In mathematics, a subcategory of a category C is a category S whose objects are objects in C and whose arrows $f:A\backslash to\; B$ are arrows in C (with the same source and target). Intuitively, a subcategory of C is therefore a category obtained from C by "removing" objects and arrows.

A full subcategory S of a category C is a subcategory of C such that for each objects A and B of S,

$\backslash mathrm\{Hom\}\_S(A,B)=\backslash mathrm\{Hom\}\_C(A,B)$

The natural functor from S of C that acts as the identity on objects and arrows is called the inclusion functor. It is always a faithful functor. The inclusion functor is full if and only if S is a full subcategory.

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

$0\backslash to\; M\text{'}\backslash to\; M\backslash to\; M\text{'}\text{'}\backslash to\; 0$

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

See also

• Reflective subcategory

Category theory

Subcategoría