Symmetric monoidal category

A symmetric monoidal category $𝖢$ is a braided monoidal category for which the braiding is involutive in the sense that $𝜏 𝑦, 𝑥 𝜏 𝑥, 𝑦 = 1 𝑥 \otimes 𝑦$ . #m/def/cat Thus it is precisely a monoidal category equipped with a natural isomorphism with components $𝜏 𝑥, 𝑦 : 𝑥 \otimes 𝑦 \to 𝑦 \otimes 𝑥$ in $𝖢 𝖢 \times 𝖢$ such that the hexagon identity

see braided monoidal category

commutes and $𝜏 𝑦, 𝑥 𝜏 𝑥, 𝑦 = 1 𝑥 \otimes 𝑦$ for all objects $𝑥, 𝑦, 𝑧 \in 𝖢$ . The hexagon identity ensures $(\otimes)$ is commutative up to natural isomorphism, by the Coherence theorem for symmetric monoidal categories.

Further terminology

If $𝖢$ is a strict category, it called strict symmetric iff $𝜏 𝑥, 𝑦 = 1 𝑥 \otimes 𝑦$ for all objects $𝑥, 𝑦$ , i.e. iff $𝑥 \otimes 𝑦 = 𝑦 \otimes 𝑥$ .

Diagrammatics

The diagrammatics of a symmetric monoidal category are single faced string diagrams in $3 + 1$ dimensions.

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