nLab
zero object

Definition

A zero object, or null object, is an object of a category that is both an initial object and a terminal object. Equivalently, a category has a zero object iff it has an initial object $\perp$ and a terminal object $\top$ and the unique morphism $\perp \to \top$ is an isomorphism.

Examples

• The category of pointed sets has a zero object, namely any one-element set.

• The trivial group is a zero object in the categories of groups and of abelian groups.

• However, the zero ring is not a zero object in the category of rings, at least as long as rings are required to have units (and ring homomorphisms to preserve them).

• For every category $C$ with a terminal object $\mathrm{pt}$ the under category $\mathrm{pt}\downarrow C$ of pointed objects in $C$ has a zero object: the morphism ${\mathrm{Id}}_{\mathrm{pt}}$.

• In any category enriched over pointed sets or abelian groups, any initial or terminal object is automatically a zero object. In this case a zero object $0$ can also be characterized by the identity $1_0=0_{\mathrm{0,0}}$, i.e. the identity morphism of the zero object is equal to the zero morphism from it to itself. This is a special case of an absolute limit?. In particular, any additive category has a zero object.

Consequences

In a category with a zero object 0, there is always a canonical morphism from any object $A$ to any other object $B$ called the zero morphism, given by the composite $A\to 0\to B$. Thus, such a category becomes enriched over pointed sets, a partial converse to the last example above.