In mathematics, specificallycategory theory, a subcategory of a categoryC is a category S whose objects 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 subcategoryS of C is given by
for every pair of morphisms f and g in hom the compositef o g is in hom whenever it is defined.
These conditions ensure that S is a category in its own right: its collection of objects is ob, its collection of morphisms is hom, and its identities and composition are as in C. There is an obvious faithful functorI : 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 ofC 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.
For a fieldK, the category of K-vector spaces forms a full subcategory of the category of K-modules.
Embeddings
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 ifS is a full subcategory. Some authors define an embedding to be a full and faithful functor. Such a functor is necessarily injective on objects up to isomorphism. For instance, the Yoneda embedding is an embedding in this sense. Some authors define an embedding to be a full and faithful functor that is injective on objects. Other authors define a functor to be an embedding if it is faithful and injective on objects. Equivalently, F is an embedding if it is injective on morphisms. A functor F is then called a full embedding if it is a full functor and an embedding. With the definitions of the previous paragraph, for any embedding F : B → C the image ofF is a subcategory S of C, and F induces an isomorphism of categories between B and S. If F is not injective on objects then the image of F is equivalent to B. In some categories, one can also speak of morphisms of the category being embeddings.
Types of subcategories
A subcategory S of C is said to be isomorphism-closed or replete if every isomorphismk : X → Y in C such that Y is in S also belongs toS. An isomorphism-closed full subcategory is said to be strictly full. A subcategory of C is wide or lluf if it contains all the objects of C. A wide subcategory is typically not full: the only wide full subcategory of a category is that category itself. A Serre subcategory is a non-empty full subcategory S of an abelian categoryC such that for all short exact sequences in C, M belongs to S if and only if both and do. This notion arises from Serre's C-theory.
