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

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

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

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

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...

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

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

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

monoidal structure. A symmetric monoidal category ( C , ⊗ , I ) {\displaystyle (\mathbf {C} ,\otimes ,I)} is compact closed if every object A ∈ C {\displaystyle...

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

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

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

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

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

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

mathematics known as category theory, a cosmos is a symmetric closed monoidal category that is complete and cocomplete. Enriched category theory is often considered...

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 category theory, a branch of mathematics, a rigid category is a monoidal category where every object is rigid, that is, has a dual X* (the internal...

in category theory, where it is right adjoint to currying in closed monoidal categories. A special case of this are the Cartesian closed categories, whose...

product functor defining a monoidal category. The isomorphism is natural in both X and Z. In other words, in a closed monoidal category, the internal Hom functor...

notion of product, Ab is a closed symmetric monoidal category. Ab is not a topos since e.g. it has a zero object. Category of modules Abelian sheaf —...

In mathematics, a fusion category is a category that is abelian, k {\displaystyle k} -linear, semisimple, monoidal, and rigid, and has only finitely many...

is monoidal closed, if one defines both the monoidal product A ⊗ B and the internal hom A ⇒ B by the cartesian product of sets. It is also a monoidal category...

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

e c t {\displaystyle \mathbb {K} -\mathrm {SVect} } is also a closed monoidal category with the internal Hom object, H o m ( V , W ) {\displaystyle \mathbf...

preadditive category). The category of rings is a symmetric monoidal category with the tensor product of rings ⊗Z as the monoidal product and the ring of...

certain coherence conditions (see symmetric monoidal category for details). A monoidal category is compact closed, if every object A ∈ C {\displaystyle A\in...

strictification result; namely, every monoidal category is monoidally equivalent to a strict monoidal category. It is not reasonable to expect we can...

category theory, a branch of mathematics, a dual object is an analogue of a dual vector space from linear algebra for objects in arbitrary monoidal categories...