monoidal category

In mathematics, a monoidal category (or tensor category) is a category

\mathbb C

equipped with a binary 'tensor' functor

\otimes: \mathbb C\times\mathbb C\to\mathbb C

and a unit object

I

. The tensor operation must be associative in the sense that there is a natural isomorphism

\alpha

with components

\alpha_{A,B,C}: (A\otimes B)\otimes C \to A\otimes(B\otimes C)

; and

I

must be a left and right identity in the sense that there are natural isomorphisms

\lambda

and

\rho

with components

\lambda_A: I\otimes A\to A

and

\rho_A: A\otimes I\to A

respectively. These natural transformations are subject to certain coherence conditions. All the necessary conditions are implied by the following two: for all

A

B

C

and

D

\mathbb C

, the diagrams

and

must commute. It follows from these two conditions that any such diagram commutes: this is Mac Lane's "coherence theorem".

A monoidal category may be regarded as a bicategory with one object.
Many monoidal categories have additional structure such as braiding or symmetry: the references describe this in detail.
There is a general notion of monoid object in a monoidal category, which generalizes the ordinary notion of monoid.
Monoidal categories are used to define models for linear logic.

Examples

Any category with standard categorical products and a terminal object is a monoidal category, with the categorical product as tensor product and the terminal object as identity. Also, any category with coproducts and an initial object is a monoidal category - with the coproduct as tensor product and the initial object as identity. (In both these cases, the structure is actually symmetric monoidal.) However, in many monoidal categories (such as K-Vect, given below) the tensor product is neither a categorical product nor a coproduct. 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 $\nabla:A\otimes A\rightarrow A$ and $\eta: \mathbf{K} \rightarrow A$ satisfying commutative diagrams.	A monoid is an object M together with morphisms $\circ: M \times M \rightarrow M$ and $1: \{*\} \rightarrow M$ satisfying commutative diagrams.
A coalgebra is an object C with morphisms $\Delta: C \rightarrow C \otimes C$ and $\epsilon:C\rightarrow \mathbf{K}$ satisfying commutative diagrams.	Any object of Set, S has two unique morphisms $\Delta: S \rightarrow S \times S$ and $\epsilon: S \rightarrow \{\}$ satisfying commutative diagram. In particular, ε is unique because {} is a terminal object.

References

Joyal, André; Street, Ross (1993). "Braided Tensor Categories". Advances in Mathematics 102, 20–78.
Mac Lane, Saunders (1997), Categories for the Working Mathematician (2nd ed.). New York: Springer-Verlag.

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