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This is one of the diagrams used in the definition of a monoidal cateogory. It takes care of the case for when there is an instance of an identity between two objects. commutes. A strict monoidal category is one for which the natural isomorphisms α, λ and ρ are identities. Every monoidal category is monoidally equivalent to a strict monoidal ...
A monoid object in K-Vect, the category of K-vector spaces (again, with the tensor product), is a unital associative K-algebra, and a comonoid object is a K-coalgebra. 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 ...
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 .
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
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 ...
The interpretation in a monoidal category is a defined by a monoidal functor :, which by freeness is uniquely determined by a morphism of monoidal signatures : (). Intuitively, once the image of generating objects and arrows are given, the image of every diagram they generate is fixed.
A braided monoidal category is a monoidal category equipped with a braiding—that is, a commutativity constraint that satisfies axioms including the hexagon identities defined below. The term braided references the fact that the braid group plays an important role in the theory of braided monoidal categories.
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 ...