> <mn>1</mn></msub><mi>…</mi><msub><mi>a</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">b\to a_1\dots a_n</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>A</mi><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">\bar A</annotation></semantics></math> are defined by induction on <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> as follows: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>A</mi><mo stretchy="false">¯</mo></mover><mo stretchy="false">(</mo><mi>b</mi><mo>;</mo><mo stretchy="false">)</mo><mo>=</mo><mi>Jb</mi></mrow><annotation encoding="application/x-tex">\bar A(b;)=Jb</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>A</mi><mo stretchy="false">¯</mo></mover><mo stretchy="false">(</mo><mi>b</mi><mo>;</mo><msub><mi>a</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>a</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><msup><mo>∫</mo> <mi>x</mi></msup><mover><mi>A</mi><mo stretchy="false">¯</mo></mover><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><msub><mi>a</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>a</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>⊗</mo><mi>P</mi><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>x</mi><mo>⋄</mo><msub><mi>a</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\bar A(b;a_1,\dots,a_{n+1})=\int^x\bar A(x;a_1,\dots,a_n)\otimes P(b,x\diamond a_{n+1})</annotation></semantics></math>.</p> <p>Not every co-multicategory arises from a promonoidal one in this way. Roughly, a promonoidal category is a co-multicategory whose <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>-ary co-multimorphisms are determined by the binary, unary, and nullary morphisms. In general, co-multicategories can be identified with a certain sort of “lax promonoidal category”.</p> <p>In fact, promonoidal categories correspond exactly to the <span class="newWikiWord">exponentiable<a href="/nlab/new/exponentiable">?</a></span> multicategories: see <a class="existingWikiWord" href="/nlab/show/multicategory">multicategory</a> for more information.</p> <h2 id="notes">Notes</h2> <p><a class="existingWikiWord" href="/nlab/show/Brian+Day">Brian Day</a> introduced the notion of a “premonoidal” category in <a href="#Day70">(Day 1970)</a>, and later renamed this to a “promonoidal” category in <a href="#Day74">(Day 1974)</a> while reformulating the identity and associativity isomorphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi><mo>,</mo><mi>ρ</mi><mo>,</mo><mi>α</mi></mrow><annotation encoding="application/x-tex">\lambda,\rho,\alpha</annotation></semantics></math> explicitly in terms of profunctor composition. However, note that his definition is op’d from the definition used in this article, in the sense that a Day-promonoidal structure on a category <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> corresponds to a pseudomonoid structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">C^{op}</annotation></semantics></math> in <a class="existingWikiWord" href="/nlab/show/Prof">Prof</a>. In particular, one example Day considers is that of a <a class="existingWikiWord" href="/nlab/show/closed+category">closed category</a>, which is actually a co-promonoidal category in the sense used here (analogous to the co-promonoidal structure on a multicategory described above).</p> <p>Regarding monoidal categories as promonoidal is useful in order to express extra structure on them, such as <a class="existingWikiWord" href="/nlab/show/closed+monoidal+category">closedness</a>, <a class="existingWikiWord" href="/nlab/show/star-autonomous+category"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mo>*</mo> </mrow> <annotation encoding="application/x-tex">\ast</annotation> </semantics> </math>-autonomy</a>, or <a class="existingWikiWord" href="/nlab/show/compact+closed+category">compact closedness</a>, in abstract bicategorical terms: these notions can be defined by adding extra structure to a <a class="existingWikiWord" href="/nlab/show/pseudomonoid">pseudomonoid</a> in the <a class="existingWikiWord" href="/nlab/show/monoidal+bicategory">monoidal bicategory</a> <a class="existingWikiWord" href="/nlab/show/Prof">Prof</a> (i.e. a promonoidal category), but the extra structure does not lie inside the sub-monoidal bicategory <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a>.</p> <h2 id="related_pages">Related pages</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a>, <a class="existingWikiWord" href="/nlab/show/multicategory">multicategory</a></li> <li><a class="existingWikiWord" href="/nlab/show/Day+convolution">Day convolution</a></li> <li>A (co-)promonoidal <a class="existingWikiWord" href="/nlab/show/poset">poset</a> is called a <a class="existingWikiWord" href="/nlab/show/ternary+frame">ternary frame</a>, when its Day convolution monoidal category is used to model <a class="existingWikiWord" href="/nlab/show/substructural+logic">substructural logic</a>.</li> </ul> <h2 id="references">References</h2> <ul> <li id="Day70"> <p><a class="existingWikiWord" href="/nlab/show/Brian+Day">Brian Day</a>, On closed categories of functors, <em>Lecture Notes in Mathematics</em> 137 (1970), 1-38.</p> </li> <li id="Day74"> <p><a class="existingWikiWord" href="/nlab/show/Brian+Day">Brian Day</a>, An embedding theorem for closed categories, <em>Lecture Notes in Mathematics</em> 420 (1974), 55-64.</p> </li> <li> <p>Day, Panchadcharam and Street, <em>On centres and lax centres for promonoidal categories</em>.</p> </li> </ul> <p>The relationship between <a class="existingWikiWord" href="/nlab/show/multicategories">multicategories</a>, <a class="existingWikiWord" href="/nlab/show/promonoidal+categories">promonoidal categories</a>, <a class="existingWikiWord" href="/nlab/show/lax+monoidal+categories">lax monoidal categories</a>, and <a class="existingWikiWord" href="/nlab/show/monoidal+categories">monoidal categories</a> is exposited in:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Brian+Day">Brian Day</a> and <a class="existingWikiWord" href="/nlab/show/Ross+Street">Ross Street</a>, <em>Lax monoids, pseudo-operads, and convolution</em>, Contemporary Mathematics 318 (2003): 75-96.</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on September 7, 2024 at 10:29:24. 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