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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> <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> <ul> <li><a href='#the_monoidal_2category_of_enriched_categories'>The monoidal 2-category of enriched categories</a></li> <li><a href='#enriched_monoidal_categories'>Enriched monoidal categories</a></li> </ul> <li><a href='#examples'>Examples</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>The notion of <em>enriched monoidal categories</em> is a compatible combination of the notions of <em><a class="existingWikiWord" href="/nlab/show/enriched+categories">enriched categories</a></em> and <em><a class="existingWikiWord" href="/nlab/show/monoidal+categories">monoidal categories</a></em>. The main point is that the <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a>-<a class="existingWikiWord" href="/nlab/show/functor">functor</a> on the <a class="existingWikiWord" href="/nlab/show/underlying">underlying</a> <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> is properly an <a class="existingWikiWord" href="/nlab/show/enriched+functor">enriched functor</a> with respect to the <a class="existingWikiWord" href="/nlab/show/underlying">underlying</a> <a class="existingWikiWord" href="/nlab/show/enriched+category">enriched category</a>.</p> <p>Special cases include <a class="existingWikiWord" href="/nlab/show/tensor+categories">tensor categories</a>, which are (<a class="existingWikiWord" href="/nlab/show/Vect">Vect</a>,<a class="existingWikiWord" href="/nlab/show/tensor+product+of+vector+spaces"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mo>⊗</mo> </mrow> <annotation encoding="application/x-tex">\otimes</annotation> </semantics> </math></a>)-enriched monoidal categories.</p> <h2 id="definition">Definition</h2> <h3 id="the_monoidal_2category_of_enriched_categories">The monoidal 2-category of enriched categories</h3> <p>For <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 <em><a class="existingWikiWord" href="/nlab/show/braided+monoidal+category">symmetric monoidal</a></em> <a class="existingWikiWord" href="/nlab/show/cosmos">cosmos</a>, the <a class="existingWikiWord" href="/nlab/show/2-category">2-category</a> <a class="existingWikiWord" href="/nlab/show/VCat">VCat</a> of <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+categories">enriched categories</a> becomes a <a class="existingWikiWord" href="/nlab/show/monoidal+2-category">monoidal 2-category</a> by declaring the <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>C</mi></mstyle><mo>⊗</mo><mstyle mathvariant="bold"><mi>D</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{C} \otimes \mathbf{D}</annotation></semantics></math> of a <a class="existingWikiWord" href="/nlab/show/pair">pair</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>C</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{C}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>D</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{D}</annotation></semantics></math> of <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+categories">enriched categories</a> to have</p> <ul> <li> <p>as <a class="existingWikiWord" href="/nlab/show/set">set</a> of <a class="existingWikiWord" href="/nlab/show/objects">objects</a> the <a class="existingWikiWord" href="/nlab/show/cartesian+product">cartesian product</a> of the given sets of objects:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Obj</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>C</mi></mstyle><mo>⊗</mo><mstyle mathvariant="bold"><mi>D</mi></mstyle><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mi>Obj</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>C</mi></mstyle><mo stretchy="false">)</mo><mo>×</mo><mi>Obj</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>D</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Obj(\mathbf{C} \otimes \mathbf{D}) \;\coloneqq\; Obj(\mathbf{C}) \times Obj(\mathbf{D}) </annotation></semantics></math></div></li> <li> <p>as <a class="existingWikiWord" href="/nlab/show/hom-objects">hom-objects</a> the <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> in <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> between the given <a class="existingWikiWord" href="/nlab/show/hom-objects">hom-objects</a>:</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><mstyle mathvariant="bold"><mi>C</mi></mstyle><mo>⊗</mo><mstyle mathvariant="bold"><mi>D</mi></mstyle><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">(</mo><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">(</mo><mi>c</mi><mo>′</mo><mo>,</mo><mi>d</mi><mo>′</mo><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mstyle mathvariant="bold"><mi>C</mi></mstyle><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>c</mi><mo>′</mo><mo stretchy="false">)</mo><mo>⊗</mo><mstyle mathvariant="bold"><mi>D</mi></mstyle><mo stretchy="false">(</mo><mi>d</mi><mo>,</mo><mi>d</mi><mo>′</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (\mathbf{C} \otimes \mathbf{D})\big((c,d), (c',d')\big) \;\coloneqq\; \mathbf{C}(c,c') \otimes \mathbf{D}(d,d') \,. </annotation></semantics></math></div></li> <li> <p><a class="existingWikiWord" href="/nlab/show/composition">composition</a> obtained by the given composition operations, after using the <a class="existingWikiWord" href="/nlab/show/braiding">braiding</a> in <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> to align factors:</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><mstyle mathvariant="bold"><mi>C</mi></mstyle><mo>⊗</mo><mstyle mathvariant="bold"><mi>D</mi></mstyle><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">(</mo><mo stretchy="false">(</mo><mi>c</mi><mo>′</mo><mo>,</mo><mi>d</mi><mo>′</mo><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">(</mo><mi>c</mi><mo>″</mo><mo>,</mo><mi>d</mi><mo>″</mo><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>⊗</mo><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>C</mi></mstyle><mo>⊗</mo><mstyle mathvariant="bold"><mi>D</mi></mstyle><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">(</mo><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">(</mo><mi>c</mi><mo>′</mo><mo>,</mo><mi>d</mi><mo>′</mo><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mo>∘</mo> <mrow><mstyle mathvariant="bold"><mi>C</mi></mstyle><mo>⊗</mo><mstyle mathvariant="bold"><mi>D</mi></mstyle></mrow></msub></mrow></mover></mtd> <mtd><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>C</mi></mstyle><mo>⊗</mo><mstyle mathvariant="bold"><mi>D</mi></mstyle><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">(</mo><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">(</mo><mi>c</mi><mo>″</mo><mo>,</mo><mi>d</mi><mo>″</mo><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo></mtd></mtr> <mtr><mtd><mpadded width="0" lspace="-100%width"><mrow><msup><mrow></mrow> <mo>≃</mo></msup></mrow></mpadded><mo maxsize="1.8em" minsize="1.8em">↓</mo><mphantom><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>−</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>−</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>−</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>−</mo><mo lspace="verythinmathspace" rspace="0em">−</mo></mrow></mphantom></mtd> <mtd></mtd> <mtd><mo maxsize="1.8em" minsize="1.8em">↑</mo><mpadded width="0"><mrow><msup><mrow></mrow> <mo>≃</mo></msup></mrow></mpadded></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>C</mi></mstyle><mo stretchy="false">(</mo><mi>c</mi><mo>′</mo><mo>,</mo><mi>c</mi><mo>″</mo><mo stretchy="false">)</mo><mo>⊗</mo><mstyle mathvariant="bold"><mi>D</mi></mstyle><mo stretchy="false">(</mo><mi>d</mi><mo>′</mo><mo>,</mo><mi>d</mi><mo>″</mo><mo stretchy="false">)</mo><mo>⊗</mo><mstyle mathvariant="bold"><mi>C</mi></mstyle><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>c</mi><mo>′</mo><mo stretchy="false">)</mo><mo>⊗</mo><mstyle mathvariant="bold"><mi>D</mi></mstyle><mo stretchy="false">(</mo><mi>d</mi><mo>,</mo><mi>d</mi><mo>′</mo><mo stretchy="false">)</mo><munderover><mo>→</mo><mi>braid</mi><mo>∼</mo></munderover><mstyle mathvariant="bold"><mi>C</mi></mstyle><mo