ABSTRACTS

 D. Bourn Split extension classifier and centrality: This is a joint work with F.Borceux. We show that, in any pointed protomodular category C, when an object X has a split extension classifier, as it is the case in the category Gp of groups, this classifier is actually underlying an internal groupoid structure D{\bullet}(X) which measures the obstruction to the abelianity of X in the sense that the kernel relation of its normalization jX is exactly the centre of X, i.e. the greatest central relation. Furthermore, the existence of such a classifier appears to have a more global classifying power: for instance, C is additive if and only if, for all X, we have jX=\tauX:X --> 1, and antiadditive (i.e. with no non trivial abelian object) if and only if, for all X, jX is a monomorphism. M. M. Clementino Radicals and closure operators in normal categories (joint work with Dikran Dikranjan and Walter Tholen): We study the behaviour of (normal) closure operators in normal categories and their relationship with (pre)radicals, generalizing results obtained by Dominique Bourn and Marino Gran in [1]. [1] D. Bourn and M. Gran, Torsion theories and closure operators, Cahiers du L.M.P.A. 213, Universite du Littoral, Calais 2004. [2] M.M. Clementino, D. Dikranjan and W. Tholen, Torsion theories and radicals in normal categories, Journal of Algebra (to appear). L. Sousa A deduction system for orthogonality: Varieties, quasivarieties and prevarieties are categories of algebras presented by equations, implications and limit sentences, respectively; correspondingly, we can also present them by convenient morphisms, via orthogonality, ([3], [2]). Rosu, in [4], defined a sound and complete equational deduction system for adequate morphisms. Following the idea of Rosu, the main theme of this talk, based in [1], is the presentation of a logic for epimorphisms with finitely presentable domain and codomain, which is used in order to obtain a sound and complete logic of implications, which extends the Birkhoff's equational logic. [1] J. Adámek, M. Sobral and L. Sousa, Logic of Implications, Preprint number 05-24, Pré-Publicações do Departamento de Mateática, Universidade de Coimbra, 2005. [2] J. Aámek and L. Sousa, On reflective subcategories of varieties, Journal of Algebra 276 (2004), no. 2, 685-705. [3] B. Banaschewski and H. Herrlich, Subcategories defined by implications, Houston Journal of Mathematics 2, 149-171 (1976). [4] G. Rosu, Complete Categorical Equational Deduction, Lecture Notes in Comput. Sci. 2142 (2001), 528--538. I. Stubbe Suplattices over Q are precisely Q-modules: It is well known that the internal suplattices in the topos of sheaves on a locale are precisely the modules on that locale. I shall show a generalization of this result to the case of (ordered) sheaves on a quantaloid. E. Vitale External derivations of internal groupoids: Let G be a group and \varphi \colon G --> Aut H a G-group. A derivation of G in H is a map d \colon G --> H such that d(xy) = d(x) + x \cdot d(y). If H is a G-module, i.e. if H is abelian, the set Der(G,H) of derivations is an abelian group w.r.t. the point-wise sum. If H is not abelian, in general Der(G,H) is just a pointed set. J.H.C. Whitehead discovered the following fact. \begin{theorem} Let (\xymatrix{H \ar[r]^{\partial} & G \ar[r]^{\varphi} & Aut H}) be a crossed module of groups. The set Der(G,H) is a monoid w.r.t. (d_1+d_2)(x)=d_1 ( \partial ( d_2(x))x) + d_2(x). \end{theorem} The aim of this talk is to understand in a more conceptual way Whitehead product of derivations. The idea is to replace crossed modules of groups by the equivalent notion of internal groupoids in the category of groups. Using the language of internal groupoids, Whitehead product becomes clear: it is nothing but the composition in the internal category. The surprise is that, once expressed in terms of internal groupoids, Whitehead theorem, as well as some other basic properties of derivations, has nothing to do with groups, but holds in the very general context of internal groupoids in an arbitrary category \cal{G} with finite limits. In this way, these results hold not only for crossed module of groups (when \cal{G} is the category of groups), but also for crossed modules of Lie algebras (take for \cal{G} the category of Lie algebras), Lie groupoids (take for \cal{G} the category of smooth manifolds), and of course ordinary groupoids (take for \cal{G} the category of sets). This is part of a work in collaboration with S. Kasangian, S. Mantovani and G. Metere.