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Generalised Stone Dualities David Farrell Department of Mathematics The University of Auckland Supervisor: Pedram Hekmati A dissertation submitted in partial fulfillment of the requirements for the degree of BSc(Hons) in Mathematics, The University of Auckland, 2018. Abstract In 1938, Marshall H. Stone, motivated by his work in functional analysis, proved his representation theorem for Boolean algebras, which establishes a correspondence be- tween Boolean algebras, and a certain class of topological spaces, called Stone spaces. This correspondence, after the inception of category theory in the 40s, was recognised to be a duality of categories. More recently, it has been found that there exists a similar correspondence between the category of those topological spaces satisfying a weak separation condition called sobriety and the category of lattices, called spatial locales, which satisfy a set of axioms analogous to those for a topology. A slight weakening of the conditions on spatial locales yields locales, which can be shown to satisfy many nice properties. For example, the analogue for Tychonoff's theorem for locales has a constructive proof. The theory of locales is considered to be a model of point-free topology, since the spaces considered do not have a primitive notion of point. The duality between sober topological spaces and spatial locales is one of a number of theorems given the name Stone duality. In this dissertation I give a further gen- eralisation of the methods of Stone duality which provides a duality theorem for any category of lattices which satisfies a handful of weak conditions. Special cases include Stone's representation theorem for Boolean algebras, the duality between sober spaces and spatial locales, and a correspondence between a category of Boolean algebras and a subcategory of measurable spaces. i ii Contents Abstract i Introduction 1 1 Lattices 3 1.1 Ideals and Filters . .6 1.2 Completeness . .8 1.3 Boolean Algebras . .9 1.3.1 Basic Facts of Boolean Algebras . .9 1.3.2 Boolean Rings . 10 2 Pointless Topology 13 2.1 Frames and Locales . 14 2.2 Points . 14 2.2.1 Points as Completely Prime Filters . 14 2.2.2 Points as Global Elements . 16 2.3 A Non-Spatial Locale . 18 2.3.1 Sublocales . 19 2.4 Properties of Locales . 20 3 Frames Of Type (p; q) 21 3.1 Frames Of Type (p; q)............................ 21 3.2 K -Locales and K -spaces . 22 3.3 Spatiality . 24 3.4 Sobriety . 25 3.5 Generalised Stone Duality . 27 4 Special Cases and Consequences of Theorem 3.5.2 29 4.1 Stone's Representation Theorem for Boolean Algebras . 29 4.2 Pointless Measure Theory . 32 iii iv CONTENTS References 41 Introduction Consider the following observation. Let (X; T ) and (Y; S) be topological spaces and f : X ! Y a homeomorphism. Then f is open and hence induces a map f ∗ : T ! S by mapping an open set U 2 T to its image f(U) under f. Since f is a bijection, so is f ∗. A homeomorphism thus induces a bijection of the corresponding topologies which respects intersections, unions and inclusions in T . So structure preserving maps of the underlying spaces give rise to structure preserving maps between the structures placed upon them. Is there a way to tell which such maps arise from maps of spaces? Perhaps not, but there are special cases where all such maps do indeed arise in this way. A topology is a poset, when ordered by inclusion, and has the property that, for any finite set of pairs, there is a unique minimal set following them in the order: their union; and there is a unique set preceding them in the order: their intersection. This is a special case of a lattice, which is a poset containing suprema and infima for all of its finite subsets. A topology is then just a lattice of sets which, in addition, contains arbitrary unions of its elements. A structure preserving map between topologies should then be one which respects all unions and intersections. In general, for a continuous function f : X ! Y of topological spaces, the image f(U) of an open set is not open, and f ∗ does not respect unions and intersections. However, by definition of continuity, of U ⊂ Y is open, then f −1(U) is open in X, and inverse images respect all unions and intersections. Thus, for every continuous map f : X ! Y there is a corresponding \homomor- phism" of topologies f −1 : S ! T going the other way. In the case considered above where f is a homeomorphism, f ∗ is the inverse image map of the inverse of f. Now, if the condition that a topology consists of sets is relaxed, and we consider just lattices with arbitrary supremums, plus a distributivity condition which will be given later, we arrive at frames. Since the maps between frames go in the opposite direction to those between topological spaces, we often consider instead the opposite category to the cat- egory of frames, called the category of locales1. To what extent do locales correspond to spaces, and to what extent do maps of locales correspond to maps of spaces? There 1Short for local lattices. 