Abstract
We classify the precrossed module central extensions using the second cohomology group of precrossed modules. We relate these central extensions to the relative central group extensions of Loday, and to other notions of centrality defined in general contexts. Finally we establish a Universal Coefficient Theorem for the (co)homology of precrossed modules, which we use to describe the precrossed module central extensions in terms of the generalized Galois theory developed by Janelidze.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arias, D., Ladra, M. and R.-Grandjeán, A.: Homology of precrossed modules, Illinois J. Math. 46(3) (2002), 739–754.
Arias, D., Ladra, M. and R.-Grandjeán, A.: Universal central extensions of precrossed modules and Milnor's relative K2, J. Pure Appl. Algebra 184 (2003), 139–154.
Barr, M. and Beck, J.: Homology and standard constructions, in B. Eckmann (ed.), Seminar on Triples and Categorical Homology Theory, Lecture Notes in Math. 80, Springer, Berlin, 1969, pp. 245–335.
Bourn, D. and Gran, M.: Central extensions in semi-Abelian categories, J. Pure Appl. Algebra 175 (2002), 31–44.
Carrasco, P., Cegarra, A. M. and R.-Grandjeán, A.: (Co)Homology of crossed modules, J. Pure Appl. Algebra 168(2- 3) (2002), 147–176.
Eilenberg, S. and MacLane, S.: Cohomology theory in abstract groups I, Ann. of Math. 48(1) (1947), 51–78.
Freeze, R. and McKenzie, T.: Commutator Theory for Congruence Modular Varieties, London Math. Soc. Lecture Notes Series 125, Cambridge University Press, 1987.
Fröhlich, A.: Baer-invariants of algebras, Trans. Amer. Math. Soc. 109 (1963), 221–244.
Hilton, P. J. and Stammbach, U.: A Course in Homological Algebra, Graduate Texts in Math. 4, Springer-Verlag, 1971.
Janelidze, G.: The fundamental theorem of Galois theory, Math. USSR-Sb. 64(2) (1989), 359–374.
Janelidze, G.: Pure Galois theory in categories, J. Algebra 132 (1990), 270–286.
Janelidze, G.: Precategories and Galois theory, in Lecture Notes in Math. 1488, Springer, Berlin, 1991, pp. 157–173.
Janelidze, G. and Kelly, G. M.: Galois theory and a general notion of central extension, J. Pure Appl. Algebra 97 (1994), 135–161.
Janelidze, G. and Kelly, G. M.: Central extensions in universal algebra: A unification of three notions, Algebra Universalis 44 (2000), 123–128.
Janelidze, G., Márki, G. L. and Tholen, W.: Semi-Abelian categories, J. Pure Appl. Algebra 168 (2002), 367–386.
Janelidze, G. and Pedicchio, M. C.: Internal categories and groupoids in congruence modular varieties, J. Algebra 193 (1997), 552–570.
Loday, J.-L.: Cohomologie et groupes de Steinberg relatifs, J. Algebra 54(1) (1978), 178–202.
Lue, A. S.-T.: Baer-invariants and extensions relative to a variety, Proc. Cambridge Philos. Soc. 63 (1967), 569–578.
Mitchell, B.: Theory of Categories, Academic Press, New York, 1965.
Norrie, K. J.: Crossed modules and analogues of group theorems, Ph.D. thesis, University of London, 1987.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Arias, D., Ladra, M. Central Extensions of Precrossed Modules. Applied Categorical Structures 12, 339–354 (2004). https://doi.org/10.1023/B:APCS.0000040555.48968.64
Issue Date:
DOI: https://doi.org/10.1023/B:APCS.0000040555.48968.64