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Graded category

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In mathematics, if is a category, then a -graded category is a category together with a functor .

Monoids and groups can be thought of as categories with a single object. 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.

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Definition

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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 be an abelian category and a monoid. Let be a set of functors from to itself. If

  • is the identity functor on ,
  • for all and
  • is a full and faithful functor for every

we say that is a -graded category.

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