Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-29T17:15:10.241Z Has data issue: false hasContentIssue false
  • English
  • Français

Group cohomology with coefficients in a crossed module

Part of: Homological algebra Connections with homological algebra and category theory Categories with structure

Published online by Cambridge University Press:  17 June 2010

Behrang Noohi
Behrang Noohi
Affiliation:
King's College London, Strand, London WC2R 2LS, UK ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We compare three different ways of defining group cohomology with coefficients in a crossed module: (1) explicit approach via cocycles; (2) geometric approach via gerbes; (3) group theoretic approach via butterflies. We discuss the case where the crossed module is braided and the case where the braiding is symmetric. We prove the functoriality of the cohomologies with respect to weak morphisms of crossed modules and also prove the ‘long’ exact cohomology sequence associated to a short exact sequence of crossed modules and weak morphisms.

Keywords

equivariant crossed moduleequivariant categorical groupequivariant 2-groupgroup cohomologyprincipal 2-bundlebutterfly

MSC classification

Secondary: 18G50: Nonabelian homological algebra 18G60: Other (co)homology theories 18D10: Monoidal categories (= multiplicative categories), symmetric monoidal categories, braided categories 20J06: Cohomology of groups
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 10 , Issue 2 , April 2011 , pp. 359 - 404
Copyright
Copyright © Cambridge University Press 2010

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

1.Aldrovandi, E. and Noohi, B., Butterflies, I, Morphisms of 2-group stacks, Adv. Math. 221(3) (2009), 687–773.CrossRefGoogle Scholar
2.Aldrovandi, E. and Noohi, B., Butterflies, III, 2-butterflies and 2-group stacks, in preparation.Google Scholar
3.Arvas, Z., Kuzpinari, T. S. and Uslu, E. Ö., Three crossed modules, Homology Homotopy Applicat. 11(2) (2009), 161–187.CrossRefGoogle Scholar
4.Borovoi, M., Abelian Galois cohomology of reductive groups, Memoirs of the American Mathematical Society, Volume 132, No. 626 (American Mathematical Society, Providence, RI, 1998).Google Scholar
5.Breen, L., Bitorseurs et cohomologie non abélienne, in The Grothendieck Festschrift I, Progress in Mathematics, Volume 86, pp. 401–476 (Birkhäuser, 1990).CrossRefGoogle Scholar
6.Breen, L., Classification of 2-gerbes and 2-stacks, Astérisque, Volume 225 (Société Mathématique de France, Paris, 1994).Google Scholar
7.Breen, L., Tannakian categories, Proceedings of Symposia in Pure Mathematics, Volume 55, Part 1, pp. 337–376 (American Mathematical Society, Providence, RI, 1994).Google Scholar
8.Brown, R. and Gilbert, N. D., Algebraic models of 3-types and automorphism structures for crossed modules, Proc. Lond. Math. Soc. 59(1) (1989), 51–73.CrossRefGoogle Scholar
9.Brown, R., Golasiński, M., Porter, T. and Tonks, A., Spaces of maps into classifying spaces for equivariant crossed complexes, Indagationes Math. 8(2) (1997), 157–172.CrossRefGoogle Scholar
10.Bullejos, M., Carrasco, P. and Cegarra, A. M., Cohomology with coefficients in symmetric cat-groups, Math. Proc. Camb. Phil. Soc. 114(1) (1993), 163–189.CrossRefGoogle Scholar
11.Carrasco, P., Complejos hipercruzados: cohomologia y extensiones, PhD thesis, Cuadernos de Algebra 6, Departamento de Algebra y Fundamentos, Universidad de Granada (1987).Google Scholar
12.Carrasco, P. and Martínez-Moreno, J., Simplicial cohomology with coefficients in symmetric categorical groups, Appl. Categ. Struct. 12 (2004), 257–285.CrossRefGoogle Scholar
13.Cegarra, A. M. and Fernández, L., Cohomology of cofibered categorical groups, J. Pure Appl. Alg. 134 (1999), 107–154.CrossRefGoogle Scholar
14.Cegarra, A. M. and Garzón, A. R., Along exact sequence in nonabelian cohomology, in Category Theory, Como, 1990, pp. 79–94, Lecture Notes in Mathematics, Volume 1488 (Springer, 1991).Google Scholar
15.Conduché, D., Modules croisés généralisés de longueur 2, J. Pure Appl. Alg. 34 (1984), 155–178.CrossRefGoogle Scholar
16.Dedecker, P., Cohomologie à coeffcients non abéliens, C. R. Acad. Sci. Paris Sér. I 287 (1958), 1160–1162.Google Scholar
17.Dedecker, P., Foncteurs εxt, , et non abéliens, C. R. Acad. Sci. Paris Sér. I 258 (1964), 4891–4894.Google Scholar
18.Garzón, A. R. and del Río, A., On ℌ1 of categorical groups, Commun. Alg. 34(10) (2006), 3691–3699.CrossRefGoogle Scholar
19.Noohi, B., Notes on 2-groupoids, 2-groups and crossed modules, Homology Homotopy Applicat. 9(1) (2007), 75–106.CrossRefGoogle Scholar
20.Noohi, B., On weak maps between 2-groups, preprint (arXiv:math/0506313v3 [math.CT]; 2008).Google Scholar
21.Noohi, B., Higher Grothendieck–Schreier theory via butterflies, in preparation.Google Scholar
22.Ulbrich, K. H., Group cohomology for Picard categories, J. Alg. 91(2) (1984), 464–498.CrossRefGoogle Scholar