Monoidal Categories - Formal Definition

Formal Definition

A monoidal category is a category equipped with

• a bifunctor called the tensor product or monoidal product,
• an object called the unit object or identity object,
• three natural isomorphisms subject to certain coherence conditions expressing the fact that the tensor operation
  • is associative: there is a natural isomorphism, called associator, with components ,
  • has as left and right identity: there are two natural isomorphisms and, respectively called left and right unitor, with components and .

The coherence conditions for these natural transformations are:

• for all, and in, the pentagon diagram

commutes;

• for all and in, the triangle diagram

commutes;

It follows from these three conditions that any such diagram (i.e. a diagram whose morphisms are built using, identities and tensor product) commutes: this is Mac Lane's "coherence theorem".

A strict monoidal category is one for which the natural isomorphisms α, λ and ρ are identities. Every monoidal category is monoidally equivalent to a strict monoidal category.