ABSTRACTS FROM TALKS A category of 'spaces' that is not a category of locales
Topos ŕ l'IHES (23 au 27 novembre 2015) Description
We describe a short list of categorical axioms that make a category
behave like the category of locales. In summary the axioms assert
that the category has an object that behaves like the Sierpinski
space and this object is double exponentiable. A number of the usual
results of locale theory can be derived using the axioms: the
(weakly) closed subgroup theorem proved, closed and proper surjections
are of effective descent, parallel theories of discrete and compact
Hausdorff spaces emerge. An example is given of a category that
satisfies the axioms but which is not the category of locales for any
topos. We show how to embed the category of elementary toposes into the category whose objects are categories that satisfy the axioms.
Geometric morphisms as structure preserving maps
Category Theory  CT2014  Cambridge July 2014 Description
The `structure' which makes a cartesian category into a topos is the power set
functor, which classifies relations. However the morphisms between toposes (geometric morphisms) are not structure preserving maps; they are instead adjunctions
for which the left adjoint preserves finite limits. Relative to any topos there is
its category of locales. Some attempts to provide a categorical axiomatization of
locales hinge on assuming the existence of a double power locale functor, characterized categorically as a double exponential. The talk will provide a very brief
overview of this categorical axiomatization of locales. The focus of the talk will be
on describing how geometric morphisms can be represented as morphisms between
categories of locales (so axiomatized). It turns out that geometric morphisms correspond to adjunctions that commute with the double power locale functor (and
associated monad). In this way geometric morphisms can be reinterpreted as
'structure preserving maps'.
Reference
.pdf version of notes ‘’Towards a localic proof of the localic groupoid representation of Grothendieck toposes’’
Theoretical Computer Science Seminar, University of Birmingham, 2nd September 2011. Description
Given that geometric morphisms can be represented as adjunctions between categories of locales, it feels natural to ask whether the representation of bounded geometric morphisms (i.e. Grothendieck toposes) via localic groupoids can be carried out using only locale theory. This could allow the representation theorem to be be proved without having to deploy some of the heavy machinery of topos theory (notably, sites and the pullback stability of various properties of geometric morphisms). Although I have not been able to complete such a proof I would like to present a strategy which will give some insight into how the localic view of geometric morphisms can be used in practice. The strategy relies on having a localic characterization of bounded geometric morphisms; although it is clear what this localic characterization should look like (for the theorem to work that is!) proving that the characterization is correct appears to be technically challenging. The talk will provide background on Joyal and Tierney’s representation of Grothendieck toposes using localic groupoids and so will provide some understanding of how this important theorem works. ‘’Categorical aspects of Locale Theory’’ School of Computer Science. University of Birmingham. (Part of ‘Theory Seminars’ series.) 16th November 2007 Description
The purpose of the talk is to try and describe aspects of locale theory that are available using only categorical reasoning. “A categorical account of the localic closed subgroup theorem” Algebraic and Topological Methods in NonClassical Logics III. Mathematical Institute. University of Oxford. 59 August, 2007. Description
Given an axiomatic account of the category of locales the closed subgroup theorem is proved. The theorem is seen "A Representation Theorem for Geometric Morphisms"
83rd Peripatetic Seminar on Sheaves and Logic, Glasgow, 6&7th May 2006 Description We show that geometric morphisms between elementary toposes can be represented as adjunctions "The Patch Construction is the same as Algebraic DCPO Representation" Mathematical Institute, Oxford, 7th March 2005, Analytic Topology in Mathematics and Computer Science series Description By use of the parallel between the suplattice approach and the preframe approach to locale theory it can be shown that the "Axiomatic Localic Relational Composition" Abstract It is well known that in any regular category relational composition can be defined. This has application, for example, "Effective descent via natural transformations" Abstract Localic triquotient surjections is a class of surjections in the category of locales that covers two familiar classes; "Scott is Natural between Frames" Abstract Reporting on joint work with S. Vickers. The talk will outline a representation theorem for the Scott continuous maps "Axioms for a category of spaces: Discrete and Compact Hausdorff spaces" Abstract A detailed account was given of the regularity of the category of compact Hausdorff locales and of discrete No slides. "Axioms for a category of spaces" 79th Meeting, Peripatetic Seminar on Sheaves and Logic (Doorn, Netherlands), June 2829 2003. Abstract An axiomatic proof of the pullback stability of maps with triquotient assignement was outlined. No slides. "Axioms for a category of spaces (Birmingham)" Given as part of the 'Topology Seminar' series at the Computer Science Dept, Birmingham University, 14^{th} March 2003. Abstract A recent observation (with Vickers) states that directed join preserving (i.e. Scott continuous/dcpo) maps between frames "Axioms for a category of spaces" Internal Open University seminar. Pure Mathematics Department, 4^{th} March 2003. Description Aimed at a general mathematical audience this talk presents the background ideas from category theory needed to attempt "Locale Pullback via DCPOs" The 78^{th} meeting of the Peripatetic Seminar on Sheaves and Logic. Description This talk shows how to give a description of locale pullback in terms of a construction in the category of "Presentations for Topology" Domains VI conference. Birmingham University. 16^{th}19^{th} September 2002. Description Outlines the various different ways that a frame (and therefore a topological space) can be presented as "The Relationship between Topology and Logic" Open University Seminar, 11^{th} June 2002 Description This talk is aimed at beginners to category theory. It outlines how a category works as a mathematical universe, and gives the
