Super Vector Space - The Category of Super Vector Spaces

The Category of Super Vector Spaces

The category of super vector spaces, denoted by K-SVect, is the category whose objects are super vector spaces (over a fixed field K) and whose morphisms are even linear transformations (i.e. the grade preserving ones).

The categorical approach to super linear algebra is to first formulate definitions and theorems regarding ordinary (ungraded) algebraic objects in the language of category theory and then transfer these directly to the category of super vector spaces. This leads to a treatment of "superobjects" such as superalgebras, Lie superalgebras, supergroups, etc. that is completely analogous to their ungraded counterparts.

The category K-SVect is a monoidal category with the super tensor product as the monoidal product and the purely even super vector space K1|0 as the unit object. The involutive braiding operator

given by

on pure elements, turns K-SVect into a symmetric monoidal category. This commutativity isomorphism encodes the "rule of signs" that is essential to super linear algebra. It effectively says that a minus sign is picked up whenever two odd elements are interchanged. One need not worry about signs in the categorical setting as long as the above operator is used wherever appropriate.

K-SVect is also a closed monoidal category with the internal Hom object, Hom(V, W), given by the super vector space of all linear maps from V to W. The ordinary Hom set Hom(V, W) is the even subspace therein:

The fact that K-SVect is closed means that the functor –⊗V is left adjoint to the functor Hom(V,–), given a natural bijection:

A superalgebra over K can be described as a super vector space A with a multiplication map

Associativity and the existence of an identity can be expressed with the usual commutative diagrams, so that a unital associative superalgebra over K is a monoid in the category K-SVect.