Subcategory

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In mathematics, a subcategory of a category C is a category S whose objects are objects in C and whose arrows [f:A\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,

[\mathrm_S(A,B)=\mathrm_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\to M'\to M\to M''\to 0]

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

Subcategory

See also