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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>Contents</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> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#relation_to_multicategories'>Relation to multicategories</a></li> <li><a href='#relation_to_pros'>Relation to PROs</a></li> <li><a href='#relationships_between_categories_of_monoidal_categories'>Relationships between categories of monoidal categories</a></li> </ul> <li><a href='#string_diagrams'>String diagrams</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>A <em>monoidal functor</em> is a <a class="existingWikiWord" href="/nlab/show/functor">functor</a> between <a class="existingWikiWord" href="/nlab/show/monoidal+categories">monoidal categories</a> that preserves the monoidal structure: a <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> of monoidal categories.</p> <h2 id="definition">Definition</h2> <div class="num_defn" id="LaxMonoidalFunctor"> <h6 id="definition_2">Definition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><msub><mo>⊗</mo> <mi>𝒞</mi></msub><mo>,</mo><msub><mn>1</mn> <mi>𝒞</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}})</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒟</mi><mo>,</mo><msub><mo>⊗</mo> <mi>𝒟</mi></msub><mo>,</mo><msub><mn>1</mn> <mi>𝒟</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}} )</annotation></semantics></math> be two <a class="existingWikiWord" href="/nlab/show/monoidal+categories">monoidal categories</a>. A <strong>lax monoidal functor</strong> between them is a <a class="existingWikiWord" href="/nlab/show/functor">functor</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>F</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>𝒞</mi><mo>⟶</mo><mi>𝒟</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> F \;\colon\; \mathcal{C} \longrightarrow \mathcal{D} \,, </annotation></semantics></math></div> <p>together with coherence <a class="existingWikiWord" href="/nlab/show/maps">maps</a>:</p> <ol> <li> <p>a <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mn>1</mn> <mi>𝒟</mi></msub><mo>⟶</mo><mi>F</mi><mo stretchy="false">(</mo><msub><mn>1</mn> <mi>𝒞</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \epsilon \;\colon\; 1_{\mathcal{D}} \longrightarrow F(1_{\mathcal{C}}) </annotation></semantics></math></div></li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>μ</mi> <mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mi>𝒟</mi></msub><mi>F</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><msub><mo>⊗</mo> <mi>𝒞</mi></msub><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mu_{x,y} \;\colon\; F(x) \otimes_{\mathcal{D}} F(y) \longrightarrow F(x \otimes_{\mathcal{C}} y) </annotation></semantics></math></div> <p>for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">x,y \in \mathcal{C}</annotation></semantics></math></p> </li> </ol> <p>satisfying the following conditions:</p> <ol> <li> <p><strong>(<a class="existingWikiWord" href="/nlab/show/associativity">associativity</a>)</strong> For all objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">x,y,z \in \mathcal{C}</annotation></semantics></math> the following <a class="existingWikiWord" href="/nlab/show/commuting+diagram">diagram commutes</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mi>𝒟</mi></msub><mi>F</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mi>𝒟</mi></msub><mi>F</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mtd> <mtd><munderover><mo>⟶</mo><mo>≃</mo><mrow><msubsup><mi>a</mi> <mrow><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><mi>F</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>,</mo><mi>F</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow> <mi>𝒟</mi></msubsup></mrow></munderover></mtd> <mtd><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mi>𝒟</mi></msub><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mi>𝒟</mi></msub><mi>F</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>μ</mi> <mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></msub><mo>⊗</mo><mi>id</mi></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>id</mi><mo>⊗</mo><msub><mi>μ</mi> <mrow><mi>y</mi><mo>,</mo><mi>z</mi></mrow></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><msub><mo>⊗</mo> <mi>𝒞</mi></msub><mi>y</mi><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mi>𝒟</mi></msub><mi>F</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mi>𝒟</mi></msub><mi>F</mi><mo