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 $𝑥, 𝑦, 𝑧 \in 𝖢$ . #m/def/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.

Diagrammatics

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

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