Graded Vector Spaces

Graded Vector Space

Given a set , a -graded vector space is a vector space together with a decomposition into subspaces indexed by :

A graded linear map between two -graded vector spaces and is a linear map such that for all . -graded vector spaces with graded linear maps form a category, denoted .
Alternatively, one can think of it as a functor , where the set is treated as a discrete category and is the category of vector spaces, and a morphism of -graded vector spaces is a natural transformation between such functors.

By far the most widely-used examples are and . Indeed, the term graded vector space is often used to mean a -graded vector space with one of these choices of . The case is also important: a -graded vector space is also called a supervector space.

People are usually interested in -graded vector spaces when the set is equipped with extra structure. In general, adding algebraic structure onto will add categorical structure onto :

Set	Category
Monoid	Monoidal category
Finite Group	Fusion category
Finite abelian group	Braided fusion category