Abstracts

ABSTRACTS

E. Faro The pretrace of a small category:
The (set-)trace of a small category C is the coend of its hom-set functor. This coend can be calculated as the set of connected components of a small category called the pretrace of C. We discuss several properties of the pretrace, giving some examples of its use in the calculation of the trace. A more general situation is that of considering an arbitrary endoprofunctor f : C^op x C –> Set instead of the hom-set functor of C. Also in this general case a pretrace category can be defined whose set of connected components is the coend of f.

R. González Weak Galois extensions:
It is a well-know fact that in the definition of Hopf-Galois extensions, the bijectivity of the canonical morphism is an essential part. In this talk we show a criterion under which the surjectivity of the canonical morphism implies the biyectivity for weak Galois extensions in a monoidal setting (for example Hopf-Galois extensions asociated to weak braided Hopf algebras), generalizing and improving recent results of Schauenburg and Schneider and Brzezinski, Turner and Wrightson.
References:
[1] J.N. Alonso Álvarez, J.M. Fernández Vilaboa, R. González Rodríguez, P. López López, A characterization of projective weak Galois extensions, Israel J. of Math. (in press) (2008).
[2] T. Brzezinski, R.B. Turner, A. P. Wrightson, The structure of weak coalgebra-Galois extensions, Comm. Algebra 34(4) (2006) 1489-1519.
[3] P. Schauenburg, H.J. Schneider, On generalized Hopf Galois extensions, J. of Algebra 202(1) (2005), 168-194.

G. Janelidze Ideal determined categories (Joint work with L. Márki, A. Ursini, and W. Tholen):
We introduce a new class of categories, which we call ideal determined, since it is a categorical counterpart of the class of ideal determined varieties of universal algebras in the sense of A. Ursini. We explain its role in categorical algebra and in particular show how it solves an axiomatization problem for semi-abelian categories.

T. Janelidze New algebraic examples of relative semi-abelian categories:
The equivalent definitions of a relative semi-abelian category [T. Janelidze, Relative semi-abelian categories, Applied Categorical Structures, accepted and available online] and the reasons for introducing it will be recalled. The purpose of this talk is to describe new examples of such categories and particularly those related to generalized central extensions in classical and universal algebra.

P. Resende Self-portraits of a topos:
Let E be an etendue, and let G be an etale groupoid that represents it (ie, E=BG). The objects of E are the G-sheaves and, in particular, G itself with the left regular action is an object of E. Denoting by O(G) the inverse quantal frame of G, E is equivalent to a suitable category of left modules O(X) over O(G) [arXiv: 0807.3859], which can be described in more than one way [arXiv: 0807.4848]. In this talk I will give an argument showing that O(G) is the quantale of global sections of the quantale of binary relations P(GxG) in E and, similarly, any module O(X) is the global sections of the P(GxG)-module P(GxX), leading us to an equivalence of categories between E and its category of internal P(GxG)-modules of the form P(GxX). Time allowing I will also discuss the possibility of extending these results to general Grothendieck toposes and more general (but not all) open groupoids G, which depends among other things on the existence of an equivalence of categories between Loc(BG) and the category of continuous G-actions.

M. Sobral Descent for compact 0-dimensional spaces (Joint work with G. Janelidze):
We describe effective descent morphisms of compact 0-dimensional topological spaces, call them fibrations, and then show that all surjections are effective descent morphisms with respect to those fibrations.

T. Van der Linden The third cohomology group classifies double central extensions :
We prove that the third cohomology group H₃(Z, A) of an object Z with coefficients in an abelian object A of a Moore category classifies the double central extensions of Z by A. Thus the connections between two branches of non-abelian (co)homology are made explicit: the direction approach to cohomology in Moore categories established by Bourn and Rodelo on one hand, and the approach to semi-abelian homology based on categorical Galois theory initiated by Janelidze and worked out by Everaert, Gran, Rossi and Van der Linden on the other.

