Borel and analytic sets in locales Ruiyuan Chen Abstract We systematically develop analogs of basic concepts from classical descriptive set theory in the context of pointless topology. Our starting point is to take the elements of the free complete Boolean algebra generated by the frame O(X) of opens to be the \1-Borel sets" in a locale X. We show that several known results in locale theory may be interpreted in this framework as direct analogs of classical descriptive set-theoretic facts, including e.g., the Lusin separation, Lusin{Suslin, and Baire category theorems for locales; we also prove several extensions of these results, such as an ordered Novikov separation theorem. We give a detailed analysis of various notions of image, and prove that a continuous map need not have an 1-Borel image. We introduce the category of \analytic 1-Borel locales" as the regular completion under images of the category of locales and 1-Borel maps (as a unary site), and prove analogs of several classical results about analytic sets, such as a boundedness theorem for well-founded analytic relations. We also consider the \positive 1-Borel sets" of a locale, formed from opens without using :. In fact, we work throughout in the more refined context of κ-copresented κ-locales and κ-Borel sets for arbitrary regular !1 ≤ κ ≤ 1; taking κ = !1 then recovers the classical context as a special case. The basis for the aforementioned localic results is a detailed study of various known and new methods for presenting κ-frames, κ-Boolean algebras, and related algebraic structures. In particular, we introduce a new type of \two-sided posite" for presenting (κ, κ)-frames A, meaning both A and Aop are κ-frames, and use this to prove a general κ-ary interpolation theorem for (κ, κ)-frames, which dualizes to the aforementioned separation theorems. Contents 1 Introduction 2 1.1 Overview of localic results . .4 1.2 Overview of algebraic results . .8 1.3 Future directions . 11 1.4 Structure of paper . 12 arXiv:2011.00437v1 [math.LO] 1 Nov 2020 2 Frames and Boolean algebras 13 2.1 The main categories . 13 2.2 General remarks on presentations and adjunctions . 16 2.3 Ideals . 18 2.4 Quotients . 20 2.5 Products and covers . 23 2.6 Posites . 27 2.7 Colimits of frames . 30 2.8 Adjoining complements . 32 2.9 Zero-dimensionality and ultraparacompactness . 36 1 2.10 Free complete Boolean algebras . 45 2.11 Distributive polyposets . 48 2.12 Interpolation . 52 3 Locales and Borel locales 58 3.1 The main categories . 58 3.2 Spatialization . 63 3.3 The Borel hierarchy . 66 3.4 Borel images and sublocales . 69 3.5 Standard σ-locales . 79 3.6 The internal logic . 83 3.7 Positive Borel locales and partial orders . 86 3.8 Baire category . 89 4 Analytic sets and locales 91 4.1 The main categories . 92 4.2 Separation theorems . 97 4.3 Inverse limit representations . 99 4.4 Ill-founded relations . 104 A Appendix: nice categories of structures 107 A.1 Limit theories and locally presentable categories . 107 A.2 Algebraic theories and monadic categories . 111 A.3 Ordered theories and categories . 114 References 117 1 Introduction In this paper, we study the connection between descriptive set theory and locale theory, two areas which can each be thought of as providing a \non-pathological" version of point-set topology. Descriptive set theory can be broadly described as the study of simply-definable sets and functions. The standard framework is to begin with a well-behaved topological space, namely a Polish space, i.e., a separable, completely metrizable space. One then declares open sets to be “simply-definable”, and builds up more complicated sets by closing under simple set-theoretic operations, e.g., countable Boolean operations (yielding the Borel sets), continuous images (yielding the analytic sets), etc. The definability restriction rules out many general point-set pathologies, yielding powerful structural and classification results which nonetheless apply to typical topological contexts encountered in mathematical practice. This has led in recent decades to an explosion of fruitful connections and applications to many diverse branches of mathematics, such as dynamical systems, operator algebras, combinatorics, and model theory. See [Kec95], [Kec99], [Mos09], [Gao09], [KM20] for general background on descriptive set theory and its applications. Locale theory, also known as pointless or point-free topology, is the \dual" algebraic study of topological spaces and generalizations thereof via their lattices of open sets; see [Joh82], [PP12]. A frame is a complete lattice with finite meets distributing over arbitrary joins; the motivating example is the frame O(X) of open sets of a topological space X. In locale theory, one formally 2 regards an arbitrary frame O(X) as the \open sets" of a generalized \space" X, called a locale. One then proceeds to study algebraically-defined analogs of various topological