A monoidal category is a category equipped with extra structure that lets you combine objects in a coherent way.
Data
A monoidal category consists of:
- A category
C. - A tensor product operation
tensor : C x C -> C. - A unit object
I. - Natural isomorphisms that express associativity and unit laws.
Coherence idea
The associativity and unit constraints ensure that different parenthesizations lead to the same result, so composition in enriched categories is well-defined.