Abstracts

ABSTRACTS

B. Banaschewski Baire sets versus the Boolean reflection of sigma-frames:
We present an analysis of the relation between the sigma-field BX of Baire sets of a Tychonoff space X (= the sigma-field of subsets of X generated by Coz X, the sigma-ring of cozero sets of X) and the abstractly given Boolean sigma-frame reflection of Coz X.

M. M. Clementino Lawvere completeness for lax proalgebras (Joint work with D. Hofmann):
In this talk, using the approach to quasi-uniform spaces as lax proalgebras, in the sense of [2], we will show that Cauchy-completeness can be viewed as (categorical) Lawvere-completeness (as detailed in [1]) and we will generalize Salbany's completion monad [3].
References:
[1] M.M. Clementino and D. Hofmann, Lawvere completeness in Topology, Appl. Categ. Structures 17 (2009), 175-210.
[2] M.M. Clementino, D. Hofmann and W. Tholen, One setting for all: metric, topology, uniformity, approach structure, Appl. Categ. Structures 12 (2004), 127-154.
[3] S. Salbany, The completion monad and its algebra, Comment. Math. Univ. Carolin. 23 (1982), 301-311.

J. Gutiérrez García On extended real-valued functions in pointfree topology (Joint work with J. Picado, in progress):
We continue the development of the localic theory of real-valued functions, recently introduced by Gutiérrez García, Kubiak and Picado [Localic real-valued functions: a general setting, Journal of Pure and Applied Algebra 213 (2009) 1064-1074]. In this talk we intend to present our last results in the study of extended real-valued functions in this setting.

D. Hofmann An unnatural duality:
In this talk we extend the notion of complete distributivity from ordered sets to (topological) spaces. Furthermore, we construct a dual adjunction between topological spaces and completely distributive spaces and relate this to the "traditional" adjunction between spaces and frames.

N. Marcus On atomless Boolean algebras:
In this talk we shall discuss some atomless Boolean algebras obtained by way of topological spaces introduced by Husek [2, 3] in his study of Herrlich's notion of k-compactness [1].
References:
[1] H. Herrlich, Fortsetzbarkeit stetiger Abbilungen und Kompaktheidsgrad topologisher Raume, Math. Zeitschr. 96 (1967) 64-72.
[2] M. Husek, On a problem of H. Herrlich, Comment. Math. Univ. Carolin. 9 (1968) 571-572
[3] M. Husek, The class of k-compact spaces is simple, Math. Z. 110 (1969) 123-126.

P. Resende Stably Gelfand quantales:
The notion of Gelfand quantale introduced by Mulvey is part of an attempt to characterize the quantales that arise from C^*-algebras. In this talk I will show that a stronger notion, that of stably Gelfand quantale, of which the quantales of C^*-algebras are still examples, has very interesting properties both in connection with C^*-algebras and their relation to groupoids via convolution of C^*-algebras, and in connection with notions of sheaf on quantales. An involutive quantale Q is stably Gelfand if it satisfies the condition
aa*a \leq a <=> aa*a=a for all a in Q.

The first appearance of this axiom, for involutive quantaloids, is in a paper of Dale Garraway who calls such quantaloids pseudo-rightsided.

A. H. Roque Descent in categories of models of a binary relational implication:
We characterize descent morphisms in categories of models of a binary relational implication and give sufficient conditions for these morphisms to be of effective descent..

B. Banaschewski	Baire sets versus the Boolean reflection of sigma-frames: We present an analysis of the relation between the sigma-field BX of Baire sets of a Tychonoff space X (= the sigma-field of subsets of X generated by Coz X, the sigma-ring of cozero sets of X) and the abstractly given Boolean sigma-frame reflection of Coz X.
M. M. Clementino	Lawvere completeness for lax proalgebras (Joint work with D. Hofmann): In this talk, using the approach to quasi-uniform spaces as lax proalgebras, in the sense of [2], we will show that Cauchy-completeness can be viewed as (categorical) Lawvere-completeness (as detailed in [1]) and we will generalize Salbany's completion monad [3]. References: [1] M.M. Clementino and D. Hofmann, Lawvere completeness in Topology, Appl. Categ. Structures 17 (2009), 175-210. [2] M.M. Clementino, D. Hofmann and W. Tholen, One setting for all: metric, topology, uniformity, approach structure, Appl. Categ. Structures 12 (2004), 127-154. [3] S. Salbany, The completion monad and its algebra, Comment. Math. Univ. Carolin. 23 (1982), 301-311.
J. Gutiérrez García	On extended real-valued functions in pointfree topology (Joint work with J. Picado, in progress): We continue the development of the localic theory of real-valued functions, recently introduced by Gutiérrez García, Kubiak and Picado [Localic real-valued functions: a general setting, Journal of Pure and Applied Algebra 213 (2009) 1064-1074]. In this talk we intend to present our last results in the study of extended real-valued functions in this setting.
D. Hofmann	An unnatural duality: In this talk we extend the notion of complete distributivity from ordered sets to (topological) spaces. Furthermore, we construct a dual adjunction between topological spaces and completely distributive spaces and relate this to the "traditional" adjunction between spaces and frames.
N. Marcus	On atomless Boolean algebras: In this talk we shall discuss some atomless Boolean algebras obtained by way of topological spaces introduced by Husek [2, 3] in his study of Herrlich's notion of k-compactness [1]. References: [1] H. Herrlich, Fortsetzbarkeit stetiger Abbilungen und Kompaktheidsgrad topologisher Raume, Math. Zeitschr. 96 (1967) 64-72. [2] M. Husek, On a problem of H. Herrlich, Comment. Math. Univ. Carolin. 9 (1968) 571-572 [3] M. Husek, The class of k-compact spaces is simple, Math. Z. 110 (1969) 123-126.
P. Resende	Stably Gelfand quantales: The notion of Gelfand quantale introduced by Mulvey is part of an attempt to characterize the quantales that arise from C^-algebras. In this talk I will show that a stronger notion, that of stably Gelfand quantale, of which the quantales of C^-algebras are still examples, has very interesting properties both in connection with C^-algebras and their relation to groupoids via convolution of C^-algebras, and in connection with notions of sheaf on quantales. An involutive quantale Q is stably Gelfand if it satisfies the condition aaa \leq a <=> aaa=a for all a in Q. The first appearance of this axiom, for involutive quantaloids, is in a paper of Dale Garraway who calls such quantaloids pseudo-rightsided.
A. H. Roque	Descent in categories of models of a binary relational implication: We characterize descent morphisms in categories of models of a binary relational implication and give sufficient conditions for these morphisms to be of effective descent..