1st GALAI Workshop, Fri, Jan 26

Invited Speakers:

Alexandru Baltag (University of Amsterdam)
Caleb Schultz Kisby (Indiana University Bloomington)
Sonja Smets (University of Amsterdam)

Speakers from Chapman Univeristy:

Drew Moshier
Peter Jipsen
Jose Gil-Ferez
Alexander Kurz

Local Organizer: Alexander Kurz

Drew Moshier

Relations in Point-Free Topology

We report on progress on a program of investigating relations on point-free topological spaces (locales). As the category of locales is order-enriched, a natural notion of a relation from X to Y is a jointly epi cospan X->C<-Y. Dualizing, these manifest as special spans on the category of frames. Namely, we see that they correspond to the familiar notion of weakening relations between frames that are compatible with the frame structure in an obvious way. Thus, the study of point-free relations is the study of these so-called frame relations.

In the talk today, we build some of the needed techniques to deal with frame relations systematically. As one application, we show that on any frame L, the preorder frame relations on L constitute another frame, dually isomorphic to the coframe of point-free subspaces of L. We then prove directly a known result of Plewe that this frame is ultraparacompact, and also that is satisfies a certain density result. Together, these two properties are strong, and appear to be very nearly characterizing frames that occur as frames of nuclei.