Normalizers in the non-pointed context

Dominique Bourn

The aim of this work is to point out a strong structural phenomenon hidden behind the existence of normalizers through the investigation of this property in the non-pointed context: given any category E, a certain property of the fibration of points \P_E: Pt(E) --> E guarentees the existence of normalizers. This property becomes a characterization of this existence when E is quasi-pointed and protomodular. This property is also showed to be equivalent to a property of the category Grd E of internal groupoids in E which is almost opposite, for the monomorphic internal functors, of the comprehensive factorization.

Keywords: equivalence relation, equivalence class, normal subobject, normalizers, Mal'tsev and protomodular categories, internal categories and groupoids, comprehensive factorization, non-pointed additive categories

2020 MSC: 18A05, 18B99, 18E13, 08C05, 08A30, 08A99

Theory and Applications of Categories, Vol. 40, 2024, No. 2, pp 32-62.

Published 2024-03-22.

TAC Home