nLab
SVect

The category $S\mathrm{Vect}$ 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

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

and

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

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

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

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

related concepts

• monoids in $S\mathrm{Vect}$ are super algebras.

• manifolds modeled on $S\mathrm{Vect}$ are supermanifolds

• etc.