Let $\mathscr{A}$ be a category. Then we say that $\mathscr{A}$ is a category with zero morphisms if for every $A,A'\in\mathscr{A}$ there is a zero morphism $0_{AA'}\in\mathscr{A}(A,A')$, and the zero morphisms obey a particular commutative diagram (see wiki). Now suppose $\mathscr{A}$ has a zero object $0$. Then $\mathscr{A}$ is a category with zero morphisms, and every zero morphism factors through the zero object uniquely. So how about the converse? If $\mathscr{A}$ is a category with zero morphisms, does it necessarily have a zero object? If not, is there any simple counterexample(s)?
Category with zero morphisms implies zero object
category-theory
Best Answer
No, a category with zero morphisms need not have a zero object. A simple counterexample is to consider a nonzero ring $R$ considered as a one-object category (even a one-object $\text{Ab}$-enriched / pre-additive category), or more generally a monoid with a zero element / absorbing element and at least one other nonzero element (but nonzero rings are nice as a common and familiar example of these).
What is true is that given a category with zero morphisms there is a unique way to adjoin a zero object to it if it doesn't already have one: it has a unique morphism to and from every other object, and every composition involving these morphisms is zero. This construction is the left adjoint of the inclusion of (categories with zero objects) into (categories with zero morphisms), where in both cases morphisms are functors that preserve zero morphisms.
Also, if a category with zero morphisms has either an initial or terminal object, that object is automatically a zero object, and a functor between two categories-with-zero-objects that preserves zero morphisms automatically preserves zero objects. I go into a bit more detail in this blog post.