Year: 2026
Type: article
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 $$\textrm{Hom}$$ Hom -bifunctor in either argument. Since $$\textrm{Hom}$$ 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 $$\delta $$ 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 $$\infty $$ ∞ -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 $$(\mathscr {C},\mathbb {E},\mathfrak {s})$$ ( C , E , s ) , we determine the maximal relative extriangulated structure, with respect to which the sequence is a $$\delta $$ δ -functor. We also find several equivalent criteria for the existence of enough projective or injective morphisms in a given extriangulated category.
Source: Mathematische Zeitschrift
Authors: Mikhail Gorsky, Hiroyuki Nakaoka, Yann Palu
Institutions: Lyon 1 Université, École Centrale de Lyon, Centre National de la Recherche Scientifique, University of Vienna, Institut Camille Jordan, Université Jean Monnet, Institut National des Sciences Appliquées de Lyon, Université Jean Monnet, University of the Ryukyus, Laboratoire Amiénois de Mathématique Fondamentale et Appliquée, Université de Picardie Jules Verne
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,290
Awards: G3010527862, SFB 1624: Höhere Strukturen, Modulräume und Integrabilität, Development of unified homological algebra encompassing exact categories and triangulated categories from viewpoint of higher categories and enriched categories, G6572250646, Development of unified homological algebra encompassing exact categories and triangulated categories from viewpoint of higher categories and enriched categories, G813609259, Clusters, Homological Algebra, Representations and Mirror Symmetry
OpenAlex ID: W4287254645