Hostname: page-component-cb9f654ff-mwwwr Total loading time: 0 Render date: 2025-09-01T16:37:57.814Z Has data issue: false hasContentIssue false
  • English
  • Français

Cohomologically triangulated categories II

Published online by Cambridge University Press:  01 December 2008

H.-J. Baues  and
F. Muro
H.-J. Baues
Affiliation:
Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany, baues@mpim-bonn.mpg.de.
F. Muro
Affiliation:
Universitat de Barcelona, Departament d'Àlgebra i Geometria, Gran via de les corts catalanes 585, 08007 Barcelona, Spain, fmuro@ub.edu.
Get access

Abstract

A cohomologically triangulated category is an additive category A together with a translation functor t and a cohomology class Δ ∈ H3(A,t) such that any good translation track category representing Δ is a triangulated track category. In this paper we give purely cohomological conditions on Δ which imply that (A,t,Δ) is a cohomologically triangulated category, avoiding the use of track categories. This yields a purely cohomological characterization of triangulated cohomology classes.

Keywords

triangulated categoriespretriangulated categoriescohomology of categories

Information

Type
Research Article
Information
Journal of K-Theory , Volume 3 , Issue 1 , February 2009 , pp. 1 - 52
Copyright
Copyright © ISOPP 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Bau89.Baues, H.-J.. Algebraic Homotopy. Cambridge University Press, 1989CrossRefGoogle Scholar
Bau97.Baues, H.-J.. On the cohomology of categories, universal Toda brackets and homotopy pairs. K-Theory 11 (3) (1997):259–285Google Scholar
Bau06.Baues, H.-J.. Triangulated track categories. Georgian Math. J. 13 (4) (2006):607–634Google Scholar
BD89.Baues, H.-J. and Dreckmann, W.. The cohomology of homotopy categories and the general linear group. K-Theory 3 (4) (1989) :307–338CrossRefGoogle Scholar
BHP97.Baues, H.-J., Hartl, M., and Pirashvili, T.. Quadratic categories and square rings. J. Pure Appl. Algebra 122 (1997):1–40CrossRefGoogle Scholar
BM07a.Baues, H.-J. and Muro, F.. Cohomologically triangulated categories I. Journal of K-Theory 1 (1) (2008), 3–48Google Scholar
BM07b.Baues, H.-J. and Muro, F.. The homotopy category of pseudofunctors and translation cohomology. J. Pure Appl. Algebra 211 (3) (2007):821–850,CrossRefGoogle Scholar
BP06.Baues, H.-J. and Pirashvili, T.. Comparison of Mac Lane, Shukla and Hochschild cohomologies. J. Reine Angew. Math. 598 (2006):25–69Google Scholar
BT96.Baues, H.-J. and Tonks, A.. On sum-normalised cohomology of categories, twisted homotopy pairs and universal Toda brackets. Quart. J. Math. Oxford Ser. (2) 188 (1996):405–433CrossRefGoogle Scholar
BW85.Baues, H.-J. and Wirsching, G.. Cohomology of small categories. J. Pure Appl. Algebra 38 (2–3) (1985):187–211,CrossRefGoogle Scholar
DP61.Dold, A. and Puppe, D.. Homologie nicht-additiver Funktoren, Anwendungen. Ann. Inst. Fourier, 11 (6) (1961):201–312CrossRefGoogle Scholar
Hov99.Hovey, M.. Model categories, Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 1999Google Scholar
JP91.Jibladze, M. and Pirashvili, T.. Cohomology of algebraic theories. Journal of Algebra 137 (1991):253–296Google Scholar
Mit72.Mitchell, B.. Rings with several objects. Advances in Mathematics 8 (1972):1–161CrossRefGoogle Scholar
MSS07.Muro, F., Schwede, S., and Strickland, N.. Triangulated categories without models. Invent. Math. 170 (2) (2007):231–241CrossRefGoogle Scholar
Mur06.Muro, F.. On the functoriality of cohomology of categories. J. Pure Appl. Algebra 204 (3) (2006):455–472CrossRefGoogle Scholar
Mur07.Muro, F.. A triangulated category without models. Preprint, arXiv:math/0703311, 2007Google Scholar
Nee91.Neeman, A.. Some new axioms for triangulated categories. J. Algebra 139 (1) (1991):221–255,CrossRefGoogle Scholar
PR05.Pirashvili, T. and Redondo, M. J.. Cohomology of the Grothendieck construction. arXiv:math.CT/0504282, 2005Google Scholar
Pup62.Puppe, D.. On the formal structure of stable homotopy theory. In Colloquium on Algebraic Topology, pages 65–71. Matematisk Institut, Aarhus Universitet, Aarhus, 1962Google Scholar
Ver77.Verdier, J.-L.. Catégories derivées. In Séminaire de Géométrie Algébrique du Bois-Marie SGA 4½, Lecture Notes in Math. 569, 262–308. Springer-Verlag, 1977Google Scholar