Subcategory

In mathematics, specifically , a subcategory of a C is a category S whose are objects in C and whose morphisms are morphisms in C with the same identities and composition of morphisms. Intuitively, a subcategory of C is a category obtained from C by "removing" some of its objects and arrows. 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: 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 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 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. The category of finite sets forms a full subcategory of the . The category whose objects are sets and whose morphisms are bijections forms a non-full subcategory of the category of sets. The forms a full subcategory of the . The category of rings (whose morphisms are unit-preserving ring homomorphisms) forms a non-full subcategory of the category of rngs. For a field K, the category of K-vector spaces forms a full subcategory of the category of (left or right) K-modules. Given a subcategory S of C, the inclusion functor I : S → C is both a faithful functor and injective on objects. It is full if and only if S is a full subcategory. Some authors define an embedding to be a full and faithful functor.