# Monoidal category

In mathematics, a monoidal category (or tensor category) is a category [itex]\mathbb C[itex] equipped with a binary 'tensor' functor [itex]\otimes: \mathbb C\times\mathbb C\to\mathbb C[itex] and a unit object [itex]I[itex]. The tensor operation must be associative in the sense that there is a natural isomorphism [itex]\alpha[itex] with components [itex]\alpha_{A,B,C}: (A\otimes B)\otimes C \to A\otimes(B\otimes C)[itex]; and [itex]I[itex] must be a left and right identity in the sense that there are natural isomorphisms [itex]\lambda[itex] and [itex]\rho[itex] with components [itex]\lambda_A: I\otimes A\to A[itex] and [itex]\rho_A: A\otimes I\to A[itex] respectively.

These natural transformations are subject to certain coherence conditions. All the necessary conditions are implied by the following two: for all [itex]A[itex], [itex]B[itex], [itex]C[itex] and [itex]D[itex] in [itex]\mathbb C[itex], 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-VectSet
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 [itex]\nabla:A\otimes A\rightarrow A[itex] and [itex]\eta: \mathbf{K} \rightarrow A[itex] satisfying
Missing image
Algebra.png
commutative diagrams

.
A monoid is an object M together with morphisms [itex]\circ: M \times M \rightarrow M[itex] and [itex]1: \{*\} \rightarrow M[itex] satisfying
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Monoid.png
commutative diagrams

.
A coalgebra is an object C with morphisms [itex]\Delta: C \rightarrow C \otimes C[itex] and [itex]\epsilon:C\rightarrow \mathbf{K}[itex] satisfying
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Coalg.png
commutative diagrams

.
Any object of Set, S has two unique morphisms [itex]\Delta: S \rightarrow S \times S[itex] and [itex]\epsilon: S \rightarrow \{*\}[itex] 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.

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