Hostname: page-component-758b78586c-72lk7 Total loading time: 0 Render date: 2023-11-29T19:06:10.668Z Has data issue: false Feature Flags: { "corePageComponentGetUserInfoFromSharedSession": true, "coreDisableEcommerce": false, "useRatesEcommerce": true } hasContentIssue false

On Graded Categorical Groups and Equivariant Group Extensions

Published online by Cambridge University Press:  20 November 2018

A. M. Cegarra
Departamento de Algebra, Facultad de Ciencias, Universidad de Granada, 18071, Granada Spain, email:
J. M. Garćia-Calcines
Departamento de Algebra Facultad de Ciencias Universidad de Granada 18071, Granada Spain, email:
J. A. Ortega
Departamento de Matemática Fundamental, Universidad de La Laguna, 38271, La Laguna, Spain, email:
Rights & Permissions [Opens in a new window]


Core share and HTML view are not possible as this article does not have html content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this article we state and prove precise theorems on the homotopy classification of graded categorical groups and their homomorphisms. The results use equivariant group cohomology, and they are applied to show a treatment of the general equivariant group extension problem.

Research Article
Copyright © Canadian Mathematical Society 2002


[1] Breen, L., Theorie de Schreier superiere. Ann. Sci. École Norm. Sup. 25 (1992), 465514.Google Scholar
[2] Breen, L., Monoidal Categories and Multiextensions. Compositio Math. 117 (1999), 295335.Google Scholar
[3] Bredon, G. E., Equivariant Cohomology Theories. Lecture Notes in Math. 34, Springer, Berlin, 1967.Google Scholar
[4] Bullejos, M., Cabello, J. and Faro, E., On the equivariant 2-type of a G-space. J. Pure Appl. Algebra 129 (1998), 215245.Google Scholar
[5] Carrasco, P. and Cegarra, A. M., (Braided) Tensor structures on homotopy groupoids and nerves of (braided) categorical groups. Comm. Algebra 24 (1996), 39954058.Google Scholar
[6] Cegarra, A. M. and Fernández, L., Cohomology of cofibred categorical groups. J. Pure Appl. Algebra 143 (1999), 107154.Google Scholar
[7] Cegarra, A. M., Garćà-Calcines, J. M. and Ortega, J. A., Cohomology of groups with operators. Homology Homotopy Appl. (1) 4 (2002), 123.Google Scholar
[8] Eilenberg, S. and MacLane, S., Cohomology theory in abstract groups. II, Group Extensions with a non-Abelian Kernel. Ann. of Math. 48 (1947), 326341.Google Scholar
[9] Fröhlich, A. and Wall, C. T. C., Graded monoidal categories. Compositio Math. 28 (1974), 229285.Google Scholar
[10] Grothendieck, A., Catégories fibrées et déscente. SGA I, exposé VI, Lecture Notes in Math. 224, Springer, Berlin, 1971, 145194.Google Scholar
[11] Grothendieck, A., Biextension de faisceaux de groupes. SGA 7 I, exposé VII, Lecture Notes in Math. 288, Springer, Berlin, 1972, 133217.Google Scholar
[12] Joyal, A. and Street, R., Braided tensor categories. Adv. Math. (1) 82 (1991), 2078.Google Scholar
[13] MacLane, S., Homology. Die Grundleheren der Math.Wiss. in Einzel. 114, Springer, 1963.Google Scholar
[14] Moerdick, I. and Svensson, J. A., The equivariant Serre spectral sequence. Proc. Amer.Math. Soc. (1) 118 (1993), 263277.Google Scholar
[15] Saavedra, N., Catégories Tannakiennes. Lecture Notes in Math. 265, Springer, Berlin, 1972.Google Scholar
[16] Schreier, O., Über die Erweiterung von Gruppen I. Monatsh.Math. Phys. 34 (1926), 165180.Google Scholar
[17] Sinh, H. X., Gr-catégories. Thése de Doctorat, Université Paris VII, (1975).Google Scholar