"Natural" categories where monomorphisms differ from injective morphisms I started reading Algebra Chapter 0 on my own, and ran into the definition of a monomorphism for a category. My initial thought was that this was simply the same as the definition of injective. However, it seems at this point in the book, he has not defined injectivity for morphisms in general, only for functions between sets.
In the functions between sets case, monomorphisms are injective functions. However, in general it looks like monomorphisms are a weaker than injective morphism. Monomorphisms seems to be "left cancelable" whereas injectivity suggests there is another morphism that forces the morphism to be "left cancelable".
As I said, injective morphism has not been defined, but I would say the definition in my head is a morphism $f \in \text{Hom}_C(A, B)$ is injective if there exists $g \in \text{Hom}_C(B, A)$ such that $gf = 1_A$.
The definition of monomorphism given is $f \in \text{Hom}_C(A, B)$ is a monomorphism if for $\alpha', \alpha'' \in \text{Hom}_C(Z, A)$ we have $f\alpha' = f\alpha''$ implies $\alpha'= \alpha''$.
Clearly an injective morphism is a monomorphism, but the converse seems likely false. My counter example is a category with two objects $A, B$ with a single morphism $f \in \text{Hom}_C(A, B)$ (and of course the identity morphisms for $A$ and $B$). The composition rule will be easy since I can only compose $f$ with identity morphisms.
Now $f$ is clearly a monomorphism since the only thing $\alpha'$ and $\alpha''$ can be in the definition is $1_A$, thus necessarily the same. But $f$ is definitely not injective, as there is no morphism at all in $\text{Hom}_C(B, A).$
This counter example feels really contrived. Most the categories I know about seem to behave much like category of sets with functions as morphisms, i.e. monomorphisms are injective morphisms.
Is there an example of a category that we deal with all the time (even though we might not call it a category in polite company) where the monomorphisms are not necessarily injective?
Thanks! Any insights are appreciated.
 A: In category theory there is no notion of injectivity for morphisms; that's why we have monomorphisms instead. What you call injectivity is sometimes called being a split monomorphism (see e.g. this blog post), and in general it is much stronger than being a monomorphism. As a simple example, in the category of fields, every morphism is a monomorphism, but the split monomorphisms are precisely the isomorphisms. 
For a large class of examples, in an abelian category $C$, if $f : A \to B$ is a monomorphism then it fits into a short exact sequence
$$0 \to A \to B \to B/A \to 0$$
and $f$ is a split monomorphism if and only if this short exact sequence splits. The condition that every short exact sequence splits, or equivalently that every monomorphism is a split monomorphism, is a strong condition on $C$ called semisimplicity that is rarely satisfied in practice: for example, if $C = \text{Mod}(R)$ is the category of modules over a ring $R$, then it holds if and only if $R$ is semisimple as a ring. 
It's more typical to define injectivity with respect to a forgetful functor $F : C \to \text{Set}$, as follows.

Definition: A morphism $f$ is injective, or maybe more precisely $F$-injective, if $F(f)$ is injective. 

This condition is sometimes called "injective on underlying sets" for clarity. In this setting we have the following results.

Proposition: If $F$ is faithful, then every injective morphism is a monomorphism.

This condition is usually part of the definition of a forgetful functor, or more precisely part of the definition of a concrete category, so it's typically automatic.

Proposition: If $F$ preserves pullbacks, then every monomorphism is injective.

This condition is frequently satisfied in practice; $F$ is frequently representable, in which case it preserves all limits, not just pullbacks. 
This is why injective morphisms and monomorphisms typically coincide in familiar examples of concrete categories. (The situation is very different for surjective morphisms and epimorphisms.) So, at this point, we might ask: 

Is there a "natural" example of a concrete category in which not every monomorphism is injective in this sense? 

It's a bit tricky to find such examples because nearly every forgetful functor that comes up in practice preserves pullbacks. But here is one: consider the category of pointed, path connected, locally path connected topological spaces, with the obvious forgetful functor passing through topological spaces. Every covering map is a monomorphism in this category; this is a restatement of one of the lifting properties of covering maps. But most covering maps are not injective. 
