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On Sheaves and Duality On sheaves and duality PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Radboud Universiteit Nijmegen op gezag van de rector magnificus prof. mr. S. C. J. J. Kortmann, volgens besluit van het college van decanen in het openbaar te verdedigen op dinsdag 8 april 2014 om 12.30 uur precies door Samuel Jacob van Gool geboren op 1 november 1987 te Amsterdam Promotor: prof. dr. M. Gehrke Manuscriptcommissie: prof. dr. I. Moerdijk prof. dr. A. Jung University of Birmingham dr. V. Marra Universita` degli Studi di Milano dr. P.-A. Mellies` Universite´ Paris Diderot - Paris 7 prof. dr. Y. Venema Universiteit van Amsterdam On sheaves and duality S. J. van Gool c 2014 by S. J. van Gool. All rights reserved. ISBN: 978-1-291-75985-3 Contents Introduction 7 Two approaches to logic ........................ 7 Overview of the thesis ......................... 12 Relevant publications .......................... 13 1 Stone duality and completions 15 1.1 Stone duality and its successors ................. 15 1.2 Order completions: duality in algebraic form ......... 20 2 Duality and canonical extensions for stably compact spaces 25 2.1 Compact ordered and stably compact spaces ......... 26 2.2 Continuous retracts and the Karoubi envelope ........ 30 2.3 Proximity lattices ......................... 33 2.4 Canonical extensions of proximity lattices ........... 41 2.5 Duality for stably compact spaces ................ 50 3 Duality for sheaves of distributive-lattice-ordered algebras 57 3.1 Preliminaries on sheaves and topology ............. 58 3.2 Decompositions from sheaves .................. 60 3.3 Sheaves associated to decompositions ............. 65 4 Sheaf representations of MV-algebras and `-groups 73 4.1 MV-algebras and their spectra .................. 76 4.2 Lifting operations and inequalities to the canonical extension 81 4.3 The structure of the lattice spectrum of an MV-algebra .... 84 4.4 The decomposition of the lattice spectrum: the map k .... 88 4.5 Kaplansky’s Theorem for MV-algebras: the map m ...... 91 4.6 Sheaf representations from k and m ............... 95 5 A non-commutative Priestley duality 103 5.1 Strongly distributive left-handed skew lattices ........ 105 5.2 Sheaves over local Priestley spaces ............... 108 5.3 The functor from spaces to algebras .............. 110 5.4 Reconstructing a space from its dual algebra ......... 112 5.5 Proof of the duality theorem ................... 118 5 6 Contents 6 Distributive envelopes and topological duality for lattices 123 6.1 Distributive envelopes ...................... 124 6.2 Topological duality ........................ 135 6.3 Quasi-uniform spaces associated with a lattice ........ 149 Bibliography 155 Notation 167 Index 171 Abstract 175 Samenvatting 177 Acknowledgements 179 About the author 182 Introduction This thesis is concerned with sheaves and duality, using order completions as an essential tool. The interplay between duality, sheaves and order com- pletions will be a recurring theme in this thesis. In this introduction, we discuss our motivation for studying these topics, and give an overview of the chapters to come. Two approaches to logic Logic provided the main motivation and inspiration for many of the top- ics that are addressed in this thesis. In the broadest possible terms, logic studies the structure of arguments. To do so, several kinds of mathematical methods can be employed, which we will broadly divide in algebraic and spatial approaches to logic. An algebraic approach to logic starts from the inherent syntactic structure of logical arguments. Algebra can be a useful tool to abstractly study such syntactic structure. For example, just as commutative rings are an abstrac- tion of the integers with the operations of addition, multiplication, and sub- traction, Boolean algebras are an abstraction of propositional sentences with the operations of classical disjunction (‘or’), conjunction (‘and’) and nega- tion (‘not’). Algebra has a great generalizing power; for example, if, in- stead of integers, one wants to study matrices with integer entries, then one can still use the framework of rings, dropping the axiom of commutativ- ity. Similarly, if one is interested in logics with quantifiers, modalities, or non-classical operations, then one can consider generalizations of Boolean algebras. In this approach to logic, the relation of logical derivability is formalized as a partial order in the algebra, and logical equivalence thus corresponds to equality. On the other hand, a spatial approach to logic starts from a collection, or space, of ‘models’ that can interpret logical formulas. The ‘meaning’ of a logical formula is then defined using this space. For example, in classical propositional logic, a point in the space is an assignment of truth values to propositional variables. Any propositional formula then defines the subset of points (models) where that formula is true. In this spatial approach, two logical formulas are considered equivalent if they are true in exactly the same models. 