发表机构
University of Coimbra; University of the Algarve（科英布拉大学; 阿尔加维大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出原模性的相对概念（$\mathscr{M}$-版本），分析其条件、性质（如分裂短五引理）与例子，推广了普通原模范畴。
AI 中文摘要
本文研究原模性的相对概念。我们说，当（关于点纤维化的）换基函子相对于范畴 $\mathsf{C}$ 的一类态射 $\mathscr{M}$ 是保守的时，有限完备范畴 $\mathsf{C}$ 满足 $\mathscr{M}$-版本的原模性。若 $\mathscr{M}$ 是 $\mathsf{C}$ 的所有态射类或所有单态射类，则该概念恰为“普通”原模范畴。我们分析类 $\mathscr{M}$ 应具备何种条件，以获得原模范畴某些熟知性质的相对 $\mathscr{M}$-版本，例如分裂短五引理（当 $\mathsf{C}$ 也是带点范畴时）及其与 Mal'tsev 范畴的关系。我们还给出了不同 $\mathscr{M}$ 选择下满足 $\mathscr{M}$-版本原模性的各类范畴的例子。
英文摘要
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 $\mathscr{M}$.
