Abstract
In this paper we study Baer invariants of precrossed modules relative to the subcategory of crossed modules, following Fröhlich and Furtado-Coelho’s general theory on Baer invariants in varieties of Ω-groups and Modi’s theory on higher dimensional Baer invariants. Several homological invariants of precrossed and crossed modules were defined in the last two decades. We show how to use Baer invariants in order to connect these various homology theories. First, we express the low-dimensional Baer invariants of precrossed modules in terms of a new non-abelian tensor product of a precrossed module. This expression is used to analyze the connection between the Baer invariants and the homological invariants of precrossed modules defined by Conduché and Ellis. Specifically we prove that the second homological invariant of Conduché and Ellis is in general a quotient of the first component of the Baer invariant we consider. The definition of classical Baer invariants is generalized using homological methods. These generalized Baer invariants of precrossed modules are applied to the construction of five term exact sequences connecting the generalized Baer invariants with the cohomology theory of crossed modules considered by Carrasco, Cegarra and R.-Grandjeán and the cohomology theory of precrossed modules.
Similar content being viewed by others
References
Arias, D., Ladra, M.: Central extensions of precrossed modules. Appl. Categor. Struct. 12(4), 339–354 (2004)
Arias, D., Ladra, M., R.-Grandjeán, A.: Homology of precrossed modules. Ill. J. Math. 46(3), 739–754 (2002)
Barr, M., Beck, J.: Homology and standard constructions. In: Eckmann, B. (ed.) Seminar on Triples and Categorical Homology Theory. Lecture Notes in Mathematics, vol. 80, pp. 245–335. Springer, Berlin (1969)
Baues, H.J., Conduché, D.: The central series for Peiffer commutators in groups with operators. J. Algebra 133(1), 1–34 (1990)
Beck, J.: Triples, algebras and cohomology. Repr. Theory Appl. Categ. 2, 1–59 (2003). Ph.D. thesis. Columbia University (1967)
Bourn, D., Gran, M.: Central extensions in semi-abelian categories. J. Pure Appl. Algebra 175(1–3), 31–44 (2002)
Bourn, D., Janelidze, G.: Extensions with abelian kernels in protomodular categories. Georgian Math. J. 11(4), 645–654 (2004)
Brown, R., Loday, J.-L.: Van Kampen theorems for diagrams of spaces. Topology 26(3), 311–335 (1987)
Carrasco, P., Cegarra, A.M., R.-Grandjeán, A.: (Co)Homology of crossed modules. J. Pure Appl. Algebra 168(2–3), 147–176 (2002)
Casas, J.M., Van der Linden, T.: A relative theory of universal central extensions. arXiv:0908.3762v3 [math.AT] (2011)
Cegarra, A.M., Bullejos, M.: Cohomology and higher dimensional Baer invariants. J. Algebra 132(2), 321–339 (1990)
Conduché, D.: Question de Whitehead et modules précroisés. Bull. Soc. Math. France 124(3), 401–423 (1996)
Conduché, D., Ellis, G.J.: Quelques propriétés homologiques des modules précroisés. J. Algebra 123(2), 327–335 (1989)
Everaert, T.: An approach to non-abelian homology based on Categorical Galois Theory. Ph.D. thesis, Vrije Universiteit Brussel (2007)
Everaert, T.: Relative commutator theory in varieties of Ω-groups. J. Pure Appl. Algebra 210(1), 1–10 (2007)
Everaert, T., Gran, M.: Precrossed modules and Galois theory. J. Algebra 297(1), 292–309 (2006)
Everaert, T., Gran, M., Van der Linden, T.: Higher Hopf formulae for homology via Galois Theory. Adv. Math. 217(5), 2231–2267 (2008)
Everaert, T., Van der Linden, T.: Baer invariants in semi-abelian categories I: general theory. Theory Appl. Categ. 12(1), 1–33 (2004)
Everaert, T., Van der Linden, T.: Baer invariants in semi-abelian categories II: homology. Theory Appl. Categ. 12(4), 195–224 (2004)
Franco, L.: Baer invariants of crossed modules. J. Algebra 160(1), 50–56 (1993)
Fröhlich, A.: Baer-invariants of algebras. Trans. Am. Math. Soc. 109, 221–244 (1963)
Furtado-Coelho, J.: Homology and generalized Baer invariants. J. Algebra 40(2), 596–609 (1976)
Goedecke, J., Van der Linden, T.: On satellites in semi-abelian categories. Homology without projectives. Math. Proc. Camb. Philos. Soc. 147(3), 629–657 (2009)
Gran, M., Van der Linden, T.: On the second cohomology group in semi-abelian categories. J. Pure Appl. Algebra 212(3), 636–651 (2008)
Janelidze, G., Kelly, G.M.: Galois theory and a general notion of central extension. J. Pure Appl. Algebra 97(2), 135–161 (1994)
Janelidze, G., Márki, L., Tholen, W.: Semi-abelian categories. J. Pure Appl. Algebra 168(2–3), 367–386 (2002)
Kervaire, M.: Multiplicateurs de Schur et K-théorie. In: Haefliger, A., Narasimban, R. (eds.) Essays on Topology and Related Topics (Mémoires dédiés à Georges de Rham), pp. 212–225. Springer, New York (1970)
Keune, F.: Homotopical algebra and algebraic K-theory. Ph.D. thesis, University of Amsterdam (1972)
Loday, J.-L.: Spaces with finitely many non-trivial homotopy groups. J. Pure Appl. Algebra 24(2), 179–202 (1982)
Modi, K.: Simplicial methods and the homology of groups. Ph.D. thesis, University of London (1976)
Paoli, S.: On the non-balanced property of the category of crossed modules in groups. J. Pure Appl. Algebra 197(1–3), 19–22 (2005)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Arias, D., Ladra, M. Baer Invariants and Cohomology of Precrossed Modules. Appl Categor Struct 22, 289–304 (2014). https://doi.org/10.1007/s10485-013-9307-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10485-013-9307-x