Symmetric monoidal category

A symmetric monoidal category $𝖢$ is a braided monoidal category for which the braiding is involutive in the sense that $𝜏 𝑦, 𝑥 𝜏 𝑥, 𝑦 = 1 𝑥 \otimes 𝑦$ . 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

commutes and $𝜏 𝑦, 𝑥 𝜏 𝑥, 𝑦 = 1 𝑥 \otimes 𝑦$ for all objects $𝑥, 𝑦, 𝑧 \in 𝖢$ . A symmetric monoidal category is called strict iff $𝜏 𝑥, 𝑦 = 1 𝑥 \otimes 𝑦$ for all objects $𝑥, 𝑦$ , i.e. iff $𝑥 \otimes 𝑦 = 𝑦 \otimes 𝑥$ .

The hexagon identity ensures $(\otimes)$ 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 $3 + 1$ dimensions.

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Symmetric monoidal category

Diagrammatics

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