Contributed Talks SOME ASPECTS OF ULTRAFILTER CONVERGENCE IN TOPOLOGY

In this talk we wish to present ultrafilter characterisations of special classes of continuous maps between topological spaces, and, if time permits, how this work led to the development of a general setting presenting spaces as categories. Our interest was motivated by the work of G. Janelidze and M. Sobral on (Grothendieck) descent theory in finite topological spaces, where, according to them, descriptions of various kind of descent maps “become very simple and natural as soon as they are expressed in the language of finite preorders”. The preorder relation of a (finite) space is just the point convergence shadow of the ultrafil-ter convergence relation, hence one might expect to obtain results about arbitrary spaces by “just ” replacing point convergence by ultrafilter convergence. Following this idea, we will present characterisations of the classes of (effective) descent maps, triquotient maps and local homeomorphisms in terms of lifting properties of chains of convergent ultrafilters [1, 3]. For instance, triquotient maps were introduced by E. Michael as those continuous maps f: X → Y for which there exists a mapping (_)] : OX → OY between the frames of open subsets which satisfies certain con