Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-24T18:28:26.422Z Has data issue: false hasContentIssue false
  • English
  • Français

Extensions, crossed modules and internal categories in categories of groups with operations

Published online by Cambridge University Press:  20 January 2009

T. Porter
T. Porter
Affiliation:
Department of Pure Mathematics, University College of North Wales, Bangor, Gwynedd LL57 2UW, Wales
Rights & Permissions [Opens in a new window]

Extract

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

In [6] Brown and Spencer noted that internal categories within the category of groups are equivalent to crossed modules. As they remarked, this result was known to various others before them, but it had not until then appeared in print. That paper led me to investigate the question of which algebraic categories, C, were such that a similar result held i.e. internal categories in C are equivalent to crossed modules of the appropriate type. The resulting work was written up in 1980 but was not submitted for publication.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 30 , Issue 3 , October 1987 , pp. 373 - 381
Copyright
Copyright © Edinburgh Mathematical Society 1987

References

REFERENCES

1.Aznar Garcia, E. R., Cohomologia no abeliana en categories de intres (Ph.D. Thesis, Univ. Santiago de Compostela, 1981), available as Alxebra 33.Google Scholar
2.Brown, R., Some non-abelian methods in homotopy theory and homological algebra, Proc. Conf. categorial topology, Toledo, Ohio, 1983 (Heldermann Verlag 1984).Google Scholar
3.Brown, R. and Higgins, P. J., Colimits of relative homotopy groups, J. Pure Appl. Algebra 22 (1981), 11–41.CrossRefGoogle Scholar
4.Brown, R. and Loday, J.-L., van Kampen theorems for diagrams of spaces, I, U.C.N.W., Pure Maths. Preprint 84.7, Topology, to appear.Google Scholar
5.Brown, R. and Spencer, C. B., Double groupoids and crossed modules, Cahiers Top. Geom. Diff. 17 (1976), 343–362.Google Scholar
6.Brown, R. and Spencer, C. B., G-groupoids, crossed modules and the fundamental group of a topological group, Proc. Kon. Ned. Akad. v. Wetensch. 79 (1976), 296–302.CrossRefGoogle Scholar
7.Dedecker, P., Cohomologie non abelienne (Lille, 1965), mimeographed 78 pp.Google Scholar
8.Ellis, G. J., Crossed Modules and their Higher Dimensional Analogues (University of Wales Ph.D. Thesis, August, 1984).Google Scholar
9.Ellis, G. J. and Steiner, R., Higher dimensional crossed modules and the homotopy groups of (n+l)—ads, U.C.N.W., Pure Maths. Preprint 85.6, 9pp.Google Scholar
10.Gerstenhaber, M., On the deformation of rings and algebras, Ann. of Math. 84 (1966), 1–19.CrossRefGoogle Scholar
11.Gerstenhaber, M., A uniform cohomology theory for algebras, Proc. Nat. Acad. Sci. U.S.A. 51 (1964), 624–629.CrossRefGoogle ScholarPubMed
12.Higgins, P. J., Groups with multiple operators, Proc. London Math. Soc. (3) 6 (1956), 366–416.CrossRefGoogle Scholar
13.Kassell, C. and Loday, J.-L., Extensions centrales d'algebres de Lie (Preprint 175, P-99, I.R.M.A., Strasbourg, 1982).Google Scholar
14.Lavendhomme, R. and Roisin, J., Cohomologie non-abélienne de structures algébriques, J. A Igebra 67 (1980), 355–414.Google Scholar
15.Loday, J.-L., Spaces with finitely many homotopy groups, J. Pure Appl. Algebra 24 (1982), 179–202.CrossRefGoogle Scholar
16.Lichtenbaum, S. and Schlessinger, M., The cotangent complex of a morphism, Trans. Amer. Math. Soc. 128 (1867), 41–70.CrossRefGoogle Scholar
17.Lue, A. S.-T., Non-abelian cohomology of associative algebras, Quart. J. Math. Oxford (2) 19 (1968), 159–180.CrossRefGoogle Scholar
18.Lue, A. S.-T., Cohomology of algebras relative to a variety, Math. Z. 121 (1971), 220–232.CrossRefGoogle Scholar
19.Maclane, S., Categories for the Working Mathematician (Graduate texts in Maths Vol. 5, Springer, 1971).Google Scholar
20.Orzech, G., Obstruction theory in algebraic categories I and II, J. Pure Appl. Algebra 2 (1972) 287–314 and 315–340.CrossRefGoogle Scholar
21.Porter, T., Internal Categories and Crossed Modules, in Category Theory, Proceedings, Gummersbach 1981 (Lecture Notes in Mathematics Vol. 962, 1982, Springer-Verlag), 245–255.Google Scholar
22.Porter, T., Crossed modules in Cat and a Brown-Spencer Theorem for 2-categories, Cahier. Top. Géom. Diff. Catégories 26 (1985), 381–388.Google Scholar
23.Porter, T., Homology of commutative algebras and an invariant of Simis and Vasconcelos, J. Algebra 99 (1986), 458–465.CrossRefGoogle Scholar
24.Whitehead, J. H. C., Combinatorial homotopy II, Bull. Amer. Math. Soc. 55 (1949), 453–496.CrossRefGoogle Scholar