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 idX 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. There is an obvious 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
A full subcategory is one that includes all morphisms 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.
Read more about this topic: Subcategory
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