Subcategory - Imported Knowlpedia Content

Let $\mathcal C$ be a category.

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:

(identities) for each $A\in\mathrm{Ob}(\mathcal D)$ , the identity morphism $\mathrm{id}_A$ lies in $\mathrm{Hom}_{\mathcal D}(A,A)$ ;
(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

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.

$\mathbf{Ab}$ 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 $\mathbf{Grp}$ .

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).