Subobject

Definition

Let $X:\mathcal{C}$ be an object of a category $\mathcal{C}$.

The subobject category of $X$ is the skeleton of the full subcategory of the slice category $\mathcal{C}/X$ spanned by the monomorphisms into $X$:

$${\mathrm{Sub}}_{\mathcal{C}}(X):=\mathrm{sk}((\mathcal{C}/X{)}_{\mathrm{mono}}).$$

Equivalently, its objects are monomorphisms

$$m:A\hookrightarrow X$$

up to isomorphism over $X$, and there is a morphism

$$m\to n$$

exactly when $m$ factors through $n$.

Remarks

This is valid in categories without a subobject classifier. A subobject classifier gives a classifier morphism corresponding ${\chi }_m:\mathcal{C}[X,\Omega ]$ to the subobject $m:A\rightarrowtail X$.