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 a full and faithful functor but not necessarily 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.