Graded Vector Spaces

Graded Vector Space

Given a set $G$, a $G$-graded vector space is a vector space $V$ together with a decomposition into subspaces indexed by $G$:
$V={⨁}_{g\in G}{V}_g.$
A graded linear map between two $G$-graded vector spaces $V$ and $W$ is a linear map $f:V\to W$ such that $f(V_g)\subseteq W_g$ for all $g\in G$. $G$-graded vector spaces with graded linear maps form a category, denoted ${\text{Vect}}_{\text{F}}^G$.
Alternatively, one can think of it as a functor $V:G\to {\mathsf{Vect}}_{\mathbb{F}}$, where the set $G$ is treated as a discrete category and ${\mathsf{Vect}}_{\text{F}}$ is the category of vector spaces, and a morphism of $G$-graded vector spaces is a natural transformation between such functors.

By far the most widely-used examples are \newcommand{\Z}{\mathbb{Z}}G=\Z and \newcommand{\N}{\mathbb{N}}G=\N. Indeed, the term graded vector space is often used to mean a $G$-graded vector space with one of these choices of $G$. The case $G=\mathbb{Z}/2$ is also important: a $\mathbb{Z}/2$-graded vector space is also called a supervector space.

People are usually interested in $G$-graded vector spaces when the set $G$ is equipped with extra structure. In general, adding algebraic structure onto $G$ will add categorical structure onto ${\text{Vect}}_{\text{F}}^G$:

$G$	${\text{Vect}}_{\text{F}}^G$
Set	Category
Monoid	Monoidal category
Finite Group	Fusion category
Finite abelian group	Braided fusion category