additive and abelian categories
(AB1) pre-abelian category
(AB2) abelian category
(AB5) Grothendieck category
left/right exact functor
In a category, an object is called a zero object, or null object if it is both an initial object and a terminal object.
A category with a zero object is sometimes called a pointed category.
This means that $0 \in \mathcal{C}$ is a zero object precisely if for every other object $A$ there is a unique morphism $A \to 0$ to the zero object as well as a unique morphism $0 \to A$ from the zero object.
There is also a notion of zero object in algebra which does not always coincide with the category-theoretic terminology. For example the zero ring $\{0\}$ is not an initial object in the category of unital rings (this is instead the integers $\mathbb{Z}$); but it is the terminal object. However, the zero ring is the zero object in the category of nonunital rings (although it happens to be unital).
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 in the category Ab of abelian groups.
For $R$ a ring, the trivial $R$-module (that whose underlying abelian group is the trivial group) is the zero-object in $R$Mod.
In particular for $R = k$ a field, the $k$-vector space of dimension 0 is the zero object in Vect.
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 $pt \downarrow C$ of pointed objects in $C$ has a zero object: the morphism $Id_{pt}$.
In any category $C$ enriched over the category of pointed sets $(Set_*, \wedge)$ with tensor product the smash product, any object that is either initial or terminal is automatically both and hence a zero object.
Write $* \in Set_*$ 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
By the axioms of an $(Set_*, \wedge)$-enriched category, since $Id_{t}$ 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.
This is a special case of an absolute limit.
Categories enriched in $(Set_*, \wedge)$ include in particular Ab-enriched categories. So any additive category, hence every abelian category has a zero object.
A category has a zero object precisely if it has an initial object $\emptyset$ and a terminal object $*$ and the unique morphism $\emptyset \to *$ is an isomorphism.
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 prop 2.