We first compare several algebraic notions of normality, from a categorical viewpoint. Then we introduce an intrinsic description of Higgins' commutator for ideal-determined categories, and we define a new notion of normality in terms of this commutator. Our main result is to extend to any semi-abelian category the following well-known characterization of normal subgroups: a subobject K is normal in A if, and only if, [A,K]<K.
Normalities and commutators / S. Mantovani, G. Metere. - In: JOURNAL OF ALGEBRA. - ISSN 0021-8693. - 324:9(2010 Nov 01), pp. 2568-2588. [10.1016/j.jalgebra.2010.07.043]
Normalities and commutators
S. MantovaniPrimo
;G. MetereUltimo
2010
Abstract
We first compare several algebraic notions of normality, from a categorical viewpoint. Then we introduce an intrinsic description of Higgins' commutator for ideal-determined categories, and we define a new notion of normality in terms of this commutator. Our main result is to extend to any semi-abelian category the following well-known characterization of normal subgroups: a subobject K is normal in A if, and only if, [A,K]
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