stretchy="false">(</mo><mi>y</mi><msub><mo>⊗</mo> <mi>𝒞</mi></msub><mi>z</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>μ</mi> <mrow><mi>x</mi><msub><mo>⊗</mo> <mi>𝒞</mi></msub><mi>y</mi><mo>,</mo><mi>z</mi></mrow></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>μ</mi> <mrow><mi>x</mi><mo>,</mo><mi>y</mi><msub><mo>⊗</mo> <mi>𝒞</mi></msub><mi>z</mi></mrow></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>F</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>x</mi><msub><mo>⊗</mo> <mi>𝒞</mi></msub><mi>y</mi><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mi>𝒞</mi></msub><mi>z</mi><mo stretchy="false">)</mo></mtd> <mtd><munder><mo>⟶</mo><mrow><mi>F</mi><mo stretchy="false">(</mo><msubsup><mi>a</mi> <mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi></mrow> <mi>𝒞</mi></msubsup><mo stretchy="false">)</mo></mrow></munder></mtd> <mtd><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><msub><mo>⊗</mo> <mi>𝒞</mi></msub><mo stretchy="false">(</mo><mi>y</mi><msub><mo>⊗</mo> <mi>𝒞</mi></msub><mi>z</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ (F(x) \otimes_{\mathcal{D}} F(y)) \otimes_{\mathcal{D}} F(z) &\underoverset{\simeq}{a^{\mathcal{D}}_{F(x),F(y),F(z)}}{\longrightarrow}& F(x) \otimes_{\mathcal{D}}( F(y)\otimes_{\mathcal{D}} F(z) ) \\ {}^{\mathllap{\mu_{x,y} \otimes id}}\downarrow && \downarrow^{\mathrlap{id\otimes \mu_{y,z}}} \\ F(x \otimes_{\mathcal{C}} y) \otimes_{\mathcal{D}} F(z) && F(x) \otimes_{\mathcal{D}} F(y \otimes_{\mathcal{C}} z) \\ {}^{\mathllap{\mu_{x \otimes_{\mathcal{C}} y , z} } }\downarrow && \downarrow^{\mathrlap{\mu_{ x, y \otimes_{\mathcal{C}} z }}} \\ F( ( x \otimes_{\mathcal{C}} y ) \otimes_{\mathcal{C}} z ) &\underset{F(a^{\mathcal{C}}_{x,y,z})}{\longrightarrow}& F( x \otimes_{\mathcal{C}} ( y \otimes_{\mathcal{C}} z ) ) } </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>a</mi> <mi>𝒞</mi></msup></mrow><annotation encoding="application/x-tex">a^{\mathcal{C}}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>a</mi> <mi>𝒟</mi></msup></mrow><annotation encoding="application/x-tex">a^{\mathcal{D}}</annotation></semantics></math> denote the <a class="existingWikiWord" href="/nlab/show/associators">associators</a> of the monoidal categories;</p> </li> <li> <p><strong>(<a class="existingWikiWord" href="/nlab/show/unitality">unitality</a>)</strong> For all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">x \in \mathcal{C}</annotation></semantics></math> the following <a class="existingWikiWord" href="/nlab/show/commuting+diagram">diagrams commute</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mn>1</mn> <mi>𝒟</mi></msub><msub><mo>⊗</mo> <mi>𝒟</mi></msub><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>ϵ</mi><mo>⊗</mo><mi>id</mi></mrow></mover></mtd> <mtd><mi>F</mi><mo stretchy="false">(</mo><msub><mn>1</mn> <mi>𝒞</mi></msub><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mi>𝒟</mi></msub><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msubsup><mi>ℓ</mi> <mrow><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow> <mi>𝒟</mi></msubsup></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>μ</mi> <mrow><msub><mn>1</mn> <mi>𝒞</mi></msub><mo>,</mo><mi>x</mi></mrow></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟵</mo><mrow><mi>F</mi><mo stretchy="false">(</mo><msubsup><mi>ℓ</mi> <mi>x</mi> <mi>𝒞</mi></msubsup><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>F</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo>⊗</mo> <mi>𝒞</mi></msub><mi>x</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ 1_{\mathcal{D}} \otimes_{\mathcal{D}} F(x) &\overset{\epsilon \otimes id}{\longrightarrow}& F(1_{\mathcal{C}}) \otimes_{\mathcal{D}} F(x) \\ {}^{\mathllap{\ell^{\mathcal{D}}_{F(x)}}}\downarrow && \downarrow^{\mathrlap{\mu_{1_{\mathcal{C}}, x }}} \\ F(x) &\overset{F(\ell^{\mathcal{C}}_x )}{\longleftarrow}& F(1 \otimes_{\mathcal{C}} x ) } </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mi>𝒟</mi></msub><msub><mn>1</mn> <mi>𝒟</mi></msub></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>id</mi><mo>⊗</mo><mi>ϵ</mi></mrow></mover></mtd> <mtd><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mi>𝒟</mi></msub><mi>F</mi><mo stretchy="false">(</mo><msub><mn>1</mn> <mi>𝒞</mi></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msubsup><mi>r</mi> <mrow><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow> <mi>𝒟</mi></msubsup></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>μ</mi> <mrow><mi>x</mi><mo>,</mo><msub><mn>1</mn> <mi>𝒞</mi></msub></mrow></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟵</mo><mrow><mi>F</mi><mo