E. Faro	The pretrace of a small category: The (set-)trace of a small category C is the coend of its hom-set functor. This coend can be calculated as the set of connected components of a small category called the pretrace of C. We discuss several properties of the pretrace, giving some examples of its use in the calculation of the trace. A more general situation is that of considering an arbitrary endoprofunctor f : C^op x C –> Set instead of the hom-set functor of C. Also in this general case a pretrace category can be defined whose set of connected components is the coend of f.
R. González	Weak Galois extensions: It is a well-know fact that in the definition of Hopf-Galois extensions, the bijectivity of the canonical morphism is an essential part. In this talk we show a criterion under which the surjectivity of the canonical morphism implies the biyectivity for weak Galois extensions in a monoidal setting (for example Hopf-Galois extensions asociated to weak braided Hopf algebras), generalizing and improving recent results of Schauenburg and Schneider and Brzezinski, Turner and Wrightson. References: [1] J.N. Alonso Álvarez, J.M. Fernández Vilaboa, R. González Rodríguez, P. López López, A characterization of projective weak Galois extensions, Israel J. of Math. (in press) (2008). [2] T. Brzezinski, R.B. Turner, A. P. Wrightson, The structure of weak coalgebra-Galois extensions, Comm. Algebra 34(4) (2006) 1489-1519. [3] P. Schauenburg, H.J. Schneider, On generalized Hopf Galois extensions, J. of Algebra 202(1) (2005), 168-194.
G. Janelidze	Ideal determined categories (Joint work with L. Márki, A. Ursini, and W. Tholen): We introduce a new class of categories, which we call ideal determined, since it is a categorical counterpart of the class of ideal determined varieties of universal algebras in the sense of A. Ursini. We explain its role in categorical algebra and in particular show how it solves an axiomatization problem for semi-abelian categories.
T. Janelidze	New algebraic examples of relative semi-abelian categories: The equivalent definitions of a relative semi-abelian category [T. Janelidze, Relative semi-abelian categories, Applied Categorical Structures, accepted and available online] and the reasons for introducing it will be recalled. The purpose of this talk is to describe new examples of such categories and particularly those related to generalized central extensions in classical and universal algebra.
P. Resende	Self-portraits of a topos: Let E be an etendue, and let G be an etale groupoid that represents it (ie, E=BG). The objects of E are the G-sheaves and, in particular, G itself with the left regular action is an object of E. Denoting by O(G) the inverse quantal frame of G, E is equivalent to a suitable category of left modules O(X) over O(G) [arXiv: 0807.3859], which can be described in more than one way [arXiv: 0807.4848]. In this talk I will give an argument showing that O(G) is the quantale of global sections of the quantale of binary relations P(GxG) in E and, similarly, any module O(X) is the global sections of the P(GxG)-module P(GxX), leading us to an equivalence of categories between E and its category of internal P(GxG)-modules of the form P(GxX). Time allowing I will also discuss the possibility of extending these results to general Grothendieck toposes and more general (but not all) open groupoids G, which depends among other things on the existence of an equivalence of categories between Loc(BG) and the category of continuous G-actions.
M. Sobral	Descent for compact 0-dimensional spaces (Joint work with G. Janelidze): We describe effective descent morphisms of compact 0-dimensional topological spaces, call them fibrations, and then show that all surjections are effective descent morphisms with respect to those fibrations.
T. Van der Linden	The third cohomology group classifies double central extensions : We prove that the third cohomology group H₃(Z, A) of an object Z with coefficients in an abelian object A of a Moore category classifies the double central extensions of Z by A. Thus the connections between two branches of non-abelian (co)homology are made explicit: the direction approach to cohomology in Moore categories established by Bourn and Rodelo on one hand, and the approach to semi-abelian homology based on categorical Galois theory initiated by Janelidze and worked out by Everaert, Gran, Rossi and Van der Linden on the other.