University of Birmingham > Talks@bham > Theoretical computer science seminar > Geometric constructions for (op)fibrations

Geometric constructions for (op)fibrations

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  • UserBertfried Fauser, University of Birmingham
  • ClockFriday 22 June 2012, 16:00-17:00
  • HouseCS 217.

If you have a question about this talk, please contact Paul Levy.

(Joint work with Steve Vickers)

Recent approaches to physics using presheaf toposes have been developed in both contra- and covariant forms. We use the localic-bundle-theorem to reinterpret these as a topos-valid bundle based approach that extends readily to sheaf toposes, using techniques of geometricity to ease the topos-theoretic calculations.

In bundle form we find that the distinction between contra- and covariance appears as a distinction between opfibrations and fibrations (in the sense of 2-category theory), and that the use of one or the other arises out of general and fundamental methodological assumptions in the different topos approaches.

The categorically abstract work reported here has as an application that geometric constructions such as the valuation locale (corresponding to the Giry monad and introducing probabilistic aspects of quantum physics) preserve opfibrations and fibrations and hence remain within one or other of the two classes of bundles.

We start from some results of Street characterizing (op)fibrations in representable 2-categories as maps supporting certain pseudo algebra structures. We extend this framework to arrow 2-categories. This allows us to use techniques of Vickers and Townsend to capture geometricity in a categorical manner, of “preserving horizontality”, removing the need to backtrack to the usage of frames. We prove that suitable arrow 2-functors preserve (op)fibrations by way of some lifting theorems for Eilenberg-Moore categories of pseudo algebras. Specializing to the arrow 2-category of locales we obtain:

1) A mutual characterization of the specialization orders on the display space and base space of the bundle, providing a better understanding of the topologies on bundle spaces.

2) The fact that several geometric power constructions, among them power locales and the geometric (probability) valuation monad of Vickers, preserve (op)fibrations.

Our research sheds some light on the similarities and differences of the contra- and covariant topos approaches. Moreover, the results are paramount in the study of Born maps and Born sections thought to model quantum probabilities and in a search for a correct manifold topology, contrary to the usually adopted discrete topology, for spaces of classical contexts of the same `type’ used in the contra/covariant approaches.

This talk is part of the Theoretical computer science seminar series.

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