Subcategory

Let \mathcal{C} be a category. A subcategory \mathcal{S} of \mathcal{C} is given by • a subcollection of objects of \mathcal{C}, denoted \operatorname{ob}(\mathcal{S}), • a subcollection of morphisms of \mathcal{C}, denoted \operatorname{mor}(\mathcal{S}). such that • for every X in \operatorname{ob}(\mathcal{S}), the identity morphism idX is in \operatorname{mor}(\mathcal{S}), • for every morphism f:X\to Y in \operatorname{mor}(\mathcal{S}), both the source X and the target Y are in \operatorname{ob}(\mathcal{S}), • for every pair of morphisms f and g in \operatorname{mor}(\mathcal{S}) the composite f\circ g is in \operatorname{mor}(\mathcal{S}) whenever it is defined. These conditions ensure that \mathcal{S} is a category in its own right: its collection of objects is \operatorname{ob}(\mathcal{S}), its collection of morphisms is \operatorname{mor}(\mathcal{S}), and its identities and composition are as in \mathcal{C}. There is an obvious faithful functor I:\mathcal{S}\to\mathcal{C}, called the inclusion functor which takes objects and morphisms to themselves. Let \mathcal{S} be a subcategory of a category \mathcal{C}. We say that \mathcal{S} is a of \mathcal{C} if for each pair of objects X and Y of \mathcal{S}, :\mathrm{Hom}_\mathcal{S}(X,Y)=\mathrm{Hom}_\mathcal{C}(X,Y). A full subcategory is one that includes all morphisms in \mathcal{C} between objects of \mathcal{S}. For any collection of objects A in \mathcal{C}, there is a unique full subcategory of \mathcal{C} whose objects are those in A. == Examples ==

• The category of finite sets forms a full subcategory of the category of sets. • The category whose objects are sets and whose morphisms are bijections forms a non-full subcategory of the category of sets. • The category of abelian groups forms a full subcategory of the category of groups. • 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. == Embeddings ==

Given a subcategory \mathcal{S} of \mathcal{C}, the inclusion functor I:\mathcal{S}\to\mathcal{C} is both a faithful functor and injective on objects. It is full if and only if \mathcal{S} 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 (full) embedding F:\mathcal{B}\to\mathcal{C} the image of F is a (full) subcategory \mathcal{S} of \mathcal{C}, and F induces an isomorphism of categories between \mathcal{B} and \mathcal{S}. If F is a full and faithful functor but not necessarily injective on objects, then the image of F is equivalent to \mathcal{B}. In some categories, one can also speak of morphisms of the category being embeddings. == Types of subcategories ==

A subcategory \mathcal{S} of \mathcal{C} is said to be isomorphism-closed or replete if every isomorphism k:X\to Y in \mathcal{C} such that Y is in \mathcal{S} also belongs to \mathcal{S}. An isomorphism-closed full subcategory is said to be strictly full. A subcategory of \mathcal{C} is wide or lluf (a term first posed by Peter Freyd) if it contains all the objects of \mathcal{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 \mathcal{S} of an abelian category \mathcal{C} such that for all short exact sequences :0\to M'\to M\to M''\to 0 in \mathcal{C}, M belongs to \mathcal{S} if and only if both M' and M'' do. This notion arises from Serre's C-theory. == See also ==