stretchy="false">(</mo><mi>c</mi><mo>′</mo><mo>,</mo><mi>c</mi><mo>″</mo><mo stretchy="false">)</mo><mo>⊗</mo><mstyle mathvariant="bold"><mi>C</mi></mstyle><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>c</mi><mo>′</mo><mo stretchy="false">)</mo><mo>⊗</mo><mstyle mathvariant="bold"><mi>D</mi></mstyle><mo stretchy="false">(</mo><mi>d</mi><mo>′</mo><mo>,</mo><mi>d</mi><mo>″</mo><mo stretchy="false">)</mo><mo>⊗</mo><mstyle mathvariant="bold"><mi>D</mi></mstyle><mo stretchy="false">(</mo><mi>d</mi><mo>,</mo><mi>d</mi><mo>′</mo><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mo>∘</mo> <mstyle mathvariant="bold"><mi>C</mi></mstyle></msub><mo>⊗</mo><msub><mo>∘</mo> <mstyle mathvariant="bold"><mi>D</mi></mstyle></msub></mrow></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>C</mi></mstyle><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>c</mi><mo>″</mo><mo stretchy="false">)</mo><mo>⊗</mo><mstyle mathvariant="bold"><mi>D</mi></mstyle><mo stretchy="false">(</mo><mi>d</mi><mo>,</mo><mi>d</mi><mo>″</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ (\mathbf{C} \otimes \mathbf{D}) \big((c',d'), (c'', d'')\big) \otimes (\mathbf{C} \otimes \mathbf{D}) \big((c,d), (c',d')\big) &\overset {\circ_{\mathbf{C} \otimes \mathbf{D}}}{ \longrightarrow } & (\mathbf{C} \otimes \mathbf{D}) \big((c,d), (c'',d'')\big) \\ \mathllap{{}^\simeq}\Big\downarrow \phantom{---------} && \Big\uparrow\mathrlap{{}^{\simeq}} \\ \mathbf{C}(c',c'') \otimes \mathbf{D}(d',d'') \otimes \mathbf{C}(c,c') \otimes \mathbf{D}(d,d') \underoverset {braid} {\sim} {\to} \mathbf{C}(c',c'') \otimes \mathbf{C}(c,c') \otimes \mathbf{D}(d',d'') \otimes \mathbf{D}(d,d') &\overset{ \circ_{\mathbf{C}} \otimes \circ_{\mathbf{D}} }{\longrightarrow}& \mathbf{C}(c,c'') \otimes \mathbf{D}(d,d'') } </annotation></semantics></math></div></li> </ul> <h3 id="enriched_monoidal_categories">Enriched monoidal categories</h3> <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>-enriched monoidal category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>C</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{C}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/pseudomonoid">pseudomonoid</a> <a class="existingWikiWord" href="/nlab/show/internalization">internal</a> to the above monoidal 2-category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mi>Cat</mi></mrow><annotation encoding="application/x-tex">V Cat</annotation></semantics></math>.</p> <p>This means mainly that</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>C</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{C}</annotation></semantics></math> 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+category">enriched category</a></p> </li> <li> <p>whose <a class="existingWikiWord" href="/nlab/show/underlying">underlying</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> is equipped with <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a>-<a class="existingWikiWord" href="/nlab/show/structure">structure</a>, hence with a <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</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><mo>⊗</mo><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>C</mi><mo>×</mo><mi>C</mi><mo>⟶</mo><mi>C</mi></mrow><annotation encoding="application/x-tex"> \otimes \,\colon\, C \times C \longrightarrow C </annotation></semantics></math></div></li> <li> <p>which is compatibly lifted for each <a class="existingWikiWord" href="/nlab/show/pair">pair</a> of <a class="existingWikiWord" href="/nlab/show/pairs">pairs</a> of <a class="existingWikiWord" href="/nlab/show/objects">objects</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> to a <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> on <a class="existingWikiWord" href="/nlab/show/hom-objects">hom-objects</a> in <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>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mo>⊗</mo> <mstyle mathvariant="bold"><mi>C</mi></mstyle></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">(</mo><mi>c</mi><mo>′</mo><mo>,</mo><mi>d</mi><mo>′</mo><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mstyle