Monoidal category

In mathematics refers to a category monoidale a category that is equipped with a double-digit functor and a unit object. The link must be associative in the sense that it gives a natural equivalence; must be left -and right- neutral in the sense that there are natural equivalences and given by and.

These natural transformations are to be coherent. All the necessary coherence conditions follow from the commutativity of the following two diagrams:

And

From these two conditions it follows that every such diagram commutes: This is Mac Lane's " coherence theorem ".

• Monoidale a category can be seen as an object Bikategorie.
• In a monoidal category can the notion of monoid object define the generalized to the monoid.

Examples

Each category with finite products and a final object contains is symmetric monoidal: The two-character functor is given by a natural selection of products, and the final object is the device object. Analogously, we can choose an initial object as a double-digit functor a coproduct and a unit object.

We now show the parallel structure of two such monoidaler categories:

For which the following diagrams commute:

Swell

• 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.

• Category theory