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 $α$ 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 $λ$ and $ρ$ 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 $D$ in $\mathbb C$ , the diagrams

Missing image
Monoidal-category-pentagon.png
Image:monoidal-category-pentagon.png

and Missing image
Monoidal-category-triangle.png
Image:monoidal-category-triangle.png

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 Missing image Algebra.png commutative diagrams .	A monoid is an object M together with morphisms $\circ: M \times M \rightarrow M$ and $1: \{\} \rightarrow M$ satisfying Missing image* Monoid.png commutative diagrams .
A coalgebra is an object C with morphisms $\Delta: C \rightarrow C \otimes C$ and $\epsilon:C\rightarrow \mathbf{K}$ satisfying Missing image Coalg.png commutative diagrams .	Any object of Set, S has two unique morphisms $\Delta: S \rightarrow S \times S$ and $\epsilon: S \rightarrow \{\}$ satisfying Missing image* Diag.png 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.

Monoidal category

Examples

References

See also: Monoidal category, Bicategory, Braided monoidal category, Categorical product, Category theory, Commutative ring, Coproduct, Field (mathematics), Identity element, Initial object