mathvariant="bold"><mi>C</mi></mstyle><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>c</mi><mo>′</mo><mo stretchy="false">)</mo><mo>⊗</mo><mstyle mathvariant="bold"><mi>C</mi></mstyle><mo stretchy="false">(</mo><mi>d</mi><mo>,</mo><mi>d</mi><mo>′</mo><mo stretchy="false">)</mo><mo>⟶</mo><mstyle mathvariant="bold"><mi>C</mi></mstyle><mo stretchy="false">(</mo><mi>c</mi><msub><mo>⊗</mo> <mi>C</mi></msub><mi>d</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>c</mi><mo>′</mo><msub><mo>⊗</mo> <mi>C</mi></msub><mi>d</mi><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \otimes_{\mathbf{C}}\big( (c,d),\, (c',d') \big) \,\colon\, \mathbf{C}(c,c') \otimes \mathbf{C}(d,d') \longrightarrow \mathbf{C}(c \otimes_C d ,\, c' \otimes_C d') </annotation></semantics></math></div></li> </ul> <p>in a compatible way.</p> <h2 id="examples">Examples</h2> <p>(…)</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cartesian+closed+enriched+category">cartesian closed enriched category</a>, <a class="existingWikiWord" href="/nlab/show/locally+cartesian+closed+enriched+category">locally cartesian closed enriched category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+category">enriched category</a>, <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+monoidal+model+category">enriched monoidal model category</a>, <a class="existingWikiWord" href="/nlab/show/simplicial+monoidal+model+category">simplicial monoidal model category</a></p> </li> </ul> <h2 id="references">References</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, Section 1.6 in: <em>Derived Algebraic Geometry II: Noncommutative Algebra</em> [<a href="https://arxiv.org/abs/math/0702299">arXiv:math/0702299</a>]</p> <p>which is Def. 4.1.7.7 in <em><a class="existingWikiWord" href="/nlab/show/Higher+Algebra">Higher Algebra</a></em> [<a href="https://www.math.ias.edu/~lurie/papers/HA.pdf">pdf</a>]</p> <blockquote> <p>(focus on <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a>-enrichment for <a class="existingWikiWord" href="/nlab/show/simplicial+monoidal+model+categories">simplicial monoidal model categories</a>)</p> </blockquote> </li> <li id="BataninMarkl12"> <p><a class="existingWikiWord" href="/nlab/show/Michael+Batanin">Michael Batanin</a>, <a class="existingWikiWord" href="/nlab/show/Martin+Markl">Martin Markl</a>, Section 2 of: <em>Centers and homotopy centers in enriched monoidal categories</em>, Advances in Mathematics <strong>230</strong> 4–6 (2012) 1811-1858 [<a href="https://doi.org/10.1016/j.aim.2012.04.011">doi:10.1016/j.aim.2012.04.011</a>, <a href="https://arxiv.org/abs/1109.4084">arXiv:1109.4084</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Michael+Ching">Michael Ching</a>, Def. 1.10 in: <em>Bar constructions for topological operads and the Goodwillie derivatives of the identity</em>, Geom. Topol. <strong>9</strong> (2005) 833-934 [<a href="https://arxiv.org/abs/math/0501429">arXiv:math/0501429</a>, <a href="http://dx.doi.org/10.2140/gt.2005.9.833">doi:10.2140/gt.2005.9.833</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Rune+Haugseng">Rune Haugseng</a>, answer to <em>Definitions of enriched monoidal category</em>, <a href="https://mathoverflow.net/a/315075/381">MO:a/315075</a> (2018)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Scott+Morrison">Scott Morrison</a>, <a class="existingWikiWord" href="/nlab/show/David+Penneys">David Penneys</a>, <em>Monoidal Categories Enriched in Braided Monoidal Categories</em>, International Mathematics Research Notices <strong>2019</strong> 11 June 2019 3527–3579 [<a href="https://doi.org/10.1093/imrn/rnx217">doi:10.1093/imrn/rnx217</a>, <a href="https://arxiv.org/abs/1701.00567">arXiv:1701.00567</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Liang+Kong">Liang Kong</a>, Wei Yuan, Zhi-Hao Zhang, Hao Zheng, <em>Enriched monoidal categories I: centers</em> [<a href="https://arxiv.org/abs/2104.03121">arXiv:2104.03121</a>]</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on June 1, 2024 at 14:38:40. 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