Graded Category
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If \mathcal is a
category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) ...
, then a \mathcal-graded category is a category \mathcal together with a
functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...
F\colon\mathcal \rightarrow \mathcal.
Monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoid ...
s and
groups A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
can be thought of as categories with a single
object Object may refer to: General meanings * Object (philosophy), a thing, being, or concept ** Object (abstract), an object which does not exist at any particular time or place ** Physical object, an identifiable collection of matter * Goal, an ...
. 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 Composition or Compositions may refer to: Arts and literature *Composition (dance), practice and teaching of choreography *Composition (language), in literature and rhetoric, producing a work in spoken tradition and written discourse, to include v ...
, 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 graded abelian category is as follows: Let \mathcal be an
abelian category In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category of ...
and \mathbb a
monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoid ...
. Let \mathcal=\ be a set of
functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...
s from \mathcal to itself. If * S_ is the
identity functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...
on \mathcal, * S_S_=S_ for all g,h \in \mathbb and * S_ is a
full and faithful functor In category theory, a faithful functor is a functor that is injective on hom-sets, and a full functor is surjective on hom-sets. A functor that has both properties is called a full and faithful functor. Formal definitions Explicitly, let ''C'' ...
for every g\in \mathbb we say that (\mathcal,\mathcal) is a \mathbb-graded category.


See also

*
Differential graded category In mathematics, especially homological algebra, a differential graded category, often shortened to dg-category or DG category, is a category whose morphism sets are endowed with the additional structure of a differential graded \Z-module. In detai ...
* Graded (mathematics) * Graded algebra *
Slice category In mathematics, specifically category theory, an overcategory (and undercategory) is a distinguished class of categories used in multiple contexts, such as with covering spaces (espace etale). They were introduced as a mechanism for keeping track ...


References

Category theory {{cattheory-stub