# PhD students of Steve Vickers: Sina Hazratpour

Sina Hazratpour completed his PhD thesis "A Logical Study of some 2-categorical Aspects of Toposes" in 2019 at the School of Computer Science, University of Birmingham, under my supervision.

Some of the most important results were written up as a joint paper with me, "Fibrations of AU-contexts beget fibrations of toposes".

### THESIS ABSTRACT

#### "A Logical Study of some 2-categorical Aspects of Toposes"

PhD Thesis, School of Computer Science, University of Birmingham, 2019. 324 pages.

Summary

There are two well-known topos-theoretic models of point-free generalized spaces: the original Grothendieck toposes (relative to classical sets), and a relativized version (relative to a chosen elementary topos S with a natural number object) in which the generalized spaces are the bounded geometric morphisms from an elementary topos E to S, and they form a 2-category BTop/S. However, often it is not clear what a preferred choice for the base S should be.

In this work, we review and further investigate a third model of generalized spaces, based on the 2-category Con of 'contexts for Arithmetic Universes (AUs)' presented by AU-sketches which originally appeared in Vickers's work in [Vic19] and [Vic17].

We show how to use the AU techniques to get simple proofs of conceptually stronger, base-independent, and predicative (op)fibration results in ETop, the 2-category of elementary toposes equipped with a natural number object, and arbitrary geometric morphisms. In particular, we relate the strict Chevalley fibrations, used to define fibrations of AU-contexts, to non-strict Johnstone fibrations, used to define fibrations of toposes.

Our approach brings to light the close connection of (op)fibration of toposes, conceived as generalized spaces, with topological properties. For example, every local homeomorphism is an opfibration and every entire map (i.e. fibrewise Stone) is a fibration.