Normalizers, Centralizers and Action Representability in Semi-Abelian Categories
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- 作者：J. R. A. Gray (1)
- 关键词：Normalizer ; Action representability ; Split extension ; Centralizer ; Semi ; abelian category ; 18A05 ; 18A99 ; 18C05 ; 18C10 ; 18C35 ; 18B99 ; 18D99
- 刊名：Applied Categorical Structures
- 出版年：2014
- 出版时间：October 2014
- 年：2014
- 卷：22
- 期：5-6
- 页码：981-1007
- 全文大小：5,479 KB
- 参考文献：1. Borceux, F., Bourn, D.: Mal’cev, protomodular, homological and semi-abelian categories. Kluwer Academic Publishers (2004)
2. Borceux, F., Janelidze, G., Kelly, G.M.: On the representability of actions in a semi-abelian category. Theory and Applications of Categories 14(11), 244-86 (2005)
3. Bourn, D.: Mal’cev categories and fibration of pointed objects. Appl. Categ. Struct. 4(2-), 307-27 (1996) CrossRef
4. Bourn, D.: Centralizer and faithful groupoid. Journal of Algebra 328(1), 43-6 (2011) CrossRef
5. Bourn, D., Gray, J.R.A.: Aspects of algebraic exponentiation. Bulletin of the Belgian Mathematical Society 19(5), 821-44 (2012)
6. Bourn, D., Janelidze, G.: Centralizers in action accessible categories. Cahiers de Topologie et Géométrie Différentielles Catégoriques 50(3), 211-32 (2009)
7. Carboni, A., Lambeck, J., Pedicchio, M.C.: Internal graphs and internal groupoids in Mal’cev categories. CMS Conference Proceedings 13, 97-09 (1992)
8. Gray, J.R.A.: Algebraic exponentiation and internal homology in general categories. Ph.D. thesis, University of Cape Town (2010)
9. Gray, J.R.A.: Algebraic exponentiation in general categories. Appl. Categ. Struct. 20(6), 543-67 (2012) CrossRef
10. Huq, S.A.: Commutator, nilpotency and solvability in categories. Q. J. Math. 19(1), 363-89 (1968) CrossRef
11. Huq, S.A.: Upper central series in a category. Journal für die reine und angewandte Mathematik 252, 209-14 (1971)
12. Janelidze, G., Márki, L., Tholen, W.: Semi-abelian categories. Journal of Pure and Applied Algebra 168, 367-86 (2002) CrossRef
13. Janelidze, G., Márki, L., Tholen, W., Ursini, A.: Ideal determined categories. Cahiers de topologie et geométrie différentielle catégoriques 51, 115-27 (2010)
14. Martins-Ferreira, N.: Low-dimensional internal categorial structures in weakly mal’cev sesquicategories. Ph.D. thesis, University of Cape Town (2008)
15. Orzech, G.: Obstruction theory in algebraic categories I, II. Journal of Pure and Applied Algebra 2(4), 287-14 and 315-40 (1972) CrossRef - 作者单位：J. R. A. Gray (1)
1. University of South Africa, Pretoria, South Africa - ISSN：1572-9095
文摘
We introduce the notion of normalizer as motivated by the classical notion in the category of groups. We show for a semi-abelian category ?that the following conditions are equivalent: ?is action representable and normalizers exist in ? the category Mono(? of monomorphisms in ?is action representable; the category ?sup class="a-plus-plus">2 of morphisms in ?is action representable; for each category \(\mathbb {D}\) with a finite number of morphisms the category \({\mathbb {C}} ^{\mathbb {D}}\) is action representable. Moreover, when in addition ?is locally well-presentable, we show that these conditions are further equivalent to: ?satisfies the amalgamation property for protosplit normal monomorphism and ?satisfies the axiom of normality of unions; for each small category \(\mathbb {D}\) , the category \({\mathbb {C}} ^{\mathbb {D}}\) is action representable. We also show that if ?is homological, action accessible, and normalizers exist in ? then ?is fiberwise algebraically cartesian closed.