Hostname: page-component-848d4c4894-4hhp2 Total loading time: 0 Render date: 2024-06-04T01:06:08.584Z Has data issue: false hasContentIssue false
  • English
  • Français

Injectivity and injective hulls of Abelian groups in a localic topos

Published online by Cambridge University Press:  17 April 2009

Kiran R. Bhutani
Kiran R. Bhutani
Affiliation:
The Catholic University of AmericaWashington, D.C., United States of America.
Rights & Permissions [Opens in a new window]

Abstract

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.

We prove the analogue of the Baer Criterion for injectivity in the category AbShℒ of abelian groups in a topos of sheaves on a locale, that is, we show A is injective in AbShℒ if and only if it is injective relative to all S ↣ Zℒ where Zℒ is the group of integers in Shℒ. for a well-ordered locale we describe the injective hulls in AbShℒ in terms of injective hulls in Ab. Further we show that the global functor A → AE preserves injective hulls if and only if ℒ is a finite boolean locale. Finally We characterise injectives in AbShℒ for some special locales.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 37 , Issue 1 , February 1988 , pp. 43 - 59
Copyright
Copyright © Australian Mathematical Society 1988

References

[1]Balbes, R. and Dwinger, P., ‘Distributive lattices” (University of Missouri Press 1975).Google Scholar
[2]Banaschewski, B., ‘When are divisible abeian groups injective?’, Quaestiones Math. 4 (1981), 285–307.CrossRefGoogle Scholar
[3]Banaschewski, B., ‘Injective sheaves of Abelian groups: a counterexample’, Canad. J. Math. 32 (1980), 1518–1521.CrossRefGoogle Scholar
[4]Banaschewski, B., ‘Coherent Frames’, Lecture notes in Math. 871.Google Scholar
[5]Banaschewski, B., ‘Recovering a space from its abelain Sheaves’: Seminar talks, (McMaster University 09 1980).Google Scholar
[6]Ebrahimi, M., ‘Algebra in a topos of sheaves’ (Doctoral dissertation McMaster University 1980).Google Scholar
[7]Fourman, M.P. and Scott, D.S., ‘Sheaves and logic’, in Application of sheaves Proceedings, Durham 1977: Lecture Notes in Math. (Springer Verlag, Berlin Heidelberg New York 1979).Google Scholar
[8]Freyd, P., ‘Abelian Categories’ (Harper and Row Publishers 1964).Google Scholar
[9]Fuchs, L., ‘Infinite abelian groups’ (Academic press, Vol. 1 1970).Google Scholar
[10]Godement, R., ‘Theorie des Fisceaux’ (Hermann, Paris 1958).Google Scholar
[11]Harting, R., ‘A remark on injective sheaves of abelain groups’ (to appear).Google Scholar
[12]Johnstone, P.T., ‘Topos Theory’ (Academic Press, London 1977).Google Scholar
[13]Johnstone, P.T., ‘Stone spaces’: Cambridge studies in Advanced Mathematics 3 (Cambridge Univeristy Press, Cambridge 1982).Google Scholar
[14]Kaplansky, I., ‘Infinite abelain groups’ (University of Michigan Press, Ann Arbor, Michigan 1968).Google Scholar
[15]MacLane, S., ‘Categories for the working mathematician’ 5: Graduate text in Mathematics (Springer Verlag 1971).Google Scholar
[16]Popenscu, N., ‘Abelian categories with applications to rings and modules’ (Academic Press 1973).Google Scholar