# PhD students of Steve Vickers: Kostas Viglas

Kostas Viglas completed his PhD thesis "Topos Investigations of the Extended Priestley Duality" (.ps, .pdf) in 2004 at the Department of Computing, Imperial College, under my supervision.

### ABSTRACT

#### "Topos Investigations of the Extended Priestley Duality"

PhD Thesis, Department of Computing, Imperial College, 2004. 192 pages.

This is a brief summary of the original results in the thesis.

First, there is a localic version of the correspondence between perfect and patch continuous monotone maps. To this end, Escardo's localic patch construction for a stably compact locale is used. Given a stably compact locale *X*, we define constructively the order on its patch locale. We also introduce localically the notion of a monotone patch continuous function in this context. The fact that lax pullbacks of perfect maps produce proper maps in **Loc** is proved. Vickers' preframe techniques are used throughout. Beck-Chevalley conditions for lax-coequalizers are also proved.

When working in **Top**, the 2-category of Grothendieck topoi and geometric morphisms, it is natural to consider functors between the (generalized) points of topoi. A 2-categorical criterion of an adjunction *F* -| *G* between
*X* and *Y* in **Top** is proved by constructing the classifying topoi of maps *Fx* -> *y* and *x* -> *Gy*,
where *x* and *y* are points of *X* and *Y* respectively and identifying them with inserters in **Top**.

Next, it is demonstrated that relative tidiness (in the sense of Moerdijk and Vermeulen) is the right topos-theoretic generalization of perfectness. Vickers has shown that the exponential of topoi [*set*]^{X}, where *X* is a stably compact locale, classifies the geometric theory of B-sheaves which implies that a point of [*set*]^{X} at stage *Z* is a B-sheaf in the sheaves over *Z*.
For *f*: *X* -> *Y* a perfect map between two stably compact locales, a description of the map
[*set*]^{f}: [*set*]^{X}
-> [*set*]^{Y} is given and is shown to have a right adjoint.
The definitions of the geometric morphisms are given by geometric constructions on the points of the exponential topos, i.e. the B-sheaves. The geometricity of these constructions is guaranteed by the fact that we can represent perfect maps by strong homomorphisms between strong proximity lattices. The adjunction is proved by application of the 2-categtorical criterion in the 2-category **Top**. The main result of this chapter is that for a map *f*: *X* -> *Y* between stably compact locales, *f* is perfect if and only if *f* is relatively tidy.

Finally, there are investigations with a possible topos analogue of the patch construction. Some results are given on relatively tidy maps between structures that are examples of "stably compact topoi". It is argued by example, that "stably compact topoi" and relatively tidy maps should convey the notion of local partial ordering in the same sense that stably compact locales and perfect maps amount to (globally) partially ordered locales and monotone continuous maps.