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. A category with a zero object is sometimes called a pointed category.

Examples

Proposition

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

• The trivial group is a zero object in the category Grp of groups and Ab 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 $*$ the under category $\mathrm{pt}\downarrow C$ of pointed objects in $C$ has a zero object: the morphism ${\mathrm{Id}}_{\mathrm{pt}}$.

Proposition

In any category $C$ enriched over the category of pointed sets $({\mathrm{Set}}_{*},\wedge )$ with tensor product the smash product, any object that is either initial or terminal object is automatically both and hence a zero object.

Proof

Write $*\in {\mathrm{Set}}_{*}$ for the singleton pointed set. Suppose $t$ is terminal. Then $C(x,t)=*$ for all $x$ and so in particular $C(t,t)=*$ and hence the identity morphism on $t$ is the basepoint in the pointed hom-set. By the axioms of a category, every morphism $f:t\to x$ is equal to the composite

$$f:t\stackrel{\mathrm{Id}}{\to }t\stackrel{f}{\to }x.$$f : t \stackrel{Id}{\to} t \stackrel{f}{\to} x \,.

By the axioms of an $({\mathrm{Set}}_{*},\wedge )$-enriched category, since ${\mathrm{Id}}_t$ is the basepoint in $C(t,t)$, also this composite is the basepoint in $C(t,x)$ and is hence the zero morphism. So $C(t,x)=*$ for all $x$ and therefore $t$ is also an initial object.

Analogously from assuming $t$ to be initial it follows that it is also terminal.

Remark

This is a special case of an absolute limit.

Remark

Categories enriched in $({\mathrm{Set}}_{*},\wedge )$ include in particular Ab-enriched categories. So any additive category, hence every abelian category has a zero object.

Properties

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 the category of pointed sets, a partial converse to the last example above.

Related concepts

• pointed category
• zero object in a derivator