nLab
SVect

The category SVectS Vect of super vector spaces is the symmetric monoidal category which as a monoidal category is the ordinary monoidal category of 2\mathbb{Z}_2-graded vector spaces for which

(VW) ev:=V evW evV oddW odd (V \otimes W)^{ev} := V^{ev}\otimes W^{ev} \oplus V^{odd} \otimes W^{odd}

and

(VW) odd:=V evW oddV oddW 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

VWσ V,WWV V \otimes W \stackrel{\sigma_{V,W}}{\to} W \otimes V

that is given on homogeneously graded elements v,wv,w of degree |v|,|w| 2|v|, |w| \in \mathbb{Z}_2 as

vw(1) |v||w|wv. 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)