Subcategory

Let $\mathcal C$ be a category .

Definition

A subcategory $\mathcal D$ of $\mathcal C$ consists of:

• a collection of objects $\mathrm{Ob}(\mathcal D)\subseteq \mathrm{Ob}(\mathcal C)$,
• for every $A,B\in \mathrm{Ob}(\mathcal D)$, a subset of morphisms $\mathrm{Hom}_{\mathcal D}(A,B)\subseteq \mathrm{Hom}_{\mathcal C}(A,B),$ such that:

1. (identities) for each $A\in\mathrm{Ob}(\mathcal D)$, the identity morphism $\mathrm{id}_A$ lies in $\mathrm{Hom}_{\mathcal D}(A,A)$;
2. (closure under composition) if $f\in\mathrm{Hom}_{\mathcal D}(A,B)$ and $g\in\mathrm{Hom}_{\mathcal D}(B,C)$, then their composite $g\circ f$ (computed in $\mathcal C$) lies in $\mathrm{Hom}_{\mathcal D}(A,C)$.

In this situation, $\mathcal D$ is itself a category , with composition and identities inherited from $\mathcal C$.

A particularly important case is a full subcategory , where $\mathcal D$ contains all morphisms in $\mathcal C$ between its objects.

Examples

1. Injective maps inside $\mathbf{Set}$: Let $\mathcal C=\mathbf{Set}$.
   Define $\mathcal D$ to have the same objects as $\mathbf{Set}$, but only injective functions as morphisms.
   This is a subcategory of $\mathbf{Set}$ (closed under composition and contains identities), but it is not full.

2. $\mathbf{Ab}$ inside $\mathbf{Grp}$: The category $\mathbf{Ab}$ of abelian groups is a subcategory of $\mathbf{Grp}$ by restricting to those objects that happen to be abelian.
   Moreover, it is a full subcategory : between two abelian groups, a group homomorphism is the same morphism whether viewed in $\mathbf{Ab}$ or in $\mathbf{Grp}$.

3. Hausdorff spaces inside $\mathbf{Top}$: Let $\mathcal D$ be the category whose objects are Hausdorff spaces and whose morphisms are continuous maps.
   Then $\mathcal D$ is a subcategory of $\mathbf{Top}$, and in fact it is full (all continuous maps between Hausdorff spaces are included).