nLab
super vector space

Idea

A super vector space is an object in the monoidal category SVect: as an object it is just a ${\mathbb{Z}}_2$-graded vector space, but when tensoring them one uses the non-trivial symmetric monoidal structure on ${\mathbb{Z}}_2$-graded vector spaces. In simple terms, this means that when switching two ‘odd’ vectors one introduces a minus sign:

$$v\otimes w\mapsto (-1{)}^{\mathrm{deg}(v)\mathrm{deg}(w)}w\otimes v$$v \otimes w \mapsto (-1)^{deg(v) deg(w)} w \otimes v

Related concepts

• super-algebra, super Lie algebra

• supertrace