Year: 2026
Type: preprint
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 $\mathscr{M}$.
Source: arXiv (Cornell University)
Authors: Clementino, Maria Manuel, Rodelo, Diana
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: Centro de Investigação e Desenvolvimento em Matemática e Aplicações, Centro de Matemática da Universidade de Coimbra, Centro de Investigação e Desenvolvimento em Matemática e Aplicações, G6265307419, G7049327362, Centro de Matemática da Universidade de Coimbra, G7318052850, G7318643082, G8321563775, G8511605407
OpenAlex ID: W7220976861