"Sketches for arithmetic universes" (list arithmetic pretoposes)

"Sketches for arithmetic universes" (list arithmetic pretoposes)

[Steve Vickers]

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/ ]