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The category of all endofunctors on a category C is a strict monoidal category with the composition of functors as the product and the identity functor as the unit. Just like for any category E , the full subcategory spanned by any given object is a monoid, it is the case that for any 2-category E , and any object C in Ob( E ), the full 2 ...
For any category C, the category [C, C] of its endofunctors has a monoidal structure induced by the composition and the identity functor I C. A monoid object in [C, C] is a monad on C. For any category with a terminal object and finite products, every object becomes a comonoid object via the diagonal morphism Δ X : X → X × X.
More generally, in category theory, the morphisms of an object to itself form a monoid, and, conversely, a monoid may be viewed as a category with a single object. In computer science and computer programming , the set of strings built from a given set of characters is a free monoid .
If R is commutative, the category of left R-modules is symmetric monoidal. The latter example class includes the category of all vector spaces over a given field. Given a field k and a group (or a Lie algebra over k), the category of all k-linear representations of the group (or of the Lie algebra) is a symmetric monoidal category. Here the ...
A closed monoidal category is a monoidal category such that for every object the functor given by right tensoring with . has a right adjoint, written ().This means that there exists a bijection, called 'currying', between the Hom-sets
tensor category Usually synonymous with monoidal category (though some authors distinguish between the two concepts.) tensor triangulated category A tensor triangulated category is a category that carries the structure of a symmetric monoidal category and that of a triangulated category in a compatible way. tensor product
The category of small categories is a closed monoidal category in exactly two ways: with the usual categorical product and with the funny tensor product. [6] Given two categories and , let be the category with functors,: as objects and unnatural transformations: as arrows, i.e. families of morphisms {: ()} which do not necessarily satisfy the condition for a natural transformation.
Any category with finite products (a "finite product category") can be thought of as a cartesian monoidal category. In any cartesian monoidal category, the terminal object is the monoidal unit. Dually, a monoidal finite coproduct category with the monoidal structure given by the coproduct and unit the initial object is called a cocartesian ...