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2018 Abstract tilting theory for quivers and related categories
Moritz Groth, Jan Šťovíček
Ann. K-Theory 3(1): 71-124 (2018). DOI: 10.2140/akt.2018.3.71

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.

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.

Citation

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Moritz Groth. Jan Šťovíček. "Abstract tilting theory for quivers and related categories." Ann. K-Theory 3 (1) 71 - 124, 2018. https://doi.org/10.2140/akt.2018.3.71

Information

Received: 23 May 2016; Revised: 3 November 2016; Accepted: 18 February 2017; Published: 2018
First available in Project Euclid: 20 December 2017

zbMATH: 1382.55012
MathSciNet: MR3695365
Digital Object Identifier: 10.2140/akt.2018.3.71

Subjects:
Primary: 55U35
Secondary: 16E35 , 18E30 , 55U40

Keywords: reflection functor , reflection morphism , stable derivator , strong stable equivalence

Rights: Copyright © 2018 Mathematical Sciences Publishers

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