Category theory MOC

Symmetric monoidal category

A symmetric monoidal category is a braided monoidal category for which the braiding is involutive in the sense that . #m/def/cat Thus it is precisely a monoidal category equipped with a natural isomorphism with components in such that the hexagon identity

commutes and for all objects . A symmetric monoidal category is called strict iff for all objects , i.e. iff .

The hexagon identity ensures is commutative up to natural isomorphism, by the Coherence theorem for symmetric monoidal categories and the Strictification theorem for symmetric monoidal categories.

Diagrammatics

The diagrammatics of a symmetric monoidal category are single faced string diagrams in dimensions.

#state/tidy | #lang/en | #SemBr