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

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

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 category theory, a closed monoidal category (or a monoidal closed category) is a category that is both a monoidal category and a closed category in...

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

In category theory, a branch of mathematics, a monoid (or monoid object, or internal monoid, or algebra) (M, μ, η) in a monoidal category (C, ⊗, I) is...

idea of a category by replacing hom-sets with objects from a general monoidal category. It is motivated by the observation that, in many practical applications...

as category theory, a monoidal category where the monoidal ("tensor") product is the categorical product is called a cartesian monoidal category. Any...

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

Closed monoidal category Braided monoidal category Topos Category of small categories Semigroupoid Comma category Localization of a category Enriched...

In category theory, monoidal functors are functors between monoidal categories which preserve the monoidal structure. More specifically, a monoidal functor...

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...

for monoidal categories, are moreover required to hold: a monoidal category is the same as a bicategory with one 0-cell. Consider a simple monoidal category...

is Rel, the category having sets as objects and relations as morphisms, with Cartesian monoidal structure. A symmetric monoidal category ( C , ⊗ , I )...

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...

In algebra, an action of a monoidal category S on a category X is a functor ⋅ : S × X → X {\displaystyle \cdot :S\times X\to X} such that there are natural...

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

one morphism into X. symmetric monoidal category A symmetric monoidal category is a monoidal category (i.e., a category with ⊗) that has maximally symmetric...

representing morphisms in monoidal categories, or more generally 2-cells in 2-categories. They are a prominent tool in applied category theory. When interpreted...

term module category for the category of modules. This term can be ambiguous since it could also refer to a category with a monoidal-category action. The...

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

algebra. The axioms are partly chosen so that the category of H-modules is a rigid monoidal category. The unit H-module is the separable algebra HL mentioned...

there are categories in which currying is not possible; the most general categories which allow currying are the closed monoidal categories. Some programming...

Dyadics – Second order tensor in vector algebra Extension of scalars Monoidal category – Category admitting tensor products Tensor algebra – Universal construction...

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

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

Monoidal may refer to: Monoidal category, concept in category theory Monoidal functor, between monoidal categories Monoidal natural transformation, between...

notion of the center of a monoid, group, or ring to a category. The center of a monoidal category C = ( C , ⊗ , I ) {\displaystyle {\mathcal {C}}=({\mathcal...

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

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 abelian...