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PreprintCC BY 4.0arXiv

How protomodular is your favourite category?

  • Maria Manuel Clementino
  • Diana Rodelo
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Abstract

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

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Cite

BibTeXRIS
BibTeX
@misc{clementino2026protomodular,
  title = {{How protomodular is your favourite category?}},
  author = {Maria Manuel Clementino and Diana Rodelo},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.09739},
  eprint = {2610.09739},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.09739},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - How protomodular is your favourite category?
AU  - Maria Manuel Clementino
AU  - Diana Rodelo
PY  - 2026
DA  - 2026/10/07
PB  - arXiv
DO  - 10.48550/arxiv.2610.09739
UR  - https://arxiv.org/abs/2610.09739
AB  - In this work we investigate a relative notion of protomodularity. We say that a finitely complete category C satisfies the \defn{ℳ-version of protomodularity} when the change-of-base functors (concerning the fibration of points) are conservative with respect to a class of morphisms ℳ of C. If ℳ is the class of all morphisms or the class of all monomorphisms of C, then this notion is precisely that of an ``ordinary'' protomodular category. We analyse what conditions the class ℳ should have to obtain the relative ℳ-versions of some well-known properties of protomodular categories, such as the Split Short Five Lemma (when C is also pointed) and their relation with Mal'tsev categories. We also give diverse examples of categories satisfying the ℳ-version of protomodularity for different choices of ℳ.
ER  -