Subcategory

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{{#invoke:Hatnote|hatnote}} In mathematics, a subcategory of a category C is a category S whose objects are objects in C and whose morphisms are morphisms in C with the same identities and composition of morphisms. Intuitively, a subcategory of C is a category obtained from C by "removing" some of its objects and arrows.

Formal definition

Let C be a category. A subcategory S of C is given by

  • a subcollection of objects of C, denoted ob(S),
  • a subcollection of morphisms of C, denoted hom(S).

such that

  • for every X in ob(S), the identity morphism idX is in hom(S),
  • for every morphism f : XY in hom(S), both the source X and the target Y are in ob(S),
  • for every pair of morphisms f and g in hom(S) the composite f o g is in hom(S) whenever it is defined.

These conditions ensure that S is a category in its own right. There is an obvious faithful functor I : SC, called the inclusion functor which takes objects and morphisms to themselves.

Let S be a subcategory of a category C. We say that S is a full subcategory of C if for each pair of objects X and Y of S

A full subcategory is one that includes all morphisms between objects of S. For any collection of objects A in C, there is a unique full subcategory of C whose objects are those in A.

Embeddings

Given a subcategory S of C the inclusion functor I : SC is both faithful and injective on objects. It is full if and only if S is a full subcategory.

Many authors define an embedding to be a full and faithful functor.[1]

Other authors define a functor to be an embedding if it is faithful and injective on objects. Equivalently, F is an embedding if it is injective on morphisms. A functor F is then called a full embedding if it is a full functor and an embedding.

For any (full) embedding F : BC the image of F is a (full) subcategory S of C and F induces a isomorphism of categories between B and S.

In some categories, one can also speak of morphisms of the category being embeddings.

Types of subcategories

A subcategory S of C is said to be isomorphism-closed or replete if every isomorphism k : XY in C such that Y is in S also belongs to S. A isomorphism-closed full subcategory is said to be strictly full.

A subcategory of C is wide or lluf (a term first posed by P. Freyd[2]) if it contains all the objects of C. A lluf subcategory is typically not full: the only full lluf subcategory of a category is that category itself.

A Serre subcategory is a non-empty full subcategory S of an abelian category C such that for all short exact sequences

in C, M belongs to S if and only if both and do. This notion arises from Serre's C-theory.

References

  1. Template:Cite web
  2. {{#invoke:citation/CS1|citation |CitationClass=book }}

See also

es:Subcategoría nl:Deelcategorie pl:Podkategoria ru:Полная подкатегория zh:子範疇