I'm pleased to announce that my paper "Sketches for arithmetic universes" has now been accepted for the Journal of Logic and Analysis. You can see a preprint on my web page ?? http://www.cs.bham.ac.uk/~sjv/papersfull.php#AUSk The purpose of the paper is to describe in a purely finitary way a 2-category Con of generalized (in the sense of Grothendieck) point-free topological spaces and continuous maps, with the aim of providing a foundationally neutral setting for topos theory. The essential trick is to use the internal nno and list objects in arithmetic universes to capture countable colimits in Grothendieck toposes; and then to use some novel techniques adapted from sketch theory to use sketches to stand in for geometric theories. In case you're unfamiliar with "generalized spaces", much of their flavour can be understood from domain theory a la Scott. The topological structure of domains comes from order and directed joins, and in the generalized spaces these become morphisms and filtered colimits. What in set theory appears as various proper classes (e.g. of sets, or of groups) become here generalized spaces (object classifier, group classifier), so universes of various kinds appear. But we also have ungeneralized topological structures such as Euclidean space or Cantor space. It remains to be seen whether this work can capture all the topological invariants for which toposes were invented, though that is my dream, but meanwhile some deep connections with toposes are explored in a companion paper, "Arithmetic universes and classifying toposes". My student Sina Hazratpour's thesis work is on technical results concerning (op)fibrations in Con, which we believe are important for issues of local compactness, exponentiability, and "bagtoposes". All the best, Steve. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]