Monoidal-category Action
In algebra, an action of a monoidal category ''S'' on a category ''X'' is a functor In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ... :\cdot: S \times X \to X such that there are natural isomorphisms s \cdot (t \cdot x) \simeq (s \cdot t)\cdot x and e \cdot x \simeq x and those natural isomorphism satisfy the coherence conditions analogous to those in ''S''. If there is such an action, ''S'' is said to act on ''X''. For example, ''S'' acts on itself via the monoid operation ⊗. References * Monoidal categories Functors {{algebra-stub ...
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Monoidal Category
In mathematics, a monoidal category (or tensor category) is a category \mathbf C equipped with a bifunctor :\otimes : \mathbf \times \mathbf \to \mathbf that is associative up to a natural isomorphism, and an object ''I'' that is both a left and right identity for ⊗, again up to a natural isomorphism. The associated natural isomorphisms are subject to certain coherence conditions, which ensure that all the relevant diagrams commute. The ordinary tensor product makes vector spaces, abelian groups, ''R''-modules, or ''R''-algebras into monoidal categories. Monoidal categories can be seen as a generalization of these and other examples. Every (small) monoidal category may also be viewed as a "categorification" of an underlying monoid, namely the monoid whose elements are the isomorphism classes of the category's objects and whose binary operation is given by the category's tensor product. A rather different application, of which monoidal categories can be considered an abstractio ...
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