Tilting theory for trees via stable homotopy theory

Abstract

We show that variants of the classical reflection functors from quiver representation theory exist in any abstract stable homotopy theory, making them available for example over arbitrary ground rings, for quasi-coherent modules on schemes, in the differential-graded context, in stable homotopy theory as well as in the equivariant, motivic, and parametrized variant thereof. As an application of these equivalences we obtain abstract tilting results for trees valid in all these situations, hence generalizing a result of Happel. The main tools introduced for the construction of these reflection functors are homotopical epimorphisms of small categories and one-point extensions of small categories, both of which are inspired by similar concepts in homological algebra.

To appear in J. Pure Appl. Algebra, it is a sequel to arXiv:1401.6451 and continues the development of abstract tilting theory. Version 2: various improvements in the presentation. Version 3: a detailed explanation added (in Construction 9.13 and Lemma 9.15) for the key fact that both the branches of Figure 2 lead to the same category