×

Galois theory and a general notion of central extension. (English) Zbl 0813.18001

Let \({\mathcal C}\) be an exact category and \({\mathcal X}\) a so-called admissible subcategory of \({\mathcal C}\), i.e., a full reflective subcategory of \({\mathcal C}\), closed under subobjects and regular quotient objects, and whose reflector \(I\) preserves pullbacks of regular epimorphisms along unit morphisms of the reflection. For any object \(B\) in \({\mathcal C}\), a regular epimorphism \(f: A\to B\) is called an extension of \(B\). It is trivial if it is the pullback along the unit morphism \(\eta_ B: B\to I(B)\) of some extension of \(I(B)\) in \({\mathcal X}\). It is central if its pullback along some extension \(p: E\to B\) is trivial. Categories of such central extensions are investigated in the general case, and in special cases where \(\mathbb{C}\) is a Goursat or Maltsev category, a category of varieties of universal algebras, a category of \(\Omega\)-groups, etc. The link with a previous paper on a generalized Galois theory is made.

MSC:

18A32 Factorization systems, substructures, quotient structures, congruences, amalgams
18G30 Simplicial sets; simplicial objects in a category (MSC2010)
18B40 Groupoids, semigroupoids, semigroups, groups (viewed as categories)

Citations:

Zbl 0799.18002
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Carboni, A.; Kelly, G.M.; Pedicchio, M.C., Some remarks on maltsev and Goursat categories, Applied categorical structures, 1, 385-421, (1993) · Zbl 0799.18002
[2] Fröhlich, A., Baer-invariants of algebras, Trans. amer. math. soc., 109, 221-244, (1963) · Zbl 0122.25702
[3] Janelidze, G., The fundamental theorem of Galois theory, Math. USSR-sb., 64, 2, 359-374, (1989) · Zbl 0677.18003
[4] Janelidze, G., Pure Galois theory in categories, J. algebra, 132, 270-286, (1990) · Zbl 0702.18006
[5] Janelidze, G., Precategories and Galois theory, (), 157-173 · Zbl 0754.18002
[6] Johnstone, P.T., Stone spaces, (1982), Cambridge University Press Cambridge · Zbl 0499.54001
[7] Jónsson, B., Algebras whose congruence latices are distributive, Math, scand., 21, 110-121, (1967) · Zbl 0167.28401
[8] Kelly, G.M., Basic concepts of enriched category theory, (1982), Cambridge University Press Cambridge · Zbl 0478.18005
[9] Lawvere, F.W., Functorial semantics of algebraic theories, Proc. nat. acad. sci. USA, 50, 869-872, (1963) · Zbl 0119.25901
[10] Lue, A.S.-T., Bear-invariants and extensions relative to a variety, Proc. Cambridge philos. soc., 63, 569-578, (1967) · Zbl 0154.27501
[11] Mac Lane, S., Homology, (1963), Springer Berlin · Zbl 0818.18001
[12] Mac Lane, S., Categories for the working Mathematician, (1971), Springer Berlin · Zbl 0232.18001
[13] Mal’cev, A.I., On the general theory of algebraic systems, Math. sbornik N.S., 35, 3-20, (1954) · Zbl 0057.02403
[14] Mitschke, A., Implication algebras are 3-permutable and 2-distributive, Algebra universalis, 1, 182-186, (1971) · Zbl 0242.08005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.