Subcategory

In mathematics, specifically category theory, a subcategory of a category ${𝒞}$ 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 subcategory ${𝒮}$ of ${𝒞}$ is given by

• a subcollection of objects of ${𝒞}$, denoted $\mathrm{ob}({𝒮})$,
• a subcollection of morphisms of ${𝒞}$, denoted $\mathrm{mor}({𝒮})$.

such that

• for every $X$ in $\mathrm{ob}({𝒮})$, the identity morphism id_$X$ is in $\mathrm{mor}({𝒮})$,
• for every morphism $f:X\to Y$ in $\mathrm{mor}({𝒮})$, both the source $X$ and the target $Y$ are in $\mathrm{ob}({𝒮})$,
• for every pair of morphisms $f$ and $g$ in $\mathrm{mor}({𝒮})$ the composite $f\circ g$ is in $\mathrm{mor}({𝒮})$ whenever it is defined.

These conditions ensure that ${𝒮}$ is a category in its own right: its collection of objects is $\mathrm{ob}({𝒮})$, its collection of morphisms is $\mathrm{mor}({𝒮})$, and its identities and composition are as in ${𝒞}$. There is an obvious faithful functor $I:{𝒮}\to {𝒞}$, 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 ${𝒮}$,

${\mathrm{H}\mathrm{o}\mathrm{m}}_{{𝒮}}(X,Y)={\mathrm{H}\mathrm{o}\mathrm{m}}_{{𝒞}}(X,Y).$

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 $A$.

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 ${𝒮}$ of ${𝒞}$, the inclusion functor $I:{𝒮}\to {𝒞}$ 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. 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 $F:{ℬ}\to {𝒞}$ the image of $F$ is a (full) 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 isomorphism $k:X\to Y$ in ${𝒞}$ such that $Y$ is in ${𝒮}$ also belongs to ${𝒮}$. An isomorphism-closed full subcategory is said to be strictly full.

A subcategory of ${𝒞}$ is wide or lluf (a term first posed by Peter Freyd) 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 category ${𝒞}$ such that for all short exact sequences

$0\to M^{\prime }\to M\to M^{″}\to 0$

in ${𝒞}$, $M$ belongs to ${𝒮}$ if and only if both $M^{\prime }$ and $M^{″}$ do. This notion arises from Serre's C-theory.

See also

• Reflective subcategory
• Exact category, a full subcategory closed under extensions.