7 8 Introduction Duality in logic: an example Duality provides a link between the algebraic and spatial approaches to logic. To illustrate the use of duality in logic, we now briefly sketch a proof of the completeness theorem for first-order logic. In the proof of that theo- rem, the crux of the argument is to show that, if a first-order sentence j is not syntactically derivable from a set of first-order sentences G, then there exists a first-order model in which all sentences in G are true, but j is not. Duality theory for Boolean algebras, as developed in the 1930’s by M. H. Stone, says that any Boolean algebra can be represented as the collection of clopen subsets of some compact Hausdorff topological space. In particular, we can apply this fact to the collection of first-order sentences, considered up to provable equivalence. To any point x of the corresponding topolog- ical space, one may associate the set of first-order sentences which (under Stone’s representation) contain the point x. The sets of sentences obtained in this manner are known as complete consistent theories, or ultrafilters. Any such ultrafilter can be used to canonically define a first-order model, Mx. This model Mx will be especially useful if the first-order sentences that are true in Mx are exactly the sentences which contain the point x. Call a point x special if its associated model has this useful property. Since Stone’s space is compact and Hausdorff, one may use Baire category theorem from general topology to prove that the special points are dense in any closed subspace. Now, if j is a first-order sentence that is not syntactically deriv- able from a set of sentences G, then the intersection of the sentences in G is closed and not contained in j; by denseness, there is a special point x in this closed set which is not in j. The associated model Mx is the first-order model that we needed to construct.1 The proof outlined in the previous paragraph follows a general scheme, which may be described as follows. A problem from logic (finding a coun- termodel for a non-derivable sentence) is solved by translating it into a topological question (does there exist a ‘special point’?), which can be an- swered using the well-developed theory of general topology (Baire cate- gory theorem). Duality is the general mechanism which enables one to translate back and forth between algebraic and spatial approaches to logic. Mathematically speaking, a duality is a contravariant categorical equiva- lence between a category of algebras and a category of spaces. This means that any algebra has a space associated to it, and vice versa, in such a way that maps between algebras correspond to maps between the associated spaces, in the reverse direction. We will give more precise mathematical 1The completeness of first-order logic was first proved by Godel¨ [69]. The proof that we outlined in this paragraph was first given by H. Rasiowa and R. Sikorski in [129]; also see the classical reference [130]. 9 background on Stone duality in Section 1.1 of this thesis. In the above example, the collection of models Mx associated to the (spe- cial) points x in the space played a crucial role. This collection of models Mx can be made in such a way that it ‘varies continuously’ in the variable x. Sheaves, the other focal point of this thesis, are designed precisely to deal with structures that vary continuously over a topological space. These observations form the basis of the sheaf-theoretic, categorical approach to first-order logic. Using order completions Stone’s original duality theory was concerned with Boolean algebras, which model the calculus of formulas in classical propositional logic. In the above example, Stone duality was used to prove a basic theorem in classical first- order logic. Many other kinds of logic, including modal, intuitionistic, sub- structural, and multi-valued logic, can be studied using generalizations of Stone duality. Priestley duality is one such generalization: it deals with dis- tributive lattices, i.e., the algebraic structures corresponding the negation- free fragment of propositional logic, cf. Figure 1 for an example. Often, one may want to enrich the algebraic theory of distributive lattices with operations, such as implication. This can lead, for example, to a defini- tion of Heyting algebras, the algebraic structures for Brouwer’s intuitionis- tic logic. If one imposes slightly different axioms for implication, one may arrive at algebraic structures for other logics. For example, the natural al- gebraic structures for infinite-valued Łukaciewicz logic, MV-algebras, are distributive lattices enriched with operations of addition and subtraction. Chapter 4 of this thesis is devoted to sheaf representations and duality the- ory for MV-algebras. For a slightly different, but related, example, algebraic structures for modal logic are obtained by adding so-called modality oper- ators to distributive lattices or Boolean algebras.
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