(AB1) pre-abelian category
(AB2) abelian category
(AB5) Grothendieck category
A category with a zero object is sometimes called a pointed category.
This means that is a zero object precisely if for every other object there is a unique morphism to the zero object as well as a unique morphism 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 is not an initial object in the category of unital rings (this is instead the integers ); 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.
Write for the singleton pointed set. Suppose is terminal. Then for all and so in particular and hence the identity morphism on is the basepoint in the pointed hom-set. By the axioms of a category, every morphism is equal to the composite
Analogously from assuming to be initial it follows that it is also terminal.
This is a special case of an absolute limit.
In a category with a zero object 0, there is always a canonical morphism from any object to any other object called the zero morphism, given by the composite .