6 - Bohrification

Summary

Introduction

More than a decade ago, Chris Isham proposed a topos-theoretic approach to quantum mechanics, initially in the context of the Consistent Histories approach, and subsequently (in collaboration with Jeremy Butterfield) in relationship with the Kochen–Specker Theorem [21–23] (see also [20] with John Hamilton). More recently, jointly with Andreas Döring, Isham expanded the topos approach so as to provide a new mathematical foundation for all of physics [38, 39]. One of the most interesting features of their approach is, in our opinion, the so-called Daseinisation map, which should play an important role in determining the empirical content of the formalism.

Over roughly the same period, in an independent development, Bernhard Banaschewski and Chris Mulvey published a series of papers on the extension of Gelfand duality (which in its usual form establishes a categorical duality between unital commutative C*-algebras and compact Hausdorff spaces; see, e.g., [57], [65]) to arbitrary toposes (with natural numbers object) [6–8]. One of the main features of this extension is that the Gelfand spectrum of a commutative C*-algebra is no longer defined as a space but rather as a locale (i.e., a lattice satisfying an infinite distributive law [57]; see also Section 6.2). Briefly, locales describe spaces through their topologies instead of through their points, and the notion of a locale continues to make sense even in the absence of points (whence the alternative name of pointfree topology for the theory of locales).