Knowlpedia

Definition

Symmetric monoidal category

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

Definition

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 two hexagon coherence diagrams relating β\beta to the associator commute. 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.