# nLab SVect

The category $S Vect$ of super vector spaces is the symmetric monoidal category which as a monoidal category is the ordinary monoidal category of $\mathbb{Z}_2$-graded vector spaces for which

$(V \otimes W)^{ev} := V^{ev}\otimes W^{ev} \oplus V^{odd} \otimes W^{odd}$

and

$(V \otimes W)^{odd} := V^{ev}\otimes W^{odd} \oplus V^{odd} \otimes W^{ev}$

but equipped with the unique non-trivial symmetric monoidal structure

$V \otimes W \stackrel{\sigma_{V,W}}{\to} W \otimes V$

that is given on homogeneously graded elements $v,w$ of degree $|v|, |w| \in \mathbb{Z}_2$ as

$v \otimes w \mapsto (-1)^{|v| |w|} w \otimes v \,.$

# related concepts

Revised on September 23, 2009 09:08:59 by Urs Schreiber (195.37.209.182)