I'm looking for a category with the titular property. Since $f : A \to B$ is supposed to be mono, the desired factorisation is automatically unique (correct?). Thus, I would need to find a category in which the factorisation does not exist, i.e., there are no arrows ending at $A$. Therefore, I went to look at preorders, where there is sometimes no arrow ending in $A$. However, unwinding the definitions, I came up empty again because the fact that parallel arrows $k, l : B \to C$, say) are equal here, makes $f$ into an isomorphism, giving an arrow back that does let $f$ fit into an equaliser diagram.
Another category I thought of (which I know to be somewhat pathological) is Field, where all morphisms are injective and thus mono (correct?). Since fields are also rings, where oftentimes morphisms to an object don't exists, I thought this might be a nice example. However, the condition I need to prove is quite strong. Sure, I can cook up some monos that don't fit into certain equaliser diagrams, but to be regular, they should fit in no equaliser diagrams.
I have also tried some 'toy' categories with just a few objects and arrows, but then many things are vacuously true, and, again, I come up empty.
Is there anything in my attempts above that I could pursue further? Or do I really need a very exotic category that I won't come up with myself? (Admission: since in Cat, the morphisms are functors, and there are so many things to check, I haven't spent much time on that. Let alone functor categories $[\mathcal{C}, \mathcal{D}]$, where the natural transformations are the morphisms.)
I do notice that in this sort of quest, one learns many things, and finds that one does not know the answer to many seemingly simple questions (e.g., what are the monos in preorders, what are the (relevant) roles of initial and terminal objects here, can finite categories work at all, etc.? Any additional (if tangential) information on these questions would be greatly appreciated! =D)
EDIT
LOL, I look two exercises ahead: "characterise the regular monos in Pos". Clearly, the category of posets should be an example (or this would be a very trivial exercise..)
So that's what I'm going to focus on now!
Best Answer
One example is the category of topological spaces. The regular monomorphisms are subspaces, while subobjects can have a topology that is finer than the subspace topology. So concretely, the “identity map” $2 \to 2$, where the domain has the discrete topology and the codomain has the indiscrete topology, is a monomorphism which is not regular.