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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 .
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 Hom [X, 1]) and a morphism 1 → X ⊗ X * satisfying natural conditions. The category is called right rigid or left rigid according to whether it has right duals or left duals.
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
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.
In a closed monoidal category C, i.e. a monoidal category with an internal Hom functor, an alternative approach is to simulate the standard definition of a dual vector space as a space of functionals. For an object V ∈ C define V ∗ to be _ (,), where 1 C is the monoidal identity.
Extending the above-mentioned braiding on the center of an ordinary monoidal category, the center of a monoidal ∞-category becomes an -monoidal category. More generally, the center of a E k {\displaystyle E_{k}} -monoidal category is an algebra object in E k {\displaystyle E_{k}} -monoidal categories and therefore, by Dunn additivity , an E k ...