Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-13T05:00:08.936Z Has data issue: false hasContentIssue false
  • English
  • Français

Cohomologically triangulated categories I

Published online by Cambridge University Press:  03 January 2008

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

Abstract

We introduce cohomologically triangulated categories as triples (A,t,▽) given by an additive category A, an additive equivalence t:AA and a cohomology class ▽ in the translation cohomology H3(A,t). A stable homotopy theory C with A = HoC yields such a triple and the class of distinguished triangles in A is deduced from ▽.

Keywords

triangulated categoriespretriangulated categoriesgroupoid-enriched categories
Type
Research Article
Information
Journal of K-Theory , Volume 1 , Issue 1 , February 2008 , pp. 3 - 48
Copyright
Copyright © ISOPP 2008

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.)

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):259–285, 1997CrossRefGoogle Scholar
Bau05.Baues, H.-J.. Triangulated track categories. Georgian Math. J. 13 (2006), no. 4, 607–634CrossRefGoogle Scholar
Bau06.Baues, H.-J.. The algebra of secondary cohomology operations. Progress in Math. 247. Birkhäuser, 2006Google Scholar
BD89.Baues, H.-J. and Dreckmann, W.. The cohomology of homotopy categories and the general linear group. K-Theory, 3(4):307–338, 1989CrossRefGoogle Scholar
BD01.Berrick, A. J. and Davydov, A. A.. Splitting of Gysin extensions. Algebr. Geom. Topol., 1:743–762, 2001CrossRefGoogle Scholar
BHP97.Baues, H.-J., Hartl, M., and Pirashvili, T.. Quadratic categories and square rings. J. Pure Appl. Algebra, 122:1–40, 1997CrossRefGoogle Scholar
BJ06.Baues, H.-J. and Jibladze, M.. Secondary derived functors and the adams spectral sequence. Topology, 45:295–324, 2006CrossRefGoogle Scholar
BKS04.Benson, D., Krause, H., and Schwede, S.. Realizability of modules over Tate cohomology. Trans. Amer. Math. Soc., 356(9):3621–3668, 2004CrossRefGoogle Scholar
BM05.Baues, H.-J. and Muro, F.. Cohomologically triangulated categories II. Preprint of the Max-Planck-Institut für Mathematik, 2005Google Scholar
BM07.Baues, H.-J. and Muro, F.. The homotopy category of pseudofunctors and translation cohomology. J. Pure Appl. Algebra, 211(3):821–850, 2007CrossRefGoogle Scholar
Bor94.Borceux, F.. Handbook of categorical algebra 1. Number 50 in Encyclopedia of Math. and its Applications. Cambridge University Press, 1994Google Scholar
BP.Baues, H.-J. and Pirashvili, T.. Comparison of MacLane cohomology, Shukla and Hochschild. J. Reine Angew. Math. 598 (2006), 25–69Google Scholar
Hov99.Hovey, M.. Model categories, volume 63 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 1999Google Scholar
Nee91.Neeman, A.. Some new axioms for triangulated categories. J. Algebra, 139(1):221–255, 1991CrossRefGoogle Scholar
Nee01.Neeman, A.. Triangulated Categories, volume 148 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 2001Google Scholar
Pir88.Pirashvili, T.. Models for the homotopy theory and cohomology of small categories. Soobshch. Akad. Nauk Gruzin. SSR, 129:261–264, 1988. (Russian)Google 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
PW92.Pirashvili, T. and Waldhausen, F.. MacLane homology and topological Hochschild homology. J. Pure Appl. Algebra, 82(1):81–98, 1992CrossRefGoogle Scholar
Sag06.Sagave, S.. Universal Toda brackets of ring spectra. PhD thesis, University of Bonn, 2006Google Scholar
Sch02.Schlichting, M.. A note on K-theory and triangulated categories. Invent. Math., 150(1): 111–116, 2002CrossRefGoogle Scholar
Ver77.Verdier, J.-L.. Catégories derivées. In Séminaire de Géométrie Algébrique du Bois-Marie SGA 4½, number 569 in Lecture Notes in Math., pages 262–308. Springer-Verlag, 1977Google Scholar