Characterization of split-monomorphism protomodularity by pullback reflection

Characterize whether the equivalent properties that pullbacks reflect monomorphisms and that every change-of-base functor reflects monomorphisms characterize split-monomorphism-protomodular categories.

Background

For the class of split monomorphisms, the paper proves that relative protomodularity implies that every change-of-base functor reflects monomorphisms. It then proves that this property is equivalent to the categorical condition that pullbacks reflect monomorphisms.

The unresolved issue is whether these equivalent reflection properties are not merely necessary but also sufficient for split-monomorphism-protomodularity.

References

We do not know whether the equivalent properties of Lemma~\ref{lem: h* reflects monos iff pbs reflect monos} characterise $$-protomodular categories.

— How protomodular is your favourite category?  (2610.09739 - Clementino et al., 7 Oct 2026) in Remark immediately after Lemma 2.2, Section 3 (the section titled “$$-protomodular categories”)