George Janelidze

Semi-abelian categories

Abstract The notion of semi-abelian category as proposed in this paper is designed to capture typical algebraic properties valid for groups, rings and algebras, say, just as abelian categories allow for a generalized treatment of abelian-group and module theory. In modern terms, semi-abelian categories are exact in the sense of Barr and protomodular in the sense of Bourn and have finite coproducts and a zero object. We show how these conditions relate to “old” exactness axioms involving normal monomorphisms and epimorphisms, as used in the fifties and sixties, and we give extensive references to the literature in order to indicate why semi-abelian categories provide an appropriate notion to establish the isomorphism and decomposition theorems of group theory, to pursue general radical theory of rings, and how to arrive at basic statements as needed in homological algebra of groups and similar non-abelian structures.

Galois theory and a general notion of central extension

Abstract We propose a theory of central extensions for universal algebras, and more generally for objects in an exact category C , centrality being defined relatively to an “admissible” full subcategory X of C . This includes not only the classical notions of central extensions for groups and for algebras, but also their generalization by Frohlich to a pair consisting of a variety C of ω-groups and a subvariety X . Our notion of central extension is adapted to the generalized Galois theory developed by the first author, the use of which enables us to classify completely the central extensions of a given object B, in terms of the actions of an “internal Galois pregroupoid”.

Facets of Descent, II

Methods of internal-category theory are applied to show that the split epimorphisms in a category C are exactly the morphisms which are effective for descent with respect to any fibration over C (or to any C-indexed category). In the same context, composition-cancellation rules for effective descent morphisms are established and being applied to (suitably defined) locally-split epimorphisms.

Pure Galois theory in categories

Abstract Galois theory, J. Pure Appl. Algebra 19 (1980) 21-42. 2. M. BARR, Abstract Galois theory, II, J. Pure Appl. Algebra 25 (1982), 227-247. 3. M. BARR AND R. DIACONESCU, On locally simply connected toposes and their fundamen- tal groups, Cahiers Topologie Gt!om. DiffPrentielle Catigoriques 22 ( I98 I), 301-3 14. 4. S. U. CHASE, D. K. HARRISON, AND A. ROSENBERG, Galois theory and cohomology of commutative rings, Mem. Amer. Mafh. Sot. 52 (1965), 15-33. 5. S. U. CHASE AND M. E. SWEEDLER, Hopf algebras and Galois theory, in “Lecture Notes in Math.,” Vol. 97, Springer-Verlag, New York/Berlin, 1969. 6. I. DIERS, Categories of Boolean sheaves of simple algebras, in “Lecture Notes in Math.,” Vol. 1187, Springer-Verlag, New York/Berlin, 1986. 7. A. GROTHENDIECK, Revetements itales et groupe fondamental, SGA 1, in “Lecture Notes in Math.,” Vol. 269, Springer-Verlag, New York/Berlin, 1972. 8. J. F. KENNISON, Galois theory and theaters of action in a topos, J. Pure Appl. Algebra 18 (1980), 149-164. 9.

On Localization and Stabilization for Factorization Systems

If (ε, M)is a factorization system on a category C, we define new classes of maps as follows: a map f:A→B is in ε′ if each of its pullbacks lies in ε(that is, if it is stably in ε), and is in M* if some pullback of it along an effective descent map lies in M(that is, if it is locally in M). We find necessary and sufficient conditions for (ε′, M*) to be another factorization system, and show that a number of interesting factorization systems arise in this way. We further make the connexion with Galois theory, where M*is the class of coverings; and include self-contained modern accounts of factorization systems, descent theory, and Galois theory.

Finite preorders and Topological descent I

Abstract It is shown that the descent constructions of finite preorders provide a simple motivation for those of topological spaces, and new counter-examples to open problems in Topological descent theory are constructed.

Functorial factorization, well-pointedness and separability

Abstract A functorial treatment of factorization structures is presented, under extensive use of well-pointed endofunctors. Actually, the so-called weak factorization systems are interpreted as pointed lax indexed endofunctors, and this sheds new light on the correspondence between reflective subcategories and factorization systems. The second part of the paper presents two important factorization structures in the context of pointed endofunctors: concordant–dissonant and inseparable–separable.

Facets of Descent III: Monadic Descent for Rings and Algebras

We develop an elementary approach to the classical descent problems for modules and algebras, and their generalizations, based on the theory of monads.

Van Kampen theorems for categories of covering morphisms in lextensive categories

Abstract We give a form of the Van Kampen Theorem involving covering morphisms in a lextensive category. This includes the usual results for covering maps of locally connected spaces, for light maps of compact Hausdorff spaces, and for locally strong separable algebras.

Galois Groups, Abstract Commutators, and Hopf Formula

A general description of the Galois group of a “pointed” normal extension in categorical Galois theory is examined under the presence of a suitable commutator operation. In particular, using the Hopf formula for the second homology group of a group, the connection between Galois theory and group homology is clarified.