A monoidal category is a category 
equipped with a bifunctor 
, a unit object 
, and natural
isomorphisms
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(1)
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(2)
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(3)
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called the associator, left unitor, and right unitor, respectively. They satisfy the pentagon and triangle coherence identities (Mac Lane 1998, pp. 162-164). A strict monoidal category has its associator and unitors given by identity morphisms.
The category of sets with the Cartesian product and the category of vector spaces over a fixed field with the vector space tensor product are monoidal categories. Every monoid gives a strict monoidal category whose objects are the elements of the monoid and whose only morphisms are identities. The term tensor category is sometimes used synonymously, although in representation theory it often includes additional linearity and rigidity or the requirement that it be an Abelian category (Etingof et al. 2015, p. 65).
