Editing Subcategory (section)

== Formal definition ==
Let ''C'' be a category. A '''subcategory''' ''S'' of ''C'' is given by
*a subcollection of objects of ''C'', denoted ob(''S''),
*a subcollection of morphisms of ''C'', denoted hom(''S'').
such that
*for every ''X'' in ob(''S''), the identity morphism id<sub>''X''</sub> is in hom(''S''),
*for every morphism ''f'' : ''X'' → ''Y'' in hom(''S''), both the source ''X'' and the target ''Y'' are in ob(''S''),
*for every pair of morphisms ''f'' and ''g'' in hom(''S'') the composite ''f'' o ''g'' is in hom(''S'') whenever it is defined.

These conditions ensure that ''S'' is a category in its own right: its collection of objects is ob(''S''), its collection of morphisms is hom(''S''), and its identities and composition are as in ''C''. There is an obvious [[Full and faithful functors|faithful]] [[functor]] ''I'' : ''S'' → ''C'', called the '''inclusion functor''' which takes objects and morphisms to themselves.

Let ''S'' be a subcategory of a category ''C''. We say that ''S'' is a '''full subcategory of''' ''C'' if for each pair of objects ''X'' and ''Y'' of ''S'',
:<math>\mathrm{Hom}_\mathcal{S}(X,Y)=\mathrm{Hom}_\mathcal{C}(X,Y).</math>
A full subcategory is one that includes ''all'' morphisms in ''C'' between objects of ''S''. For any collection of objects ''A'' in ''C'', there is a unique full subcategory of ''C'' whose objects are those in ''A''.