stretchy="false">(</mo><msubsup><mi>r</mi> <mi>x</mi> <mi>𝒞</mi></msubsup><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><msub><mo>⊗</mo> <mi>𝒞</mi></msub><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ F(x) \otimes_{\mathcal{D}} 1_{\mathcal{D}} &\overset{id \otimes \epsilon }{\longrightarrow}& F(x) \otimes_{\mathcal{D}} F(1_{\mathcal{C}}) \\ {}^{\mathllap{r^{\mathcal{D}}_{F(x)}}}\downarrow && \downarrow^{\mathrlap{\mu_{x, 1_{\mathcal{C}} }}} \\ F(x) &\overset{F(r^{\mathcal{C}}_x )}{\longleftarrow}& F(x \otimes_{\mathcal{C}} 1 ) } </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℓ</mi> <mi>𝒞</mi></msup></mrow><annotation encoding="application/x-tex">\ell^{\mathcal{C}}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℓ</mi> <mi>𝒟</mi></msup></mrow><annotation encoding="application/x-tex">\ell^{\mathcal{D}}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>r</mi> <mi>𝒞</mi></msup></mrow><annotation encoding="application/x-tex">r^{\mathcal{C}}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>r</mi> <mi>𝒟</mi></msup></mrow><annotation encoding="application/x-tex">r^{\mathcal{D}}</annotation></semantics></math> denote the left and right <a class="existingWikiWord" href="/nlab/show/unitors">unitors</a> of the two monoidal categories, respectively.</p> </li> </ol> <p id="StrictMonoidalFunctor"> If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">\epsilon</annotation></semantics></math> and all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>μ</mi> <mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mu_{x,y}</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/isomorphisms">isomorphisms</a>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> is called a <strong>strong monoidal functor</strong>. (Note that ‘strong’ is also sometimes applied to ‘monoidal functor’ to indicate possession of a <a class="existingWikiWord" href="/nlab/show/tensorial+strength">tensorial strength</a>.) If they are even <a class="existingWikiWord" href="/nlab/show/identity+morphisms">identity morphisms</a>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> is called a <strong>strict monoidal functor</strong>.</p> </div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>In the literature often the term “monoidal functor” refers by default to what in def. <a class="maruku-ref" href="#LaxMonoidalFunctor"></a> is called a strong monoidal functor. With that convention then what def. <a class="maruku-ref" href="#LaxMonoidalFunctor"></a> calls a lax monoidal functor is called a <strong>weak monoidal functor</strong>.</p> </div> <div class="num_remark"> <h6 id="remark_2">Remark</h6> <p>Lax monoidal functors are the <a class="existingWikiWord" href="/nlab/show/lax+morphisms">lax morphisms</a> for an appropriate <a class="existingWikiWord" href="/nlab/show/2-monad">2-monad</a>.</p> </div> <div class="num_remark"> <h6 id="definition_3">Definition</h6> <p>An <strong><a class="existingWikiWord" href="/nlab/show/oplax+monoidal+functor">oplax monoidal functor</a></strong> (with various alternative names including <strong>comonoidal</strong>), is a monoidal functor from the <a class="existingWikiWord" href="/nlab/show/opposite+categories">opposite categories</a> <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> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>D</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">D^{op}</annotation></semantics></math>.</p> </div> <div class="num_remark"> <h6 id="definition_4">Definition</h6> <p>A <em><a class="existingWikiWord" href="/nlab/show/monoidal+transformation">monoidal transformation</a></em> between monoidal functors is a <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a> that respects the extra structure in an obvious way.