LFCS Seminar: 5 February 2020 - Steve Vickers

Title: Bundles as continuous space-valued maps

Abstract:

If we call a continuous map p: Y -> X a “bundle”, it’s because we think of it in reverse as a space-valued map on X, assigning to each base point x the corresponding fibre of p. If this fibre assignment is continuous, then it provides the bundle space Y with a topology, and makes p continuous.

Except ... that makes no sense in classical point-set topology. There is no point-set topological space of all point-set topological spaces, so what can it mean to say the fibre assignment is continuous?

Topos theory answers this with two complementary ideas, of *point-free topology* and of *generalized spaces*. Both involve the so-called “geometric” logic. By using maps from X to a generalized space of point-free spaces, we can capture the continuity of a fibre assignment, and now we really can reconstruct the bundle.

Some topos approaches to quantum physics (Doering-Isham; Heunen-Landsman-Spitters; Reynaud) provide an interesting illustration of this. In each case, the base space is a space of “contexts” (systems of commuting observables), and the fibres are Gelfand spectra. The construction results in a spectral bundle.

My own recent work uses Joyal’s arithmetic universes (AUs) as a substitute for the topos theory that avoids some problems that arise from using “arbitrary” colimits.