Annals of K-Theory

Abstract tilting theory for quivers and related categories

Moritz Groth and Jan Šťovíček

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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.

Our results rely on a careful analysis of the compatibility of gluing constructions for small categories with homotopy Kan extensions and homotopical epimorphisms, and on a study of the combinatorics of amalgamations of categories.

Article information

Ann. K-Theory, Volume 3, Number 1 (2018), 71-124.

Received: 23 May 2016
Revised: 3 November 2016
Accepted: 18 February 2017
First available in Project Euclid: 20 December 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55U35: Abstract and axiomatic homotopy theory
Secondary: 16E35: Derived categories 18E30: Derived categories, triangulated categories 55U40: Topological categories, foundations of homotopy theory

stable derivator reflection functor reflection morphism strong stable equivalence


Groth, Moritz; Šťovíček, Jan. Abstract tilting theory for quivers and related categories. Ann. K-Theory 3 (2018), no. 1, 71--124. doi:10.2140/akt.2018.3.71.

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