Full Subcategory

Let $\mathcal C$ be a category and $\mathcal D$ a subcategory of $\mathcal C$.

Definition

The subcategory $\mathcal D$ is full if for every pair of objects $A,B\in \mathrm{Ob}(\mathcal D)$,

In words: once you decide which objects to keep, you keep all morphisms between them.

Equivalently, the inclusion $\mathcal D\hookrightarrow \mathcal C$ is a functor that is “full on hom-sets” (it induces surjections on each hom-set).

Examples

1. $\mathbf{Ab}\subseteq \mathbf{Grp}$: The category of abelian groups is a full subcategory of the category of all groups: between abelian groups, the morphisms are exactly the same group homomorphisms as in $\mathbf{Grp}$.

2. Hausdorff spaces inside $\mathbf{Top}$: The category of Hausdorff spaces (objects: Hausdorff spaces, morphisms: continuous maps) is a full subcategory of $\mathbf{Top}$.

3. Full subcategory “spanned by” chosen objects: If $S\subseteq \mathrm{Ob}(\mathcal C)$ is any collection of objects, there is a full subcategory $\mathcal C|_S$ whose objects are exactly $S$ and whose morphisms are

   $$\mathrm{Hom}_{\mathcal C|_S}(A,B)=\mathrm{Hom}_{\mathcal C}(A,B)\quad (A,B\in S).$$

   For example, inside $\mathbf{Set}$ one can take $S=\{\emptyset,\{*\},\{0,1\}\}$ and obtain the full subcategory on those three sets.