Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-26T08:55:04.410Z Has data issue: false hasContentIssue false
  • English
  • Français

On satellites in semi-abelian categories: Homology without projectives

Published online by Cambridge University Press:  22 June 2009

JULIA GOEDECKE  and
TIM VAN DER LINDEN
JULIA GOEDECKE
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge, CB3 0WB. e-mail: [email protected]
TIM VAN DER LINDEN
Affiliation:
Vakgroep Wiskunde, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussel, Belgium. e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Working in a semi-abelian context, we use Janelidze's theory of generalised satellites to study universal properties of the Everaert long exact homology sequence. This results in a new definition of homology which does not depend on the existence of projective objects. We explore the relations with other notions of homology, and thus prove a version of the higher Hopf formulae. We also work out some examples.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 147 , Issue 3 , November 2009 , pp. 629 - 657
Copyright
Copyright © Cambridge Philosophical Society 2009

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

REFERENCES

[1]Barr, M. and Beck, J. Homology and standard constructions. Seminar on Triples and Categorical Homology Theory. Lecture Notes in Math. vol. 80 (Springer, 1969), 245–335.CrossRefGoogle Scholar
[2]Borceux, F.Handbook of Categorical Algebra. Encyclopedia of Math. Appl. vol. 50, 51 and 52 (Cambridge University Press, 1994).Google Scholar
[3]Borceux, F. and Bourn, D.Mal'cev, Protomodular, Homological and Semi-Abelian Categories. Math. Appl., vol. 566 (Kluwer Academic Publishers, 2004).CrossRefGoogle Scholar
[4]Bourn, D.On the direct image of intersections in exact homological categories. J. Pure Appl. Algebra 196 (2005), 39–52.CrossRefGoogle Scholar
[5]Everaert, T.An approach to non-abelian homology based on Categorical Galois Theory. Ph.D. thesis. (Vrije Universiteit Brussel 2007).Google Scholar
[6]Everaert, T. Higher central extensions and Hopf formulae. To appear in J. Algebra doi: 10.1016/j.jalgebra.2008.12.015 (2008).CrossRefGoogle Scholar
[7]Everaert, T., Gran, M. and Van der Linden, T.Higher Hopf formulae for homology via Galois Theory. Adv. Math. 217 (2008), 2231–2267.CrossRefGoogle Scholar
[8]Everaert, T. and Van der Linden, T.Baer invariants in semi-abelian categories I: General theory. Theory Appl. Categ. (1) 12 (2004), 1–33.Google Scholar
[9]Everaert, T. and Van der Linden, T.Baer invariants in semi-abelian categories II: Homology. Theory Appl. Categ. (4) 12 (2004), 195–224.Google Scholar
[10]Fröhlich, A.Baer-invariants of algebras. Trans. Amer. Math. Soc. 109 (1963), 221–244.CrossRefGoogle Scholar
[11]Guitart, R. and Van den Bril, L.Calcul des satellites et présentations des bimodules à l'aide des carrés exacts, I and II. Cah. Topol. Géom. Différ. Catég. (3) XXIV (1983), 299–330 and (4) XXIV (1983), 333–369.Google Scholar
[12]Guitart, R.An anabelian definition of abelian homology. Cah. Topol. Géom. Différ. Catég. (4) XLVIII (2007), 261–269.Google Scholar
[13]Gran, M. and Rosický, J.Semi-abelian monadic categories. Theory Appl. Categ. (6) 13 (2004), 106–113.Google Scholar
[14]Gran, M. and Van der Linden, T.On the second cohomology group in semi-abelian categories. J. Pure Appl. Algebra 212 (2008), 636–651.CrossRefGoogle Scholar
[15]Hopf, H.Fundamentalgruppe und zweite Bettische Gruppe. Comment. Math. Helv. 14 (1942), 257–309.CrossRefGoogle Scholar
[16]Janelidze, G.On satellites in arbitrary categories. Bulletin of the Academy of Sciences of the Georgian SSR (3) 82 (1976), 529–532, in Russian.Google Scholar
[17]Janelidze, G. Precategories and Galois theory. Category Theory, Proceedings Como 1990 (Carboni, A., Pedicchio, M. C., and Rosolini, G., eds.) Lecture Notes in Math., vol. 1488 (Springer, 1991), 157–173.CrossRefGoogle Scholar
[18]Janelidze, G. and Kelly, G. M.Galois theory and a general notion of central extension. J. Pure Appl. Algebra 97 (1994), 135–161.CrossRefGoogle Scholar
[19]Janelidze, G., Márki, L. and Tholen, W.Semi-abelian categories. J. Pure Appl. Algebra 168 (2002), 367–386.CrossRefGoogle Scholar
[20]Johnstone, P. T.Topos Theory. L.M.S. Monographs vol. 10 (Academic Press, 1977).Google Scholar
[21]Karpilovsky, G.The Schur Multiplier. London Math. Soc. Monogr. Ser. vol. 2 (Oxford University Press, 1987).Google Scholar
[22]Lane, S. MacCategories for the Working Mathematician, second ed. Grad. Texts in Math. vol. 5 (Springer 1998).Google Scholar
[23]Schur, I.Über die Darstellung der endlichen Gruppen durch gebrochene lineare Substitutionen. J. Reine Angew. Math. 127 (1904), 20–50.Google Scholar
[24]Schur, I.Untersuchungen über die Darstellung der endlichen Gruppen durch gebrochene lineare Substitutionen. J. Reine Angew. Math. 132 (1907), 85–137.Google Scholar