Towards a Joyal-Tierney Theorem for Arrow Algebras
Max Wehmeier

Abstract:
The Joyal-Tierney theorem states that morphisms into a given locale are equivalent to internal locales of its sheaf topos. This is useful in the study of topoi and can even be used in algebraic topology. Since arrow algebras generalize locales, we wanted to see if this theorem also holds for them. In this thesis, we will look at different approaches of defining the internalization functor for arrow algebras, which gives one direction of the Joyal-Tierney theorem. These approaches have varying degrees of maturity and we will discuss their strengths and weaknesses.
On the way to these results, we will also go on a journey through categorical logic with a special focus on our main tool: triposes. We will explore the tripos to topos construction and some characterizations in it. In particular, we will look at assemblies and proto-assemblies to define the evolution functor.