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 <math>f:A\to B</math> 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,
- <math>\mathrm{Hom}_S(A,B)=\mathrm{Hom}_C(A,B)</math>
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
- <math>0\to M'\to M\to M\to 0</math>
in C, M belongs to S if and only if both <math>M'</math> and <math>M</math> do. This notion arises from Serre's C-theory.