Vol. 3, No. 1, 2018

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Abstract tilting theory for quivers and related categories

Moritz Groth and Jan Šťovíček

Vol. 3 (2018), No. 1, 71–124

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.

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stable derivator, reflection functor, reflection morphism, strong stable equivalence
Mathematical Subject Classification 2010
Primary: 55U35
Secondary: 16E35, 18E30, 55U40
Received: 23 May 2016
Revised: 3 November 2016
Accepted: 18 February 2017
Published: 7 September 2017
Moritz Groth
Mathematisches Institut
Rheinische Friedrich-Wilhelms-Universität Bonn
Jan Šťovíček
Department of Algebra
Charles University in Prague
Czech Republic