PhD Thesis
University of Birmingham (UK), 2019
Hazratpour, Sina (2019). "A logical study of some 2-categorical aspects of topos theory." University of Birmingham.
🇬🇧 Link to the deposit in the British Library
Abstract: 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’ work in Vickers, 2019 and Vickers, 2017. 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.