# Categories of compact and compactly generated Hausdorff locales over a base topos

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## Date

2022-12-28

## Authors

Τσάμης, Κωνσταντίνος

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## Abstract

This thesis is focused on the categories of compact Hausdorff locales and of compactly generated (Hausdorff) ones. Our main contributions have to do with the question of whether they share with the categories of the respective
classical topological spaces the property of forming, respectively, a pretopos and a regular cartesian closed category. Locales constitute the correct substitute for topological spaces internally in toposes, where points may occur scarcely, as their existence may depend on non-constructive principles like the axiom of choice or the weaker prime ideal theorem for distributive lattices. They appear as duals of structures such as commutative rings, distributive lattices or C-algebras even when these structures lack sufficiently many prime or maximal ideals, they allow us to talk about fundamental structures like the real numbers as spaces which maintain the right properties, where their construction as Dedekind cuts or Cauchy sequence may produce nonisomorphic results in the absence of the axiom of choice. Hence we take care that all our arguments remain valid in the internal logic of a topos, meaning that we avoid using the axiom of choice or the principle of excluded middle with the sole exception of Chapter 3 which explicitly refers to categories of topological spaces.
The benefits, from the point of view of classical mathematics, of such an approach are also well-known: The category of locales over a given locale is equivalent to the category of the locales internal to the topos of sheaves on the given locale, hence results about locales that are internally valid in a topos (for example our Corollary 4.2.5 and Proposition 5.1.2 in this thesis) translate to results about continuous maps over the
given base. The method for developing a sufficiently rich theory of locales is algebraic. Algebraic structures have been in service of topology in various ways but we will focus on how distributive lattices and in particular frames play this role. The category of locales is by definition the one opposite to the category of frames. The latter allows us to construct categorical colimits in the category of locales (limits in the category of frames) in a simple way as in any category of algebraic structures.
The main results of the thesis are presented in the last two chapters of the thesis. They include a proof that the category of compact Hausdorff locales is a pretopos and that the category of compactly generated Hausdorff locales is a regular category, provided that it is coreflective in the category of Hausdorff locales. Under the same hypothesis we show that in this category products commute with
colimits, which is a necessary condition for cartesian closedness. We also investigate a possible characterization of the pretopos of compact Hausdorff locales by presenting a comparison functor from a pretopos that satisfies some extra properties to compact Hausdorff locales. This functor is faithful, full on subobjects and exact. In order to define that functor we prove first that closed quotients of compact Hausdorff locales are compact Hausdorff, generalizing the corresponding result for spaces in the localic setting.
There are a few more minor results, such as an account of the functorial behaviour of the tensor product of sup-lattices, cast in terms of the concrete description of the tensor product in [24] (Propositions 1.4.1, 1.4.2). There are also new proofs of known results, primarily a proof of the regularity of the category of weakly Hausdorff compactly generated spaces (Section 2.4), and proofs in the theory of locales that use positive formulations of the involved concepts and are valid in the internal logic of a topos, where in the literature we could only find proofs involving negative formulations (for example Proposition 3.3.4). For reasons of unity and self-sufficiency of the text we have included known proofs of most known results exposed in the introductory three first chapters. We have only omitted proofs of results that are too technical and would occupy disproportionally big space in the text.

## Description

## Keywords

Sup-lattices, Frames, Locales, Compact Hausdorrf locales, Compactly generated Hausdorff locales, Topos of Sheaves, Tensor product of frames, Regular category, Exact category, Cartesian closed category