What is a zero morphism in an abelian category

Question

I am trying to familiarize myself with some basic category theory and I am getting confused with what a $0$-morphism is.

If we are in category of say $k$-vector spaces then I am guessing $0$-morphism would be the map that sends everything to $0$. In these examples it makes more sense to me because each object has this $0$ element. But it's not really clear to me what happens in more abstract cases.

I would appreciate if someone could explain to me how I should think of these $0$-morphisms. Thanks!

Answer 1

The zero morphism $A \to B$ can be factored into

$$ A \to 0 \to B $$

where $0$ is a zero object. (i.e. it is a terminal object and an initial object)

As an aside, when you wrote it as "$0$-morphism", my first reaction was that you were referring to the concept from higher category theory; e.g. in $\mathbf{Cat}$, categories are $0$-morphisms (i.e. objects), functors are $1$-morphisms, and natural transformations are $2$-morphisms.

Answer 2

Another way to put it.

An abelian category $\mathscr M$ is in particular a category enriched over the category $\mathsf{Ab}$ of abelian groups. So for two objects $A,B$ of $\mathscr M$, the hom-set $\hom_{\mathscr M}(A,B)$ actually carries a structure of abelian group : in particular, it has a neutral, which is the zero map from $A$ to $B$.