Abstract tilting theory for quivers and related categories

TL;DR

This paper extends classical tilting theory for quivers to more general categories, establishing new equivalences in their representation theories across various homotopical contexts.

Contribution

It introduces a generalized framework for reflection functors applicable to arbitrary small categories, leading to new derived equivalences beyond classical quiver cases.

Findings

Generalized reflection functors induce equivalences in diverse homotopy-theoretic settings.

Recovers and extends Happel's derived equivalences for finite acyclic quivers.

Establishes new derived equivalences for non-finite, non-acyclic quivers.

Abstract

We generalize the construction of reflection functors from classical representation theory of quivers to arbitrary small categories with freely attached sinks or sources. These reflection morphisms are shown to induce equivalences between the corresponding representation theories with values in arbitrary stable homotopy theories, including representations over fields, rings or schemes as well as differential-graded and spectral representations. Specializing to representations over a field and to specific shapes, this recovers derived equivalences of Happel for finite, acyclic quivers. However, even over a field our main result leads to new derived equivalences for example for not necessarily finite or acyclic quivers. The results obtained here rely on a careful analysis of the compatibility of gluing constructions for small categories with homotopy Kan extensions and homotopical epimorphisms, as well as on a study of the combinatorics of amalgamations of categories.