In mathematics, especially 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 such a way that the structures are compatible.
A classic example is the category of sets, Set, where the monoidal product of sets and is the usual cartesian product , and the internal Hom is the set of functions from to . A non-cartesian example is the category of vector spaces, K-Vect, over a field . Here the monoidal product is the usual tensor product of vector spaces, and the internal Hom is the vector space of linear maps from one vector space to another.
The internal language of closed symmetric monoidal categories is linear logic and the type system is the linear type system. Many examples of closed monoidal categories are symmetric. However, this need not always be the case, as non-symmetric monoidal categories can be encountered in category-theoretic formulations of linguistics; roughly speaking, this is because word-order in natural language matters.
and 26 Related for: Closed monoidal category information
In mathematics, a monoidalcategory (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 monoidalcategory C {\displaystyle {\mathcal {C}}} is a choice of isomorphism γ A ,...
In category theory, a branch of mathematics, a symmetric monoidalcategory is a monoidalcategory (i.e. a category in which a "tensor product" ⊗ {\displaystyle...
is the simply typed lambda calculus. They are generalized by closedmonoidalcategories, whose internal language, linear type systems, are suitable for...
(i.e., making the category symmetric monoidal or even symmetric closedmonoidal, respectively).[citation needed] Enriched category theory thus encompasses...
monoidal structure. A symmetric monoidalcategory ( C , ⊗ , I ) {\displaystyle (\mathbf {C} ,\otimes ,I)} is compact closed if every object A ∈ C {\displaystyle...
there are categories in which currying is not possible; the most general categories which allow currying are the closedmonoidalcategories. Some programming...
In category theory, monoidal functors are functors between monoidalcategories which preserve the monoidal structure. More specifically, a monoidal functor...
In category theory, a traced monoidalcategory is a category with some extra structure which gives a reasonable notion of feedback. A traced symmetric...
In the mathematical field of category theory, a dagger symmetric monoidalcategory is a monoidalcategory ⟨ C , ⊗ , I ⟩ {\displaystyle \langle \mathbf...
certain coherence conditions (see symmetric monoidalcategory for details). A monoidalcategory is compact closed, if every object A ∈ C {\displaystyle A\in...
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...
is monoidalclosed, 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...
obvious example of a preadditive category is the category Ab itself. More precisely, Ab is a closedmonoidalcategory. Note that commutativity is crucial...
product functor defining a monoidalcategory. The isomorphism is natural in both X and Z. In other words, in a closedmonoidalcategory, the internal Hom functor...
notion of product, Ab is a closed symmetric monoidalcategory. Ab is not a topos since e.g. it has a zero object. Category of modules Abelian sheaf —...
one-object categories, into FinVect. DisCoCat models are monoidal functors from a pregroup grammar to FinVect. FinSet ZX-calculus category of modules...
consider a 2-category with a single object; these are essentially monoidalcategories. Bicategories are a weaker notion of 2-dimensional categories in which...
and graph homomorphisms into a symmetric closedmonoidalcategory (as opposed to merely symmetric monoidal), the other being the tensor product of graphs...
an autonomous category is a monoidalcategory where dual objects exist. A left (resp. right) autonomous category is a monoidalcategory where every object...
monoidalcategory. The construction of the derived morphisms of one variable will work in a closedmonoidalcategory. The category of sets is closed monoidal...
In mathematics, a fusion category is a category that is abelian, k {\displaystyle k} -linear, semisimple, monoidal, and rigid, and has only finitely many...
Global Information
