How protomodular is your favourite category?

Abstract: In this work we investigate a relative notion of protomodularity. We say that a finitely complete category $\mathsf{C}$ satisfies the \defn{$\mathscr{M}$-version of protomodularity} when the change-of-base functors (concerning the fibration of points) are conservative with respect to a class of morphisms $\mathscr{M}$ of $\mathsf{C}$. If $\mathscr{M}$ is the class of all morphisms or the class of all monomorphisms of $\mathsf{C}$, then this notion is precisely that of an ``ordinary'' protomodular category. We analyse what conditions the class $\mathscr{M}$ should have to obtain the relative $\mathscr{M}$-versions of some well-known properties of protomodular categories, such as the Split Short Five Lemma (when $\mathsf{C}$ is also pointed) and their relation with Mal'tsev categories. We also give diverse examples of categories satisfying the $\mathscr{M}$-version of protomodularity for different choices of $\mathscr{M}$.