SUBCATEGORY

In mathematics, a 'subcategory' of a category ''C'' is a category ''S'' whose objects are objects in ''C'' and whose arrows $f:A\; o\; B$ 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'',
:$mathrm\{Hom\}\_S(A,B)=mathrm\{Hom\}\_C(A,B).$
''S'' is called 'strictly full' if in addition to being full, it is "closed under isomorphisms": for any ''A'' in ''S'' and ''B'' in ''C'', if ''A'' is isomorphic to ''B'', then ''B'' is also in ''S''.
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
:$0\; o\; M\text{'}\; o\; M\; o\; M\text{'}\text{'}\; o\; 0$
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.

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See also

★ Reflective subcategory