Knowlpedia

Definition

Symmetric monoidal category

A monoidal category whose tensor factors can be exchanged by a coherent involutive braiding.

A symmetric monoidal category is a (C,⊗,1)(\mathcal C,\otimes,\mathbb 1) equipped with

βX,Y:X⊗Y⟶∼Y⊗X\beta_{X,Y}:X\otimes Y\overset{\sim}{\longrightarrow}Y\otimes X

such that βY,X∘βX,Y=id⁡X⊗Y\beta_{Y,X}\circ\beta_{X,Y}=\operatorname{id}_{X\otimes Y} and the hexagon identity

αY,Z,X∘βX,Y⊗Z∘αX,Y,Z=(id⁡Y⊗βX,Z)∘αY,X,Z∘(βX,Y⊗id⁡Z)\alpha_{Y,Z,X}\circ\beta_{X,Y\otimes Z}\circ\alpha_{X,Y,Z} =(\operatorname{id}_Y\otimes\beta_{X,Z})\circ\alpha_{Y,X,Z}\circ (\beta_{X,Y}\otimes\operatorname{id}_Z)

holds for all objects, with composition read from right to left. Here α\alpha is the monoidal associator. Together with the stated involutivity, this identity implies the second hexagon identity. The maps βX,Y\beta_{X,Y} form the symmetry or symmetric braiding.

Examples

For modules over a commutative ring, β(x⊗y)=y⊗x\beta(x\otimes y)=y\otimes x. For the symmetry instead includes the Koszul sign:

β(v⊗w)=(−1)∣v∣∣w∣w⊗v\beta(v\otimes w)=(-1)^{|v||w|}w\otimes v

on homogeneous vectors. This is symmetric because applying it twice gives the identity.

Related structures

A braided monoidal category has coherent maps βX,Y\beta_{X,Y} but does not require βY,XβX,Y\beta_{Y,X}\beta_{X,Y} to be the identity. Thus every symmetric monoidal category is braided, but not conversely.

References
  1. Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, and Victor Ostrik, Tensor Categories, American Mathematical Society, 2015. DOI record. Relevant: §2.1.