Categorical Galois Theory: Revision and Some Recent Developments

Janelidze, George

doi:10.1007/978-1-4020-1898-5_2

Cited by 12 publications

(10 citation statements)

References 42 publications

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“…The categorical theory of central extensions was later introduced by Janelidze and Kelly in [22] as a special case of the general Galois theory (see [3] or [21] for an account of various recent developments). This theory introduces a notion of centrality for an extension in any Barr exact category C [1] which is relative to V. Rossi (B) Dipartimento di Matematica e Informatica, Università degli Studi di Udine, Via delle Scienze 206, 33100 Udine, Italy e-mail: rossi@dimi.uniud.it the choice of a full reflective subcategory X in C satisfying a certain admissibility condition.…”

Section: Introductionmentioning

confidence: 99%

Admissible Galois Structures and Coverings in Regular Mal’cev Categories

Rossi

2006

Appl Categor Struct

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Given a regular Gumm category C such that any regular epimorphism is effective for descent, we prove that any Birkhoff subcategory X in C gives rise to an admissible Galois structure. This result allows one to consider some new applications of the categorical Galois theory in the context of topological algebras. Given a regular Mal'cev category C, we first characterize the coverings of the Galois structurē 1 induced by the subcategory C Ab of the abelian objects in C. Then we consider C as a subcategory of the category Eq(C) of the equivalence relations in C, and we characterize the coverings of the corresponding Galois structure¯ 2 . By composing the Galois structures¯ 1 and¯ 2 we obtain the Galois structure¯ induced by C Ab as a subcategory of Eq(C). We give the characterization of the¯ -coverings in terms of the coverings of¯ 1 and¯ 2 .

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Section: Introductionmentioning

confidence: 99%

Admissible Galois Structures and Coverings in Regular Mal’cev Categories

Rossi

2006

Appl Categor Struct

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“…Using again the equivalence between B-precrossed modules and internal reflexive graphs over B in Lie, we find that (21) where k ∈ Ker(f ) and k ′ ∈ Ker(g). Since the commutator in equation (22) can be treated exactly as the commutator in (20), we find that the double extension (21) where both sides are ideals of L 1 .…”

Section: Proof As Inmentioning

confidence: 98%

“…All these results apply when E is the class of regular epimorphisms in 2-Eq(C); indeed, in that case monadic extensions are exactly the same as effective descent morphisms and, moreover, Conn(C) is a strongly E-Birkhoff subcategory of 2-Eq(C) by Proposition 1. We will need, however, to restrict our attention to a smaller class of extensions, which we will call fibrations; this term was already used in [22] to denote the classes of arrows occurring in a Galois structure.…”

Section: Of Extensions Is a Double Extension If And Only If In The DImentioning

confidence: 99%

Higher commutator conditions for extensions in Mal'tsev categories

Duvieusart

Gran

2018

Journal of Algebra

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We define a Galois structure on the category of pairs of equivalence relations in an exact Mal'tsev category, and characterize central and double central extensions in terms of higher commutator conditions. These results generalize both the ones related to the abelianization functor in exact Mal'tsev categories, and the ones corresponding to the reflection from the category of internal reflexive graphs to the subcategory of internal groupoids. Some examples and applications are given in the categories of groups, precrossed modules, precrossed Lie algebras, and compact groups.

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“…For more details, we refer the interested reader to the book by Borceux and Janelidze [6], or to the recent articles [17,21].…”

Section: Conventionmentioning

confidence: 99%

Torsion theories and Galois coverings of topological groups

Gran

Rossi

2007

Journal of Pure and Applied Algebra

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For any torsion theory in a homological category, one can define a categorical Galois structure and try to describe the corresponding Galois coverings. In this article we provide several characterizations of these coverings for a special class of torsion theories, which we call quasi-hereditary. We describe a new reflective factorization system that is induced by any quasi-hereditary torsion theory. These results are then applied to study various examples of torsion theories in the category of topological groups.

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Categorical Galois Theory: Revision and Some Recent Developments

Cited by 12 publications

References 42 publications

Admissible Galois Structures and Coverings in Regular Mal’cev Categories

Admissible Galois Structures and Coverings in Regular Mal’cev Categories

Higher commutator conditions for extensions in Mal'tsev categories

Torsion theories and Galois coverings of topological groups

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