graded category について

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Monoids and groups can be thought of as categories with a single element. A monoid-graded or group-graded category is therefore one in which to each morphism is attached an element of a given monoid (resp. group), its grade. This must be compatible with composition, in the sense that compositions have the product grade.
==Definition==

There are various different definitions of a graded category, up to the most abstract one given above. A more concrete definition of a semigroup-graded Abelian category is as follows:
Let

\mathcal

be an Abelian category and

\mathbb

a semigroup. Let

\mathcal=\

be a set of functors from

\mathcal

to itself. If
*

S_

is the identity functor on

\mathcal

,
*

S_S_=S_

for all

g,h\in \mathbb

and
*

S_

is a full and faithful functor for every

g\in \mathbb

we say that

(\mathcal,\mathcal)

is a

\mathbb

-graded category.

抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
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