notions. A key insight due to Isbell [Isb72] is that such analogies are not perfect, but that this is a feature, not a bug: where it differs from point-set topology, locale theory tends to be less pathological. The starting point for this paper is an observation \explaining" this feature of locale theory: it seems to behave like a formal generalization of descriptive set theory, with countability restrictions removed. This observation is supported by many results scattered throughout the literature, e.g.: • A fundamental construction of Isbell [Isb72] is to freely adjoin complements to an arbitrary frame O(X), yielding a bigger frame N (O(X)), whose elements are in canonical bijection with 0 the sublocales of X. This is analogous to the Σ2 (i.e., Fσ) sets in a Polish space X, the closure of the open and closed sets under finite intersections and countable unions; the complements 0 0 of Σ2 sets, the Π2 (i.e., Gδ) sets, are precisely the Polish subspaces of X [Kec95, 3.11]. • Isbell [Isb72] also showed that dense sublocales of an arbitrary locale are closed under arbitrary intersections, just as dense Polish subspaces of Polish spaces are closed under countable intersections, by the Baire category theorem [Kec95, 8.4]. • Transfinitely iterating the N (−) construction yields a \Borel hierarchy" for an arbitrary locale X, the union of which is the free complete Boolean algebra generated by the frame O(X). By a classical result of Gaifman [Gai64] and Hales [Hal64], free complete Boolean algebras are generally proper classes, whence the \Borel hierarchy of locales is proper". • Ball [Bal18] developed a \Baire hierarchy of real-valued functions" on an arbitrary locale, via a transfinite construction closely related to the aforementioned N functor. • Isbell [Isb91] proved an analog of the classical Hausdorff–Kuratowski theorem [Kec95, 22.27]: 0 any complemented sublocale (i.e., \∆2 set") can be written as (((F0 n F1) [ F2) n F3) [··· for some transfinite decreasing sequence F0 ⊇ F1 ⊇ · · · of closed sublocales. • The closed subgroup theorem of Isbell{Kˇr´ıˇz{Pultr{Rosick´y[IKPR88] states that every closed localic subgroup of a localic group is closed, just as every Polish subgroup of a Polish group is closed [Gao09, 2.2.1]. Moreover, the proof of the closed subgroup theorem by Johnstone [Joh89] uses a localic version of Pettis's theorem [Gao09, 2.3.2], itself proved via an analog of the classical proof based on (what we are calling) localic Baire category. • The dual formulation of measure theory is well-known, via measure algebras. Explicit connections with general locales have been made by Simpson [Sim12] and Pavlov [Pav20]. • De Brecht [deB13] introduced quasi-Polish spaces, a well-behaved non-Hausdorff gener- alization of Polish spaces sharing many of the same descriptive set-theoretic properties. Heckmann [Hec15] proved that the category of these is equivalent to the category of locales X whose corresponding frame O(X) is countably presented. Thus, \locale = (Polish space) − (countability restrictions) − (separation axioms)". • In [Ch19a], we explicitly used this correspondence to transfer a known result from locale theory (the Joyal{Tierney representation theorem [JT84] for Grothendieck toposes via localic groupoids) to the classical descriptive set-theoretic context (a Makkai-type strong conceptual completeness theorem for the infinitary logic L!1! in terms of Borel groupoids of models). 3 • In [Ch19b], we gave an intrinsic categorical characterization of the classical category SBor of standard Borel spaces (Polish spaces with their topologies forgotten, remembering only the Borel sets). This characterization was proved for \standard κ-Borel locales" for all regular cardinals κ ≥ !, using algebraic techniques; taking κ = !1 then yielded the classical result. In spite of these and other results, the known analogy between locale theory and descriptive set theory has been on a largely ad hoc basis to date, without a complete or coherent locale-theoretic account of even basic descriptive set-theoretic notions such as Borel sets. The goal of this paper is to provide the beginnings of such an account. 1.1 Overview of localic results Our main focus of study is on the two categories κLoc of κ-locales, and κBorLoc of κ-Borel locales, for each infinite regular cardinal κ (or κ = 1). These are defined as the formal duals of the respective algebraic categories κFrm of κ-frames, meaning frames but required only to have κ-ary joins, and κBool of κ-(complete )Boolean algebras. Thus, a κ-locale X is, formally, the same thing as a κ-frame Oκ(X), whose elements we call the κ-open sets in X; while a κ-Borel 1 locale X is formally a κ-Boolean algebra Bκ(X), whose elements we call the κ-Borel sets in X. A morphism f : X ! Y in any of these categories is a homomorphism f ∗ between the respective algebras, in the opposite direction, thought of as taking preimage of sets under f.