## Annals of K-Theory

### Abstract tilting theory for quivers and related categories

#### 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.

#### Article information

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

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

https://projecteuclid.org/euclid.akt/1513774603

Digital Object Identifier
doi:10.2140/akt.2018.3.71

Mathematical Reviews number (MathSciNet)
MR3695365

Zentralblatt MATH identifier
1382.55012

#### Citation

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. https://projecteuclid.org/euclid.akt/1513774603

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