Annals of K-Theory

Abstract tilting theory for quivers and related categories

Moritz Groth and Jan Šťovíček

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/akt.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

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

Permanent link to this document
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

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

Keywords
stable derivator reflection functor reflection morphism strong stable equivalence

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


Export citation

References

  • J. Ayoub, Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique, II, Astérisque 315, Société Mathématique de France, Paris, 2007.
  • I. N. Bernšteĭn, I. M. Gel'fand, and V. A. Ponomarev, “Coxeter functors, and Gabriel's theorem”, Uspekhi Mat. Nauk 28:2(170) (1973), 19–33. In Russian; translated in Russian Math. Surveys 28:2 (1973), 17–32.
  • F. Borceux, Handbook of categorical algebra, 1: Basic category theory, Encyclopedia of Mathematics and its Applications 50, Cambridge University Press, 1994.
  • D.-C. Cisinski, “Images directes cohomologiques dans les catégories de modèles”, Ann. Math. Blaise Pascal 10:2 (2003), 195–244.
  • D. Dugger and B. Shipley, “$K$-theory and derived equivalences”, Duke Math. J. 124:3 (2004), 587–617.
  • J. Franke, “Uniqueness theorems for certain triangulated categories with an Adams spectral sequence”, preprint, 1996, http://www.math.uiuc.edu/K-theory/0139/Adams.pdf.
  • P. Gabriel, “Unzerlegbare Darstellungen, I”, Manuscripta Math. 6:1 (1972), 71–103. Correction in 6:3 (1972), 309.
  • W. Geigle and H. Lenzing, “Perpendicular categories with applications to representations and sheaves”, J. Algebra 144:2 (1991), 273–343.
  • T. G. Goodwillie, “Calculus, II: Analytic functors”, $K$-Theory 5:4 (1991), 295–332.
  • M. Groth, “A short course on $\infty$-categories”, preprint, 2010.
  • M. Groth, “Derivators, pointed derivators and stable derivators”, Algebr. Geom. Topol. 13:1 (2013), 313–374.
  • M. Groth and J. Š\v tovíček, “Abstract representation theory of Dynkin quivers of type $A$”, Adv. Math. 293 (2016), 856–941.
  • M. Groth and J. Š\v tovíček, “Tilting theory for trees via stable homotopy theory”, J. Pure Appl. Algebra 220:6 (2016), 2324–2363.
  • M. Groth and J. Š\v tovíček, “Tilting theory via stable homotopy theory”, J. Reine Angew. Math. (online publication February 2016).
  • M. Groth and J. Š\v tovíček, “Spectral Serre duality for acyclic quivers”, in preparation.
  • M. Groth, K. Ponto, and M. Shulman, “The additivity of traces in monoidal derivators”, J. K-Theory 14:3 (2014), 422–494.
  • M. Groth, K. Ponto, and M. Shulman, “Mayer–Vietoris sequences in stable derivators”, Homology Homotopy Appl. 16:1 (2014), 265–294.
  • A. Grothendieck, “Les dérivateurs”, preprint, 1991, http://www.math.jussieu.fr/~maltsin/groth/Derivateurs.html.
  • D. Happel, “Dynkin algebras”, pp. 1–14 in Sémin. d'algèbre P. Dubreil et M.-P. Malliavin, 37ème année (Paris, 1985), edited by M.-P. Malliavin, Lecture Notes in Math. 1220, Springer, 1986.
  • D. Happel, “On the derived category of a finite-dimensional algebra”, Comment. Math. Helv. 62:3 (1987), 339–389.
  • A. Heller, Homotopy theories, Memoirs of the Amer. Math. Soc. 383, 1988.
  • P. S. Hirschhorn, Model categories and their localizations, Mathematical Surveys and Monographs 99, American Mathematical Society, Providence, RI, 2003.
  • M. Hovey, Model categories, Mathematical Surveys and Monographs 63, American Mathematical Society, Providence, RI, 1999.
  • A. Joyal, “The theory of quasi-categories and its applications”, lecture notes, Centre de Recerca Matemàtica, 2008, http://mat.uab.cat/~kock/crm/hocat/advanced-course/Quadern45-2.pdf.
  • A. Joyal, “The theory of quasi-categories, I–II”, book in progress.
  • A. Joyal and M. Tierney, “Strong stacks and classifying spaces”, pp. 213–236 in Category theory (Como, 1990), edited by A. Carboni et al., Lecture Notes in Math. 1488, Springer, 1991.
  • S. Ladkani, “Universal derived equivalences of posets”, preprint, 2007.
  • J. Lurie, Higher topos theory, Annals of Mathematics Studies 170, Princeton University Press, 2009.
  • J. Lurie, “Higher algebra”, preprint, 2016, http://www.math.harvard.edu/~lurie/papers/HA.pdf.
  • S. Mac Lane, Categories for the working mathematician, 2nd ed., Graduate Texts in Mathematics 5, Springer, 1998.
  • J. MacDonald and L. Scull, “Amalgamations of categories”, Canad. Math. Bull. 52:2 (2009), 273–284.
  • G. Maltsiniotis, “Introduction à la théorie des dérivateurs”, preprint, 2001, https://webusers.imj-prg.fr/~georges.maltsiniotis/ps/m.ps.
  • G. Maltsiniotis, “Catégories triangulées supérieures”, preprint, 2005, https://webusers.imj-prg.fr/~georges.maltsiniotis/ps/triansup.ps.
  • G. Maltsiniotis, “Carrés exacts homotopiques et dérivateurs”, Cah. Topol. Géom. Différ. Catég. 53:1 (2012), 3–63.
  • K. Ponto and M. Shulman, “The linearity of traces in monoidal categories and bicategories”, Theory Appl. Categ. 31 (2016), Paper No. 23, 594–689.
  • D. G. Quillen, Homotopical algebra, Lecture Notes in Math. 43, Springer, 1967.
  • O. Renaudin, “Plongement de certaines théories homotopiques de Quillen dans les dérivateurs”, J. Pure Appl. Algebra 213:10 (2009), 1916–1935.
  • J. Rickard, “Morita theory for derived categories”, J. London Math. Soc. $(2)$ 39:3 (1989), 436–456.
  • C. M. Ringel, Tame algebras and integral quadratic forms, Lecture Notes in Math. 1099, Springer, 1984.
  • D. Simson and A. Skowroński, Elements of the representation theory of associative algebras, 2: Tubes and concealed algebras of Euclidean type, London Mathematical Society Student Texts 71, Cambridge University Press, 2007.
  • V. Trnková, “Sum of categories with amalgamated subcategory”, Comment. Math. Univ. Carolinae 6 (1965), 449–474.