1 2 CONTENTS exist locales which do not arise from topological spaces, and some which do have maps which do not arise from continuous functions of spaces. However there exist conditions called spatiality and sobriety for locales and topological spaces respectively such that one has an equivalence between the full subcategories SpLoc and SobTop of spatial locales and sober topological spaces. This equivalence is an example of a so-called Stone Duality theorem, which is a loose name for a collection of theorems which give dualities between categories thought to consist of algebraic objects, such as lattices, and categories thought to contain geometric objects, such as topological spaces. The main difference between topological spaces and locales is the topological spaces come with a \primitive" notion of points; the elements of the point set. Locales do not. Hence the study of locales is often referred to as point-free topology or pointless topol- ogy. Locales have many analogies with topological spaces, and many definitions and theorems can be translated from topological spaces to locales either directly or through Stone duality. There are, however, some distinctions. For instance, for topological spaces, the statement a product of spaces is compact if and only if each of its factors is, known as Tychonoff's theorem, is equivalent to the axiom of choice. The notion of compactness carries over for lattice almost verbatim, but the corresponding analogue of Tychonoff's theorem has a constructive proof. For this and other reasons, a large source of the motivation for pointless topology comes from constructive mathematics, hence results relying on the axiom of choice are marked with a *. In measure theory one of the main objects of study are so-called measurable spaces. Measurable spaces are, again, just sets with a particular type of lattice of subsets associated. Can Stone duality apply to measurable spaces? In fact, is the choice of type of lattice important at all? Is there a suitable notion of type of lattice such that each type comes with a Stone-like duality theorem? Are there conditions which ensure that maps of lattices correspond to maps of some type of space? In this dissertation I will answer these questions. The structure is as follows. In chapter 1, I give a brief treatment of the fundamen- tals of lattice theory, including topics such as lattices, homomorphisms, distributivity, complements, completeness and Boolean algebras. In chapter 2, I cover some introduc- tory Locale theory, including their associated point spaces and some of the nicer results. In chapter 3, I give generalisations for the concepts from locale theory, including the aforementioned notion of type of lattice, which takes the form of a choice of suitable category K of lattices. Chapter 3 culminates in the main result of this dissertation, Theorem 3.5.2, the extension of the Stone duality theorem to any choice of K . Chap- ter 4 covers the Stone representation theorem for Boolean algebras, which, with a little extra work, can be seen as a special case of Theorem 3.5.2. Chapter 5 covers the details of the application of Theorem 3.5.2 to measurable spaces. Chapter 1 Lattices Definition 1.0.1. Let L be a poset. Then L is said to be a lattice if, every finite set S ⊂ L has both an infimum ^S, called the meet of S, and a supremum _S, called the join of S. If x; y 2 L, we write x^y := ^fx; yg and x_y := _fx; yg. Hence one has associative commutative binary operations ^; _ : L × L ! L for any lattice L. A subset S ⊂ L of a lattice L is said to join in L if it has a supremum _S in L, and meet in L if it has an infimum ^S in L. One writes ^ _ ? = > and ? = ?: and observes that every lattice is bounded. Finite lattices can be given by Hasse diagrams, in which the elements are depicted as vertices with an edge running upwards from a node x to a node y if and only if y covers x, i.e., x ≤ y and x ≤ z ≤ y implies z = x or z = y. For instance, one has the lattices • • • • • • • • • • 2 = f0; 1g 23 3 4 CHAPTER 1. LATTICES where 2 is equipped with the unique order with 0 ≤ 1 and 23 has the order given by (a; b; c) ≤ (d; e; f) iff a ≤ d and b ≤ e and c ≤ f.
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