</p> </div> <h2 id="properties">Properties</h2> <div class="num_prop" id="MonoidsToMonoidsByLaxMonoidal"> <h6 id="proposition">Proposition</h6> <p><strong>(Lax monoidal functors send <a class="existingWikiWord" href="/nlab/show/monoid">monoid</a>s to monoids)</strong></p> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mo>⊗</mo><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><mi>D</mi><mo>,</mo><mo>⊗</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F : (C,\otimes) \to (D,\otimes)</annotation></semantics></math> is a lax monoidal functor and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>∈</mo><mi>C</mi><mo>,</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msub><mi>μ</mi> <mi>A</mi></msub><mo>:</mo><mi>A</mi><mo>⊗</mo><mi>A</mi><mo>→</mo><mi>A</mi><mo>,</mo><mspace width="thickmathspace"></mspace><msub><mi>i</mi> <mi>A</mi></msub><mo>:</mo><mi>I</mi><mo>→</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (A \in C,\;\; \mu_A : A \otimes A \to A, \; i_A : I \to A) </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/monoid+object">monoid object</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>, then the object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(A)</annotation></semantics></math> is naturally equipped with the structure of a monoid in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> by setting</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mrow><mi>F</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow></msub><mo>:</mo><msub><mi>I</mi> <mi>D</mi></msub><mover><mo>→</mo><mrow></mrow></mover><mi>F</mi><mo stretchy="false">(</mo><msub><mi>I</mi> <mi>C</mi></msub><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><mi>F</mi><mo stretchy="false">(</mo><msub><mi>i</mi> <mi>A</mi></msub><mo stretchy="false">)</mo></mrow></mover><mi>F</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> i_{F(A)} : I_D \stackrel{}{\to} F(I_C) \stackrel{F(i_A)}{\to} F(A) </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>μ</mi> <mrow><mi>F</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow></msub><mo>:</mo><mi>F</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>F</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msub><mo>∇</mo> <mrow><mi>F</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>,</mo><mi>F</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow></msub></mrow></mover><mi>F</mi><mo stretchy="false">(</mo><mi>A</mi><mo>⊗</mo><mi>A</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><mi>F</mi><mo stretchy="false">(</mo><msub><mi>μ</mi> <mi>A</mi></msub><mo stretchy="false">)</mo></mrow></mover><mi>F</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mu_{F(A)} : F(A) \otimes F(A) \stackrel{\nabla_{F(A), F(A)}}{\to} F(A \otimes A) \stackrel{F(\mu_A)}{\to} F(A) \,. </annotation></semantics></math></div> <p>This construction defines a functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Mon</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>:</mo><mi>Mon</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Mon</mi><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Mon(f) : Mon(C) \to Mon(D) </annotation></semantics></math></div> <p>between the <a class="existingWikiWord" href="/nlab/show/categories+of+monoids">categories of monoids</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>, respectively.</p> </div> <p>More generally, lax functors send <a class="existingWikiWord" href="/nlab/show/enriched+categories">enriched categories</a> to enriched categories, an operation known as <a class="existingWikiWord" href="/nlab/show/change+of+enriching+category">change of enriching category</a>. See there for more details.</p> <p>Similarly:</p> <div class="un_prop" id="OplaxSendsComonoidsToComonoids"> <h6 id="proposition_2">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/oplax+monoidal+functors">oplax monoidal functors</a> sends <a class="existingWikiWord" href="/nlab/show/comonoids">comonoids</a> to comonoids)</strong></p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mo>⊗</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(C,\otimes)</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>C</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}C</annotation></semantics></math> for the corresponding <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> <a class="existingWikiWord" href="/nlab/show/2-category">2-category</a>.</p> <p>Lax monoidal functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">f : C \to D</annotation></semantics></math> correspond to <a class="existingWikiWord" href="/nlab/show/lax+2-functor">lax 2-functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>F</mi><mo>:</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>C</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>D</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{B}F : \mathbf{B}C \to \mathbf{B}D \,. </annotation></semantics></math></div> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> is strong monoidal then this is an ordinary <a class="existingWikiWord" href="/nlab/show/2-functor">2-functor</a>. If it is strict monoidal, then this is a <a class="existingWikiWord" href="/nlab/show/strict+2-functor">strict 2-functor</a>.