Monoidal category

A strict monoidal category is a category with a product operation × on objects that has properties analogous to those of the tensor product (it is not assumed to be a categorical product). The product is assumed to be associative, and have an left and right identity, I. The corresponding morphisms, expressing the 'associative law', and 'identity element' properties, are subject to a system of coherence conditions: these are familiar in the cases of the examples given below, but in general require some small amount of syntax to express. The concept of braided monoidal category has been much studied from the 1980s onwards; it occurs in string theory applications, and is a more 'relaxed' theory defined by fewer such coherence conditions.

Examples of monoidal categories, illustrating the parallelism between the category of vector spaces over a field and the category of sets, are given below.

K-Vect	Set
Given a field (or commutative ring) K, the category K-Vect is a symmetric monoidal category with product ⊗ and identity K.	The category Set is a symmetric monoidal category with product × and identity {*}.
A unital associative algebra is an object of K-Vect together with morphisms and satisfying	A monoid is an object M together with morphisms and satisfying .
A coalgebra is an object B with morphisms and satisfying .	Any object of Set, S has two unique morphisms and satisfying . In particular, ε is unique because {*} is a terminal object.