Diffeological Morita Equivalence

We introduce a new notion of Morita equivalence for diffeological groupoids, generalising the original notion for Lie groupoids. For this we develop a theory of diffeological groupoid actions, -bundles and -bibundles. We define a notion of principality for these bundles, which uses the notion of a subduction, generalising the notion of a Lie group(oid) principal bundle. We say two diffeological groupoids are Morita equivalent if and only if there exists a biprincipal bibundle between them. Using a Hilsum-Skandalis tensor product, we further define a composition of diffeological bibundles, and obtain a bicategory DiffBiBund. Our main result is the following: a bibundle is biprincipal if and only if it is weakly invertible in this bicategory. This generalises a well known theorem from the Lie groupoid theory. As an application of the framework, we prove that the orbit spaces of two Morita equivalent diffeological groupoids are diffeomorphic. We also show that the property of a diffeological groupoid to be fibrating, and its category of actions, are Morita invariants.Comment: 35 page

Ordered Locales

We extend the Stone duality between topological spaces and locales to include order: there is an adjunction between the category of preordered topological spaces satisfying the so-called open cone condition, and the newly defined category of ordered locales. The adjunction restricts to an equivalence of categories between spatial ordered locales and sober T 0-ordered spaces with open cones.</p

Towards point-free spacetimes

In this thesis we propose and study a theory of ordered locales, a type of point-free space equipped with a preorder structure on its frame of opens. It is proved that the Stone-type duality between topological spaces and locales lifts to a new adjunction between a certain category of ordered topological spaces and the newly introduced category of ordered locales. As an application, we use these techniques to develop point-free analogues of some common aspects from the causality theory of Lorentzian manifolds. In particular, we show that so-called indecomposable past sets in a spacetime can be viewed as the points of the locale of futures. This builds towards a point-free causal boundary construction. Furthermore, we introduce a notion of causal coverage that leads naturally to a generalised notion of Grothendieck topology incorporating the order structure. From this naturally emerges a localic notion of domain of dependence, which is generally distinct from the traditional notion in spacetimes

Axioms for the category of Hilbert spaces and linear contractions

The category of Hilbert spaces and linear contractions is characterised by elementary categorical properties that do not refer to probabilities, complex numbers, norm, continuity, convexity, or dimension

Axioms for the category of Hilbert spaces and linear contractions

The category of Hilbert spaces and linear contractions is characterised by elementary categorical properties that do not refer to probabilities, complex numbers, norm, continuity, convexity, or dimension.Comment: 16 page