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First-Order Logical Duality

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First-Order Logical Duality First-Order Logical Duality Henrik Forssell A dissertation submitted in partial ful¯llment of the requirements for the degree of PhD in Logic, Computation and Methodology Carnegie Mellon University 2008 Abstract Generalizing Stone duality for Boolean algebras, an adjunction between Bool- ean coherent categories|representing ¯rst-order syntax|and certain topo- logical groupoids|representing semantics|is constructed. The embedding of a Boolean algebra into a frame of open sets of a space of 2-valued mod- els is replaced by an embedding of a Boolean coherent category, B, into a topos of equivariant sheaves on a topological groupoid of set-valued models and isomorphisms between them. The latter is a groupoid representation of the topos of coherent sheaves on B, analogously to how the Stone space of a Boolean algebra is a spatial representation of the ideal completion of the algebra, and the category B can then be recovered from its semantical groupoid, up to pretopos completion. By equipping the groupoid of sets and bijections with a particular topology, one obtains a particular topological groupoid which plays a role analogous to that of the discrete space 2, in be- ing the dual of the object classi¯er and the object one `homs into' to recover a Boolean coherent category from its semantical groupoid. Both parts of the adjunction, then, consist of `homming into sets', similarly to how both parts of the equivalence between Boolean algebras and Stone spaces consist of `homming into 2'. By slicing over this groupoid (modi¯ed to display an alternative setup), Chapter 3 shows how the adjunction specializes to the case of ¯rst-order single sorted logic to yield an adjunction between such theories and an inde- pendently characterized slice category of topological groupoids such that the counit component at a theory is an isomorphism. Acknowledgements I would like, ¯rst and foremost, to thank my supervisor Steve Awodey. I would like to thank the members of the committee: Jeremy Avigad, Lars Birkedal, James Cummings, and Dana Scott. I would like to thank the faculty and sta® of the department of philosophy at Carnegie Mellon University. I would like to thank Michael Warren, Ko- hei Kishida, and my fellow students at the departments of philosophy and mathematics at Carnegie Mellon University. I would like to thank Elise for her patience and support. ii Contents 1 Introduction 1 1.1 Algebra, Geometry, and Logic . 1 1.2 Logical Dualities . 4 1.2.1 Propositional Logic and Stone Duality . 4 1.2.2 Lawvere's Duality for Equational Logic . 11 1.2.3 Makkai's First-Order Logical Duality . 15 1.3 A Sheaf-Theoretical Approach . 17 1.4 Representing Topoi by Groupoids . 23 2 The Syntax-Semantics Adjunction 25 2.1 Overview . 25 2.2 The Semantical Groupoid of a Decidable Coherent Category . 30 2.2.1 Spaces of Models and Isomorphisms . 31 2.2.2 Stricti¯cation: Equipping Coherent Categories with Canonical Coherent Structure . 33 2.3 Representing Decidable Coherent Topoi by Groupoids of Models 38 2.3.1 Cover-Reflecting, Set-Valued Functors . 39 2.3.2 The Semantical Groupoid of a Theory T . 41 2.3.3 Topological Groupoids and Equivariant Sheaves . 46 2.3.4 Equivariant Sheaves on the Semantical Groupoid . 50 2.4 Syntax-Semantics Adjunction . 58 2.4.1 The Representation Theorem . 58 2.4.2 The Semantical Functor . 61 2.4.3 The Syntactical Functor . 62 2.4.4 The Syntax-Semantics Adjunction . 74 2.4.5 Restricting the Adjunction . 79 2.5 The Boolean Object Classi¯er . 87 iii 3 First-Order Logical Duality 94 3.1 Sheaves on the Space of Models . 95 3.1.1 The Space of Models . 95 3.1.2 An Open Surjection . 101 3.2 Equivariant Sheaves on the Groupoid of Models . 110 3.2.1 The Groupoid of Models . 110 3.2.2 Equivariant Sheaves on the Groupoid of Models . 113 3.3 Syntax-Semantics Adjunction . 119 3.3.1 The Category of FOL . 119 3.3.2 The Object Classi¯er . 121 3.3.3 The Semantical Groupoid Functor . 125 3.3.4 The Theory Functor . 128 3.4 Stone Fibrations . 135 3.4.1 Sites for Groupoids . 135 3.4.2 Stone Fibrations over N» . 141 3.5 Groupoids and Theories . 143 3.5.1 Applications and Future Work . 143 3.5.2 Groupoids and Theory Extensions . 145 A An Argument by Descent 153 A.1 An Argument by Descent . 153 A.1.1 The Theory of Countable T-Models . 154 A.1.2 The Theory of an Isomorphism between Two Count- able T-models . 158 A.1.3 Pullback of the Open Surjection against Itself . 162 A.1.4 The Theory of Two Composable Isomorphisms and the Triple Pullback of the Open Surjection against Itself . 166 A.1.5 The Category of G-Equivariant Sheaves on X . 169 B Table of Categories 172 B.1 A Table of Categories . 