Subcategory
From Exampleproblems
In mathematics, a subcategory S of a category C consists of subsets of the morphisms and of the objects of C, such that the subset X of morphisms is closed under composition in C, and the subset Y of objects contains the source and target of all the f in X. The inclusion functor S --> C is a faithful functor
A full subcategory is one for which X consists precisely of (the union of) the morphism sets
- MorC(A, B)
for A and B in Y. (These definitions would need a small amount of rephrasing to cope with proper classes.)
A Serre subcategory is a non-empty full subcategory B of an abelian category A such that for all short exact sequences,
- 0 → M′ → M → M′′ → 0
in A, M belongs to B if and only if both M′ and M′′ do. This notion arises from Serre's C-theory.