</p> </div> <h3 id="relation_to_multicategories">Relation to multicategories</h3> <div class="num_remark"> <h6 id="remark_3">Remark</h6> <p>Lax monoidal functors between monoidal categories are in correspondence with <a class="existingWikiWord" href="/nlab/show/morphism+of+multicategories">morphisms</a> between their underlying (<a class="existingWikiWord" href="/nlab/show/representable+multicategories">representable</a>) <a class="existingWikiWord" href="/nlab/show/multicategories">multicategories</a>.</p> </div> <h3 id="relation_to_pros">Relation to PROs</h3> <div class="num_remark"> <h6 id="remark_4">Remark</h6> <p>Strong monoidal functors between monoidal categories are in correspondence with morphisms between their underlying (representable) colored <a class="existingWikiWord" href="/nlab/show/PROs">PROs</a>.</p> </div> <div class="num_remark"> <h6 id="remark_5">Remark</h6> <p>Strict monoidal functors between monoidal categories are in correspondence with morphisms between their underlying colored <a class="existingWikiWord" href="/nlab/show/PROs">PROs</a> that preserve the distinguished isomorphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo stretchy="false">)</mo><mover><mo>→</mo><mo>∼</mo></mover><mi>I</mi></mrow><annotation encoding="application/x-tex">() \xrightarrow{\sim} I</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mo>∼</mo></mover><mo stretchy="false">(</mo><mi>A</mi><mo>⊗</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A, B) \xrightarrow{\sim} (A \otimes B)</annotation></semantics></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>,</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A, B</annotation></semantics></math>.</p> </div> <h3 id="relationships_between_categories_of_monoidal_categories">Relationships between categories of monoidal categories</h3> <div class="num_prop"> <h6 id="proposition_3">Proposition</h6> <p>The 1-category of strict monoidal categories and strict monoidal functors is not equivalent to the 1-category of monoidal categories and strong monoidal functors.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>The former has an initial object, whereas the latter does not.</p> </div> <div class="num_prop"> <h6 id="proposition_4">Proposition</h6> <p>The inclusion from the 1-category of strict monoidal categories and strong monoidal functors into the 1-category of monoidal categories and strong monoidal functors is not an equivalence.</p> </div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>As mentioned at <a class="existingWikiWord" href="/nlab/show/monoidal+category#the_2category_of_monoidal_categories">monoidal category</a>, not every skeletal monoidal category is monoidally equivalent to a strict skeletal monoidal category. Therefore the inclusion is not essentially surjective.</p> </div> <div class="num_prop"> <h6 id="proposition_5">Proposition</h6> <p>The inclusion from the 2-category of strict monoidal categories and strict monoidal functors into the 2-category of monoidal categories and strong monoidal functors is not an equivalence.</p> </div> <div class="proof"> <h6 id="proof_3">Proof</h6> <p>Not every strong monoidal functor between strict monoidal categories is equivalent to a strict one. See for example <a href="https://mathoverflow.net/questions/172815/strictifying-strong-monoidal-functors">this MathOverflow question</a>.</p> </div> <div class="num_prop"> <h6 id="proposition_6">Proposition</h6> <p>The inclusion of the the 2-category of strict monoidal categories and strong monoidal functors into the 2-category of monoidal categories and strong monoidal functors is an equivalence.</p> </div> <div class="proof"> <h6 id="proof_4">Proof</h6> <p>By the <a class="existingWikiWord" href="/nlab/show/coherence+theorem+for+monoidal+categories">coherence theorem for monoidal categories</a>, every monoidal category is strong monoidally equivalent to a strict one.</p> </div> <h2 id="string_diagrams">String diagrams</h2> <p>Just like monoidal categories, monoidal functors have a <a class="existingWikiWord" href="/nlab/show/string+diagram">string diagram</a> calculus; see <a href="https://web.archive.org/web/20191021024946/http://web.science.mq.edu.au/~mmccurdy/cms2010talk.pdf">these slides</a> for some examples.