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On closure operators and reflections on Goursat categories. (English) Zbl 1153.18003

A recent result of D. Bourn and M. Gran [J. Algebra 305, 18-47 (2006; Zbl 1123.18009)] has extended the classical correspondence between hereditary torsion theories (in abelian categories) and universal closure operators, to the context of homological categories. The first result of the paper extends this further to the context of regular categories, establishing a bijection between (regular) epireflective subcategories and of what the authors call effective closure operators on effective equivalence relations (essentially, those idempotent closure operators with \(f^{-1}(\overline{S}) = \overline{f^{-1}(S)})\) for all regular epis \(f\)).
Examples as the categories of topological models of Mal’tsev theories are discussed in some details. The authors then show that in the context of exact Goursat categories, this result restricts to a bijection between the Birkhoff subcategories and the effective closure operators satisfying \(f(\overline{S}) = \overline{f(S)}\) for all regular epis \(f\). Again examples and motivations from universal algebra are given.

MSC:

18C05 Equational categories
18G50 Nonabelian homological algebra (category-theoretic aspects)
08B05 Equational logic, Mal’tsev conditions
54B30 Categorical methods in general topology
18B30 Categories of topological spaces and continuous mappings (MSC2010)
18A40 Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.)
18E40 Torsion theories, radicals

Citations:

Zbl 1123.18009
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