nLab
SVect

and

Formal context

The category $S Vect$ of super vector spaces is the symmetric monoidal category which as a monoidal category is the ordinary monoidal category of $ℤ_{2}$ -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 \overset{σ_{V, W}}{\to} W \otimes V

that is given on homogeneously graded elements $v, w$ of degree $∣ v ∣, ∣ w ∣ \in ℤ_{2}$ as

v \otimes w \mapsto (- 1)^{∣ v ∣ ∣ w ∣} w \otimes v .

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