Monoidal category

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

These natural transformations are subject to certain coherence conditions. All the necessary conditions are implied by the following two: for all <math>A<math>, <math>B<math>, <math>C<math> and <math>D<math> in <math>\mathbb C<math>, the diagrams

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and Missing image

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.


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.

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 <math>\nabla:A\otimes A\rightarrow A<math> and <math>\eta: \mathbf{K} \rightarrow A<math> satisfying
Missing image
commutative diagrams

A monoid is an object M together with morphisms <math>\circ: M \times M \rightarrow M<math> and <math>1: \{*\} \rightarrow M<math> satisfying
Missing image
commutative diagrams

A coalgebra is an object C with morphisms <math>\Delta: C \rightarrow C \otimes C<math> and <math>\epsilon:C\rightarrow \mathbf{K}<math> satisfying
Missing image
commutative diagrams

Any object of Set, S has two unique morphisms <math>\Delta: S \rightarrow S \times S<math> and <math>\epsilon: S \rightarrow \{*\}<math> satisfying
Missing image
commutative diagram

In particular, ε is unique because {*} is a terminal object.


  • 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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