Path-lifting for Grothendieck toposes

Path-lifting for Grothendieck Toposes

A general path-lifting theorem, which fails in the context of topological spaces, is shown to hold for toposes, for locales (a slight generalization of topological spaces), and hence for complete separable metric spaces. This result generalizes the known fact that any connected locally connected topos (respectively complete separable metric space) is path-connected. In [MW] we proved that every...

Gelfand spectra in Grothendieck toposes using geometric mathematics

In the (covariant) topos approach to quantum theory by Heunen, Landsman and Spitters, one associates to each unital C*-algebra A a topos T (A) of sheaves on a locale and a commutative C*-algebra A within that topos. The Gelfand spectrum of A is a locale Σ in this topos, which is equivalent to a bundle over the base locale. We further develop this external presentation of the locale Σ, by noting...

A N Elementary Proof of the Descent Theorem for Grothendieck Toposes

Sheaves for predicative toposes

In this paper, we discuss the stability of predicative toposes under taking sheaves for an internal site. What is the most useful notion of a predicative topos has not been established yet, but we discuss sheafification for some candidates for that title. The results obtained here sharpen the results published by Moerdijk and Palmgren in [10] and [11]. We also propose a new axiom and show its u...

Sheaf toposes for realizability

We compare realizability models over partial combinatory algebras by embedding them into sheaf toposes. We then use the machinery of Grothendieck toposes and geometric morphisms to study the relationship between realizability models over different partial combinatory algebras. This research is part of the Logic of Types and Computation project at Carnegie Mellon University under the direction o...