# nLab SVect

superalgebra

and

supergeometry

## Applications

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 ${ℤ}_{2}$-graded vector spaces for which

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

and

$\left(V\otimes W{\right)}^{\mathrm{odd}}:={V}^{\mathrm{ev}}\otimes {W}^{\mathrm{odd}}\oplus {V}^{\mathrm{odd}}\otimes {W}^{\mathrm{ev}}$(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 $\mid v\mid ,\mid w\mid \in {ℤ}_{2}$ as

$v\otimes w↦\left(-1{\right)}^{\mid v\mid \mid w\mid }w\otimes v\phantom{\rule{thinmathspace}{0ex}}.$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)