## 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)

Zbl 0799.18002
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### 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
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