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Algèbre catégorique
Composition de l'équipe
Francis BORCEUX Bao Long DANG VAN Mathieu DUCKERTS Marino GRAN Julia GOEDECKE Olivette NGAHA Damien SERVAIS Enrico VITALE
1) Some topics studied in this area are : the algebraic structure of classes of localizations ; the localization associated with a closure operator ; characterization of quasi-topos and of those toposes on which the left exact presheaves constitute a topos ; a generalization of both the theory of sheaves and the theory of accessible categories to the context of enriched categories. A new impulsion to the research in this area has been given by the theory of the exact completion, applied to obtain new results on (essential) localizations of monadic categories and algebraic categories, and to give a uniform approach to some classical results (Giraud characterization of topos, Gabriel-Popescu representation theorem for Grothendieck categories and Freyd representation theorems in abelian categories). It has been used also in universal algebra, especially in connection with Malcev varieties and quasi-varieties. More recently, a complete classification of localizations and geometric morphisms of algebraic and presheaf categories has been obtained. 2) Under the impulsion of the categorical approach to Galois theory and descent theory developed by G. Janelidze, interesting progresses have been achieved in the study of central extensions for internal categories in exact Malcev categories . Strongly connected with internal categories, categorical groups have been studied as a 2-dimensional analogous of groups, obtaining applications to ring theory, group extensions, homology and homotopy groupoids and factorization systems. Moreover, cohomology theories for algebraic objects which occur as monoids in monoidal categories (as monads, operads, categories, theories) have been studied. Another categorical approach to ring theory consists in replacing R-algebras with small enriched categories. In this line, our research group has developed a theory of Azumaya categories, together with the associated Brauer group. The more delicate treatment of the Brauer-Taylor group has also been achieved, and related new problems involving "categories without units" have been investigated. 3) The classical descent theory for ring homomorphisms admits a description in terms of monads, allowing to study it in much more general contexts. We investigate in particular a possible refinement of this classical theory in terms of the pure spectrum of the ring. We have also solved the descent problem in the case of algebraic fibrations and of internal functors in a lextensive category. A book on Galois theories and descent morphisms, due for the biggest part to our research group, has been published. In connection with Galois theories, we also investigate the context of Omega-groups. We have exhibited a characterization of those categories in which the groups of automorphisms are representable (like the category of all groups).
4) The discovery of an internal notion of crossed module opens a new perspective for the developments of this theory in the context of semi-abelian categories and offers a unified approach to the study of the homological properties of crossed modules, crossed rings and other structures internal to varieties of algebras. Encouraging results in the classification of central extensions of precrossed modules have also been obtained. We have exhibited a characterization of those categories in which the groups of automorphisms are representable (like the category of all groups). We have proved a Jordan-Höelder theorem for semi-abelan categories. We have studied intensively both the semi-abelian and the protomodular topologica algebra. We have also studied and characterised the reflections of semi-abelian categories which generalise the well-known theory of localisations in the abelian case; this has been further generalised to the non-pointed case.
Publications représentatives
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