In category theory, a branch of mathematics, a symmetric monoidal category is a monoidal category (i.e. a category in which a "tensor product" ⊗ {\displaystyle...

In mathematics, a monoidal category (or tensor category) is a category C {\displaystyle \mathbf {C} } equipped with a bifunctor ⊗ : C × C → C {\displaystyle...

In the mathematical field of category theory, a dagger symmetric monoidal category is a monoidal category ⟨ C , ⊗ , I ⟩ {\displaystyle \langle \mathbf...

categories are symmetric. However, this need not always be the case, as non-symmetric monoidal categories can be encountered in category-theoretic formulations...

mathematics, a commutativity constraint γ {\displaystyle \gamma } on a monoidal category C {\displaystyle {\mathcal {C}}} is a choice of isomorphism γ A ,...

(i.e., making the category symmetric monoidal or even symmetric closed monoidal, respectively).[citation needed] Enriched category theory thus encompasses...

category theory, a traced monoidal category is a category with some extra structure which gives a reasonable notion of feedback. A traced symmetric monoidal...

{\mathcal {C}}}  : A symmetric monoidal functor is a braided monoidal functor whose domain and codomain are symmetric monoidal categories. The underlying functor...

Derived category Triangulated category Model category 2-category Dagger symmetric monoidal category Dagger compact category Strongly ribbon category Closed...

a commutative monoid; a Cartesian category with its finite products is an example of a symmetric monoidal category. For any objects X , Y ,  and  Z {\displaystyle...

of a commutative monoid; a category with finite coproducts is an example of a symmetric monoidal category. If the category has a zero object Z {\displaystyle...

respectively. The augmented simplex category, unlike the simplex category, admits a natural monoidal structure. The monoidal product is given by concatenation...

isomorphisms make the appropriate category of pointed spaces into a symmetric monoidal category with the smash product as the monoidal product and the pointed 0-sphere...

mathematics, a *-autonomous (read "star-autonomous") category C is a symmetric monoidal closed category equipped with a dualizing object ⊥ {\displaystyle...

more detail, this means that a category C is pre-abelian if: C is preadditive, that is enriched over the monoidal category of abelian groups (equivalently...

tensor product of modules ⊗, the category of modules is a symmetric monoidal category. A monoid object of the category of modules over a commutative ring...

for monoidal categories; thus, for example, a monoidal category is the same as a weak 2-category with one 0-cell.[citation needed] In higher category theory...

set, An (n + 1)-category is a category enriched over the category n-Cat. So a 1-category is just a (locally small) category. The monoidal structure of Set...

More generally, any monoidal closed category is a closed category. In this case, the object I {\displaystyle I} is the monoidal unit. Eilenberg, S.;...

closed symmetric monoidal category. Ab is not a topos since e.g. it has a zero object. Category of modules Abelian sheaf — many facts about the category of...

In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked...

obvious example of a preadditive category is the category Ab itself. More precisely, Ab is a closed monoidal category. Note that commutativity is crucial...

the simply typed lambda calculus. They are generalized by closed monoidal categories, whose internal language, linear type systems, are suitable for both...

the following "piecemeal" definition: A category is preadditive if it is enriched over the monoidal category Ab of abelian groups. This means that all...

In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic...

specifically category theory, a quasi-category (also called quasicategory, weak Kan complex, inner Kan complex, infinity category, ∞-category, Boardman complex...

consider a 2-category with a single object; these are essentially monoidal categories. Bicategories are a weaker notion of 2-dimensional categories in which...

In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products...

bells and whistles in symmetric monoidal categories". arXiv:1908.02633 [math.CT]. Freyd, Peter J.; Scedrov, Andre (1990). Categories, Allegories. North Holland...

bilinear; in other words, C is enriched over the monoidal category of abelian groups. In a preadditive category, every finitary product is necessarily a coproduct...