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A cohomology operation encoding the octahedral axiom

  • Javier Díaz Cabrera
  • Fernando Muro
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Abstract

We introduce a new secondary cohomology operation, the naturality defect product, associated with a graded natural transformation between the cohomology functors of two differential graded functors, and study its main properties. We then prove that a pretriangulated category satisfies the octahedral axiom if and only if the naturality defect product associated with the natural isomorphism given by a theorem of Heller vanishes on the homotopy category of acyclic complexes over it. This characterization extends immediately to n-angulated categories. As an application, we exhibit non-vanishing instances for the recently found examples of pretriangulated categories which are not triangulated. Finally, we show that horizontal composition of these natural isomorphisms preserves n-angulated structures, and we use it to obtain, from each of these examples and for every r >= 1, a pre-3r-angulated category which is not 3r-angulated.

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Cite

BibTeXRIS
BibTeX
@misc{cabrera2026cohomology,
  title = {{A cohomology operation encoding the octahedral axiom}},
  author = {Javier Díaz Cabrera and Fernando Muro},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.35232},
  eprint = {2610.35232},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.35232},
}
RIS
TY  - UNPB
TI  - A cohomology operation encoding the octahedral axiom
AU  - Javier Díaz Cabrera
AU  - Fernando Muro
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2610.35232
UR  - https://arxiv.org/abs/2610.35232
AB  - We introduce a new secondary cohomology operation, the naturality defect product, associated with a graded natural transformation between the cohomology functors of two differential graded functors, and study its main properties. We then prove that a pretriangulated category satisfies the octahedral axiom if and only if the naturality defect product associated with the natural isomorphism given by a theorem of Heller vanishes on the homotopy category of acyclic complexes over it. This characterization extends immediately to n-angulated categories. As an application, we exhibit non-vanishing instances for the recently found examples of pretriangulated categories which are not triangulated. Finally, we show that horizontal composition of these natural isomorphisms preserves n-angulated structures, and we use it to obtain, from each of these examples and for every r >= 1, a pre-3r-angulated category which is not 3r-angulated.
ER  -