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.


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)


Zbl 0799.18002
Full Text: DOI


[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
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