There are various different definitions of a graded category, up to the most abstract one given above. A more concrete definition of a graded abelian category is as follows:^[1]

Let ${\displaystyle {\mathcal {C}}}$  be an abelian category and ${\displaystyle \mathbb {G} }$  a monoid. Let ${\displaystyle {\mathcal {S}}=\{S_{g}:g\in \mathbb {G} \}}$  be a set of functors from ${\displaystyle {\mathcal {C}}}$  to itself. If

• ${\displaystyle S_{1}}$  is the identity functor on ${\displaystyle {\mathcal {C}}}$ ,
• ${\displaystyle S_{g}S_{h}=S_{gh}}$  for all ${\displaystyle g,h\in \mathbb {G} }$  and
• ${\displaystyle S_{g}}$  is a full and faithful functor for every ${\displaystyle g\in \mathbb {G} }$ 

we say that ${\displaystyle ({\mathcal {C}},{\mathcal {S}})}$  is a ${\displaystyle \mathbb {G} }$ -graded category.

• Differential graded category
• Graded (mathematics)
• Graded algebra
• Slice category