172 iv Chapter 1 Introduction 1.1 Algebra, Geometry, and Logic In this thesis, I present an extension of Stone Duality for Boolean Algebras from classical propositional logic to classical ¯rst-order logic. The leading idea is, in broad strokes, to take the traditional logical distinction between syntax and semantics and analyze it in terms of the classical mathematical distinction between algebra and geometry, with syntax corresponding to al- gebra and semantics to geometry. Insights from category theory allows us glean a certain duality between the two notions of algebra and geometry. We see a ¯rst glimpse of this in Stone's duality theorem for Boolean algebras, the categorical formulation of which states that a category of `algebraic' ob- jects (Boolean algebras) is the categorical dual of a category of `geometrical' objects (Stone spaces). \Categorically dual" means that the one category is opposite to the other, in that it can be obtained from the other by for- mally reversing the morphisms. In a more far reaching manner, this form of algebra-geometry duality is exhibited in modern algebraic geometry as re- formulated in the language of schemes in the Grothendieck school. E.g. in the duality between the categories of commutative rings and the category of a±ne schemes. On the other hand, we are informed by the area of category theory known as categorical logic that algebra is closely connected with logic, in the sense that logical theories can be seen as categories and suitable categories can be seen as logical theories. For instance, Boolean algebras correspond to classical propositional theories, equational theories correspond to categories 1 with ¯nite products, Boolean coherent categories correspond to classical ¯rst- order logic, and topoi|e.g. of sheaves on a space|correspond to higher-order intuitionistic logic. Thus the study of these algebraic objects has logical interpretation and, vice versa, reasoning in or about logical theories has application in their corresponding algebraic objects. With the connection between algebra and logic in hand, instances of the algebra-geometry duality can be seen to manifest a syntax-semantics duality between an algebra of syntax and a geometry of semantics. Stone duality, in its logical interpretation, manifests a syntax-semantics duality for propositional logic. As the category of Boolean algebras can be considered as the category of propositional theories modulo `algebraic' equivalence, the category of Stone spaces can be seen as the category of spaces of corresponding two-valued models. We obtain the set of models corresponding to a Boolean algebra by taking morphisms in the category of Boolean algebras from the given algebra into the two-element Boolean algebra, 2, » ModB = HomBA (B; 2) (1.1) And with suitable topologies in place, we can retrieve the Boolean algebra by taking morphisms in the category of Stone spaces from that space into the two-element Stone space, 2, » B = HomStone (ModB; 2) Here, the two-element set, 2, is in a sense living a `dual' life, and `hom- ming into 2' forms an adjunction between (the opposite of) the `syntactical' category of Boolean algebras and the category of topological spaces, which becomes an equivalence once we restrict to the `semantical' subcategory of Stone spaces. For equational (or algebraic) theories, i.e. those formulated in languages without relation symbols and with all axioms equations, an example of syntax-semantics duality occurred already in F.W. Lawvere's thesis ([16]). Such a theory A can be considered, up to `algebraic' or categorical equiva- lence, as a particular category CA with ¯nite products, and the category of set-valued models of the theory A is then the category of ¯nite product pre- / serving functors CA Sets from CA into the category of sets and functions, ModA ' HomFP (CA; Sets) (1.2) 2 Then CA can be recovered from the category ModA of models as the category / HomG(ModA; Sets) of those functors F : ModA Sets that preserve all limits, ¯ltered colimits, and regular epimorphisms, CA ' HomG(ModA; Sets) We think of those as the `continuous' maps in this context. In a wider per- spective, `homming into the dual object Sets` creates an adjunction between (the opposite of) the category of categories with ¯nite products|in which the theories live|and category of cocomplete categories|in which the cor- responding model categories live. An adjunction which can be restricted to an equivalence between the `syntactical' and `semantical' subcategories of theories and models. Full ¯rst-order theories, too, form a category when considered up to algebraic equivalence, namely the category of Boolean coherent categories and coherent functors between them. This category contains the category of sets and functions, Sets, homming into which produces the usual set- valued models that often are of particular interest for ¯rst-order theories.
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