Reasonable triangulated categories have filtered enhancements
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Abstract:
We prove that a triangulated category which is the underlying category of a stable derivator has a filtered enhancement, providing an affirmative answer to a conjecture in [J. K-Theory 6 (2010), pp. 387–504].References
- A. A. Beĭlinson, J. Bernstein, and P. Deligne, Faisceaux pervers, Analysis and topology on singular spaces, I (Luminy, 1981) Astérisque, vol. 100, Soc. Math. France, Paris, 1982, pp. 5–171 (French). MR 751966
- A. A. Beĭlinson, On the derived category of perverse sheaves, $K$-theory, arithmetic and geometry (Moscow, 1984–1986) Lecture Notes in Math., vol. 1289, Springer, Berlin, 1987, pp. 27–41. MR 923133, DOI 10.1007/BFb0078365
- M. V. Bondarko, Weight structures vs. $t$-structures; weight filtrations, spectral sequences, and complexes (for motives and in general), J. K-Theory 6 (2010), no. 3, 387–504. MR 2746283, DOI 10.1017/is010012005jkt083
- Francis Borceux, Handbook of categorical algebra. 1, Encyclopedia of Mathematics and its Applications, vol. 50, Cambridge University Press, Cambridge, 1994. Basic category theory. MR 1291599
- Luc Illusie, Complexe cotangent et déformations. I, Lecture Notes in Mathematics, Vol. 239, Springer-Verlag, Berlin-New York, 1971 (French). MR 0491680
- Samuel Eilenberg and Saunders Mac Lane, On the groups $H(\Pi ,n)$. I, Ann. of Math. (2) 58 (1953), 55–106. MR 56295, DOI 10.2307/1969820
- Halvard Fausk and Daniel C. Isaksen, t-model structures, Homology Homotopy Appl. 9 (2007), no. 1, 399–438. MR 2299805, DOI 10.4310/HHA.2007.v9.n1.a16
- Moritz Groth, Derivators, pointed derivators and stable derivators, Algebr. Geom. Topol. 13 (2013), no. 1, 313–374. MR 3031644, DOI 10.2140/agt.2013.13.313
- A. Grothendieck, Les Dérivateurs, manuscript available at https://webusers.imj-prg.fr /~georges.maltsiniotis/groth/Derivateurs.html.
- Mark Hovey, Brooke Shipley, and Jeff Smith, Symmetric spectra, J. Amer. Math. Soc. 13 (2000), no. 1, 149–208. MR 1695653, DOI 10.1090/S0894-0347-99-00320-3
- Bernhard Keller, Deriving DG categories, Ann. Sci. École Norm. Sup. (4) 27 (1994), no. 1, 63–102. MR 1258406
- Bernhard Keller and Pedro Nicolás, Weight structures and simple dg modules for positive dg algebras, Int. Math. Res. Not. IMRN 5 (2013), 1028–1078. MR 3031826, DOI 10.1093/imrn/rns009
- Henning Krause, Localization theory for triangulated categories, Triangulated categories, London Math. Soc. Lecture Note Ser., vol. 375, Cambridge Univ. Press, Cambridge, 2010, pp. 161–235. MR 2681709, DOI 10.1017/CBO9781139107075.005
- Amnon Neeman, Triangulated categories, Annals of Mathematics Studies, vol. 148, Princeton University Press, Princeton, NJ, 2001. MR 1812507, DOI 10.1515/9781400837212
- Amnon Neeman, Some new axioms for triangulated categories, J. Algebra 139 (1991), no. 1, 221–255. MR 1106349, DOI 10.1016/0021-8693(91)90292-G
- Marco Porta, The Popescu-Gabriel theorem for triangulated categories, Adv. Math. 225 (2010), no. 3, 1669–1715. MR 2673743, DOI 10.1016/j.aim.2010.04.002
- Chrysostomos Psaroudakis and Jorge Vitória, Realisation functors in tilting theory, Math. Z. 288 (2018), no. 3-4, 965–1028. MR 3778987, DOI 10.1007/s00209-017-1923-y
- O. Schnürer, Homotopy categories and idempotent completeness, weight structures and weight complex functors, preprint, arXiv:1107.1227 [math.CT], 2011.
- S. Virili, Morita theory for stable derivators, preprint, arXiv:1807.01505 [math.KT], 2018.
Additional Information
- George Ciprian Modoi
- Affiliation: Babeş–Bolyai University, Faculty of Mathematics and Computer Science , 1, Mihail Kogălniceanu, 400084 Cluj–Napoca, Romania
- MR Author ID: 671168
- Email: cmodoi@math.ubbcluj.ro
- Received by editor(s): December 7, 2017
- Received by editor(s) in revised form: September 7, 2018
- Published electronically: March 26, 2019
- Communicated by: Alexander Braverman
- © Copyright 2019 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 147 (2019), 2761-2773
- MSC (2010): Primary 18E30, 55U35, 18D05
- DOI: https://doi.org/10.1090/proc/14474
- MathSciNet review: 3973880