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><strong>monoidal functor</strong>, <strong>strong monoidal functor</strong></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/multifunctor">multifunctor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/module+over+a+monoidal+functor">module over a monoidal functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+adjunction">monoidal adjunction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/indexed+monoidal+category">indexed monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cartesian+functor">cartesian functor</a></p> </li> <li> <p><strong>lax monoidal functor</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/functor+with+smash+products">functor with smash products</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/oplax+monoidal+functor">oplax monoidal functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bilax+monoidal+functor">bilax monoidal functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Frobenius+monoidal+functor">Frobenius monoidal functor</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> <li> <p><a class="existingWikiWord" href="/nlab/show/idempotent+monoidal+functor">idempotent monoidal functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+%28%E2%88%9E%2C1%29-functor">monoidal (∞,1)-functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+%28%E2%88%9E%2Cn%29-functor">monoidal (∞,n)-functor</a></p> </li> </ul> <h2 id="references">References</h2> <ul> <li id="EK65"> <p><a class="existingWikiWord" href="/nlab/show/Samuel+Eilenberg">Samuel Eilenberg</a>, <a class="existingWikiWord" href="/nlab/show/G.+Max+Kelly">G. Max Kelly</a>, p. 473 in: <em>Closed Categories</em>, in: <a class="existingWikiWord" href="/nlab/show/Samuel+Eilenberg">S. Eilenberg</a>, <a class="existingWikiWord" href="/nlab/show/D.+K.+Harrison">D. K. Harrison</a>, <a class="existingWikiWord" href="/nlab/show/S.+MacLane">S. MacLane</a>, <a class="existingWikiWord" href="/nlab/show/H.+R%C3%B6hrl">H. Röhrl</a> (eds.): <em><a class="existingWikiWord" href="/nlab/show/Proceedings+of+the+Conference+on+Categorical+Algebra+-+La+Jolla+1965">Proceedings of the Conference on Categorical Algebra - La Jolla 1965</a></em>, Springer (1966) 421-562 [<a href="https://doi.org/10.1007/978-3-642-99902-4">doi:10.1007/978-3-642-99902-4</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Saunders+MacLane">Saunders MacLane</a>, §XI.2 of: <em><a class="existingWikiWord" href="/nlab/show/Categories+for+the+Working+Mathematician">Categories for the Working Mathematician</a></em>, Graduate Texts in Mathematics <strong>5</strong> Springer (second ed. 1997) [<a href="https://link.springer.com/book/10.1007/978-1-4757-4721-8">doi:10.1007/978-1-4757-4721-8</a>]</p> </li> <li id="EGNO15"> <p><a class="existingWikiWord" href="/nlab/show/Pavel+Etingof">Pavel Etingof</a>, <a class="existingWikiWord" href="/nlab/show/Shlomo+Gelaki">Shlomo Gelaki</a>, <a class="existingWikiWord" href="/nlab/show/Dmitri+Nikshych">Dmitri Nikshych</a>, <a class="existingWikiWord" href="/nlab/show/Victor+Ostrik">Victor Ostrik</a>, §2.4 in: <em>Tensor Categories</em>, AMS Mathematical Surveys and Monographs <strong>205</strong> (2015) [<a href="https://bookstore.ams.org/surv-205">ISBN:978-1-4704-3441-0</a>, <a href="http://www-math.mit.edu/~etingof/egnobookfinal.pdf">pdf</a>]</p> <blockquote> <p>(discussed what we call <em><a class="existingWikiWord" href="/nlab/show/strong+monoidal+functors">strong monoidal functors</a></em>)</p> </blockquote> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Marcelo+Aguiar">Marcelo Aguiar</a> and Swapneel Mahajan, <em>Monoidal functors, species and Hopf algebras</em>. (<a href="http://pi.math.cornell.edu/~maguiar/a.pdf">pdf</a>)</p> </li> </ul> <p>Exposition of basics of <a class="existingWikiWord" href="/nlab/show/monoidal+categories">monoidal categories</a> and <a class="existingWikiWord" href="/nlab/show/categorical+algebra">categorical algebra</a>:</p> <ul> <li><em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+categories+and+toposes">geometry of physics – categories and toposes</a></em>, Section 2: <em><a href="geometry+of+physics+--+categories+and+toposes#BasicNotionsOfCategoricalAlgebra">Basic notions of categorical algebra</a></em></li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on March 27, 2024 at 05:24:40. 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