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.
Read more about this topic: Monoidal Categories
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