Concept

# Subcategory

Résumé
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. 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: 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 inclu
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