Subcategory

In mathematics, specifically category theory, a subcategory of a category \(\mathcal{C}\) is a category \(\mathcal{S}\) whose objects are objects in \(\mathcal{C}\) and whose morphisms are morphisms in \(\mathcal{C}\) with the same identities and composition of morphisms. Intuitively, a subcategory of \(\mathcal{C}\) is a category obtained from \(\mathcal{C}\) by "removing" some of its objects and arrows.

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 id_\(X\) 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 full subcategory 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 \(A\).

• 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.

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.

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.