Braided monoidal category

A monoidal category $𝖢$ is called braided iff there exists a natural isomorphism with components $𝜏 𝑥, 𝑦 : 𝑥 \otimes 𝑦 \to 𝑦 \otimes 𝑥$ in $𝖢 𝖢 \times 𝖢$ called the braiding such that the braiding laws or hexagon identities

and

commute for all objects $𝑥, 𝑦, 𝑧 \in 𝖢$ . cat Iff the braiding is involutive in the sense that $𝜏 𝑦, 𝑥 𝜏 𝑥, 𝑦 = 1 𝑥 \otimes 𝑦$ , then the category $𝖢$ is called symmetric, and iff $𝜏 𝑥, 𝑦 = 1 𝑥 \otimes 𝑦$ then $𝖢$ is called strictly symmetric.

The braiding laws ensure the braid is well behaved in the sense of the Coherence theorem for braided monoidal categories and Strictification theorem for braided monoidal categories.

Diagrammatics

The diagrammatics of a monoidal category are single faced string diagrams in $2 + 1$ dimensions.

tidy | en | SemBr

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Braided monoidal category

Diagrammatics

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