Abstract
In this talk I will describe the context of the Patch construction as it was in 1995/6 when Escardo introduced his version. I was working on a very similar problem at the time, also at Imperial (we shared an office), but came up with a different construction. It was not until 2011 that I could answer the question: why are there different constructions? In the body of the talk I will show how this naturally comes about once we see the Patch construction process as the compact Hausdorff dual to the process of backing out the compact elements of an algebraic dcpo. I will then outline how Escardo’s Patch provides us with an example of a statement about compact Hausdorff locales that is expressible within the regular fragment of logic, but which is not mirrored by its discrete dual.
Time permitting I will point to areas where I think further research on the Patch could be considered; specifically an idea about extending the domain of definition to certain localic presheaves.
Talk Details
Speaker: Christopher Townsend (Independent researcher, UK)
Event: Types and Topology Workshop in Honour of Martin Escardo’s 60th Birthday
Date: Wednesday 17 December 2025
Time: 09:25
Slides: Available
Key Topics
- Patch construction
- Compact Hausdorff locales
- Algebraic dcpo
- Regular fragment of logic
- Localic presheaves
- Domain theory