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 . 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 monoidal category are single faced string diagrams in dimensions.

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