Monoidal Category

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Monoidal Category

A Monoidal category $ \mathcal{M}$ is a one object category equipped with a binary operation on that object, and a unit element. In the situation in which the object in $ \mathcal{M}$ is a category then, $ \mathcal{M}$ is defined as follows:

Definition 2.3   A monoidal category $ \mathcal{M}$ is a triplet (M, *, i) such that
  • M is a category
  • * is a functor $ M\times M\rightarrow M$
  • $ i\in M$ such that $ \forall\hspace{.02in}x\in M$ i * x = x * i = x

The * functor can be identified either with the tensor product, the direct sum or the direct product according to which category M one is tacking into consideration.

Cecilia Flori 2006-11-19