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Peer-reviewedCC BY 4.0Mathematische Zeitschrift

Positive and negative extensions in extriangulated categories

  • Mikhail Gorsky1,2
  • Hiroyuki Nakaoka3
  • Yann Palu4
4 affiliations
  1. 1Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, 1090, Wien, Austria
  2. 2Institut Camille Jordan UMR 5208, Université Jean Monnet, CNRS, Centrale Lyon, INSA Lyon, Université Claude Bernard Lyon 1, 20, rue Annino, 42023, Saint-Étienne, France
  3. 3Department of Mathematical Sciences, University of the Ryukyus, 1 Senbaru, Nishihara, Okinawa, 903-0213, Japan
  4. 4LAMFA, Université Picardie Jules Verne, 80039, Amiens, France
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Abstract

We initiate the study of derived functors in the setting of extriangulated categories. By using coends, we adapt Yoneda’s theory of higher extensions to this framework. We show that, when there are enough projectives and enough injectives, thus defined extensions agree with the ones defined earlier via projective/injective resolutions. For categories with enough projective or enough injective morphisms, we prove that these are right derived functors of the Hom Hom -bifunctor in either argument. Since Hom Hom is only half-exact in each argument, it is natural to expect “negative extensions", i.e. its left derived functors, to exist and not necessarily vanish. We define negative extensions with respect to the first and to the second argument and show that they give rise to universal δ δ -functors when there are enough projective or injective morphisms, respectively. In general, they are not balanced. However, for topological extriangulated categories, the existence of a balanced version of negative extensions follows from combining the work of Klemenc on exact ∞ ∞ -categories with results of the second and third authors. We discuss various criteria under which one has the balance of the above bifunctors on the functorial or on the numerical levels. This happens, in particular, in the cases of exact or triangulated categories, and also in the case of the category of n -term complexes with projective components over a finite-dimensional algebra. Given a connected sequence of functors on an extriangulated category (𝒞,𝔼,s) ( C , E , s ) , we determine the maximal relative extriangulated structure, with respect to which the sequence is a δ δ -functor. We also find several equivalent criteria for the existence of enough projective or injective morphisms in a given extriangulated category.

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Cite

BibTeXRIS
BibTeX
@article{gorsky2026positive,
  title = {{Positive and negative extensions in extriangulated categories}},
  author = {Mikhail Gorsky and Hiroyuki Nakaoka and Yann Palu},
  journal = {Mathematische Zeitschrift},
  year = {2026},
  month = aug,
  doi = {10.1007/s00209-026-04051-2},
  url = {https://doi.org/10.1007/s00209-026-04051-2},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - Positive and negative extensions in extriangulated categories
AU  - Mikhail Gorsky
AU  - Hiroyuki Nakaoka
AU  - Yann Palu
PY  - 2026
DA  - 2026/08/01
JO  - Mathematische Zeitschrift
DO  - 10.1007/s00209-026-04051-2
UR  - https://doi.org/10.1007/s00209-026-04051-2
AB  - We initiate the study of derived functors in the setting of extriangulated categories. By using coends, we adapt Yoneda’s theory of higher extensions to this framework. We show that, when there are enough projectives and enough injectives, thus defined extensions agree with the ones defined earlier via projective/injective resolutions. For categories with enough projective or enough injective morphisms, we prove that these are right derived functors of the Hom Hom -bifunctor in either argument. Since Hom Hom is only half-exact in each argument, it is natural to expect “negative extensions", i.e. its left derived functors, to exist and not necessarily vanish. We define negative extensions with respect to the first and to the second argument and show that they give rise to universal δ δ -functors when there are enough projective or injective morphisms, respectively. In general, they are not balanced. However, for topological extriangulated categories, the existence of a balanced version of negative extensions follows from combining the work of Klemenc on exact ∞ ∞ -categories with results of the second and third authors. We discuss various criteria under which one has the balance of the above bifunctors on the functorial or on the numerical levels. This happens, in particular, in the cases of exact or triangulated categories, and also in the case of the category of n -term complexes with projective components over a finite-dimensional algebra. Given a connected sequence of functors on an extriangulated category (𝒞,𝔼,s) ( C , E , s ) , we determine the maximal relative extriangulated structure, with respect to which the sequence is a δ δ -functor. We also find several equivalent criteria for the existence of enough projective or injective morphisms in a given extriangulated category.
SN  - 0025-5874
SN  - 1432-1823
ER  -