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In algebra, an '''action of a [[monoidal category]]''' <math>(S, \otimes, e)</math> on a category ''X'' is a [[funct ...eory)|monoid object]] in [[Category of small categories|''Cat'']] with the monoidal product being the [[Product category|category product]]. This means that '' ...

...y''', and any finite coproduct category can be thought of as a cocartesian monoidal category. ...ed categories]].<ref>{{nlab|id=cartesian+monoidal+category|title=Cartesian monoidal category}}</ref> ...

...ory''' or '''2-rig''') is a category equipped with two [[monoidal category|monoidal structures]], one distributing over the other. * a [[symmetric monoidal category|symmetric monoidal structure]] <math>(\mathbf C, \oplus, O)</math> ...

...athcal{E}_n</math>-algebra''' in a [[symmetric monoidal category|symmetric monoidal]] [[infinity category]] ''C'' consists of the following data: ...''n'' = 1, a [[braided monoidal category]] if ''n'' = 2, and a [[symmetric monoidal category]] if ''n'' ≥ 3. ...

...neral context of [[Enriched category|enriched]] [[functor category|functor categories]]. ...math>\mathrm{Hom}(\mathbf{C},\mathbf{D})</math> for two symmetric monoidal categories <math>\mathbf{C}, \mathbf{D}</math>. ...

...[Glossary of category theory#simple object|simple objects]], such that the monoidal unit is simple. If the [[ground field]] <math>k</math> is [[Algebraically c ...ikshych|first2=Dmitri|last3=Ostrik|first3=Viktor|date=2005|title=On Fusion Categories|url=https://annals.math.princeton.edu/2005/162-2/p01|journal=Annals of Math ...

...r product are called "[[monoidal categories]]". Special types of monoidal categories exist with interesting properties.{{Disambiguation|math}} ...

...by Neantro Saavedra Rivano in his thesis on [[Tannakian category|Tannakian categories]].<ref>{{cite book |first=N. Saavedra |last=Rivano |title=Catégories Tannak *An object ''X'' of a monoidal category is called left rigid if there is an object ''Y'' and morphisms <ma ...

...4-3975}}</ref>''' is a generalisation of a [[monoidal category]] where the monoidal product need not be a [[bifunctor]], but only to be functorial in its two ...=Mario |date=2023-08-07 |title=Promonads and String Diagrams for Effectful Categories |journal=Electronic Proceedings in Theoretical Computer Science |volume=380 ...

...; namely, every [[monoidal category]] is monoidally equivalent to a strict monoidal category.<ref>{{harvnb|Schauenburg|2001}}</ref> === Coherence condition (Monoidal category) === ...

The trace is defined in the context of a [[symmetric monoidal category]] ''C'', i.e., a [[category (mathematics)|category]] equipped with where 1 is the monoidal unit and the extremal morphisms are the coevaluation and evaluation, which ...

...semisimple finite weak Hopf algebra is a [[fusion category]] (which is a [[monoidal category]] with extra properties). It was also shown by Etingof, Nikshych a ...Note that we also implicitly use [[Mac Lane]]'s coherence theorem for the monoidal category of vector spaces, identifying <math>(U \otimes V) \otimes W \cong ...

...here is a bijection between the <math>\mathcal {C}</math>-valued symmetric monoidal functors of the cobordism category and the objects of <math>\mathcal {C}</m ...er words, the bijection between <math>\mathcal {C}</math>-valued symmetric monoidal functors and the objects of <math>\mathcal {C}</math> is uniquely defined b ...

...', also called a '''tortile category''', is a particular type of [[braided monoidal category]]. A [[monoidal category]] <math>\mathcal C</math> is, loosely speaking, a [[category (math ...

...braided vector spaces entire [[braided monoidal category|braided monoidal categories]] with braidings between any objects <math>\tau_{V,W}</math>, most importa ...

...ced by [[Hoàng Xuân Sính]] in the late 1960s under the name {{nowrap|'''gr-categories'''}},<ref>{{Citation | last1=Hoàng | first1=Xuân Sính | authorlink1=Hoàng X A 2-group is a [[monoidal category]] ''G'' in which every [[morphism]] is [[isomorphism|invertible]] ...

...tation |last=Rowell |first=Eric |title=On classification of modular tensor categories |date=2009-11-09 |url=https://arxiv.org/abs/0712.1377 |access-date=2025-03- ...nsor|tensors]]. The [[Coherence condition|coherence conditions]] on fusion categories turn into compatibility conditions on the tensors. ...

...math>-ring spectrum]], is a [[commutative monoid]] in a good<ref>symmetric monoidal with respect to [[smash product]] and perhaps some other conditions; one ch Almost all reasonable categories of commutative ring spectra can be shown to be [[Quillen equivalent]] to ea ...

...groups#Representations, modules and the convolution algebra|Isomorphism of categories#Examples}} There is an [[isomorphism of categories]] between the category of representations of a group {{var|G}} over a field ...

...widely used approaches today employ the language of [[model category|model categories]].{{fact|date=January 2019}} ...y structured ring spectra as monoids in a suitable category of spectra are categories of diagram spectra. Probably the most famous one of these is the category o ...