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Realizability Categories Realizability Categories W. P. Stekelenburg August 20, 2012 ii Introduction This thesis contains a collection of results of my Ph.D. research in the area of real- izability and category theory. My research was an exploration of the intersection of these areas focused on gaining a deeper understanding rather than on answering a specific question. This gave us some theorems that help to define what realizability is, or at least what realizability categories are. To provide some context, this chapter introduces realizability and category the- ory and makes a small survey of their intersection. In the end it summarizes our contributions. 0.1 Realizability Realizability is a collection of tools in the study of constructive logic, where it tackles questions about consistency and independence that are not easily answered by other means. We have no overview of this ever growing collection and know no general criterion for what can be considered realizability and what can not. Therefore, instead of giving a definition, we will present the historical starting point of realizability, and a selection of some later developments. In [35] Kleene introduces recursive realizability. It interprets arithmetical propo- sitions by assigning sets of numbers to them. Definition 0.1.1. Let N be the natural numbers. Let (m, n) 7→ hm, ni : N × N → N and n 7→ (n0, n1): N → N×N be a recursive bijection and let (m, n) 7→ mn : N×N * N be a universal partial recursive function, i.e., for each partial recursive f : N * N there is an e ∈ N such that for all n ∈ domf, en is defined and equal to f(n). We write mn↓ if (m, n) is in the domain of the universal partial recursive function. We define the realizability relation r. as follows. • n r. x = y if and only if x = y; • n r. p ∧ q if n0 r. p and n1 r. q; • n r. p ∨ q if n0 = 0 and n1 r. p, or n0 = 1 and n1 r. q; • n r. p → q if for all n0 r. p, nn0↓ and nn0 r. q; • n r. ¬p if no n0 r. p. • n r. ∀x.p(x) if for all n0 ∈ N, nn0↓ and nn0 r. p(n0); iii iv INTRODUCTION 0 0 0 • n r. ∃x.p(x) if some n ∈ N, n0 = n and n1 r. p(n ). A proposition p is valid if there is some n ∈ N such that n r. p. The realizers encode some justification for the validity of the formulas they realize. In particular, realizers of p → q are indices of partial recursive functions that send realizers of q to realizers of p. The resulting structure has the following features: • it is a model of Heyting arithmetic; • because every proposition p either has a realizer or doesn’t, p∨¬p and (¬¬p) → p are valid; • nonetheless, there is a predicate p such that ¬(∀n.p(n) ∨ ¬p(n)) is realized. We see the paradox that q(n) = p(n) ∨ ¬p(n) is valid for all n, while ∀x.q(x) can be false, thanks to an interpretation of universal quantification quite different from the one in classical model theory. Kleene proposed a number of variations on recursive realizability. • We can consider whether the existence of realizers is formally provable in Heyt- ing arithmetic or in other formal systems. • We can restrict the set of realized negations, implications, or universal quantifi- cations to a preselected set to avoid realizing false propositions like the unde- cidability of a set of numbers. This restriction allows a more faithful approach to intuitionistic logic. • Kleene developed function realizability, where functions f : N → N take the place of numbers. There is a universal partial continuous function NN×NN * NN for the product topology in NN, which takes the place of the universal partial recursive function. • A further variation on function realizability is that a formula is valid if there is a total recursive function that realizes it [37]. This idea of using a special set of realizers to determine validity is called relative realizability. Others proposed further extensions. • Besides N and NN other sets are suitable for building realizability interpre- tations, namely Feferman’s partial applicative structures (see [18]) and their generalizations. • Instead of a single set of realizers, one can work with a system of sets of realizers. The first example of this was Troelstra’s reformulation of Kreisel’s modified realizability [38], [66]. • Troelstra extended realizability beyond arithmetic, to higher order systems [67]. • Realizability can be combined with sheaf semantics by developing it in the internal language of a Grothendieck topos [68], [22], [44]. 0.2. EFFECTIVE TOPOS v An area of application of realizability is computer science, after all, computers are inherently recursive. Practical limitations of computers, in particular the amount of time and memory required to finish a computation, gave us realizability interpre- tations for languages that are different from first order languages and realizability counter-models for weaker formal systems than classical or intuitionistic first order logic, see [14]. On the other hand, the desire to extract computational information from proofs in classical mathematics has led Krivine to introduce a realizability in- terpretation for classical set theory, see [39]. 0.2 Effective topos We combine realizability with category theory. For an introduction to category theory, see [47]. Category theory started as a part of algebraic topology, as a language for describing the connections between algebraic invariants of topological spaces, see [17]. The theory proved useful in other areas of mathematics, in particular in other parts of algebra and geometry, but also in the more remote areas. Lawvere initiated the application of category theory to logic, see [41] and [42]. Several subjects from category theory, in particular from categorical logic, play a prominent role in this thesis: elementary toposes, regular, exact and Heyting cate- gories, fibred locales, complete fibred Heyting algebras and triposes. Toposes are categories that have finite limits and power objects: an object PX is a power object of X, if there is a monomorphism m : EX → X × PX such that for each monomorphism n : U → X × Y there is a unique f : Y → PX such that n is the pullback of m along g. U / EX y n m X × Y / X × PX X×f This definition of toposes comes from Lawvere and Tierney (see [40], [64] and [63]), although a more restricted notion of toposes appeared earlier in Grothendieck’s work. See [48], [33] or [32] for more information on topos theory. Toposes have an internal language [55]: a higher order intuitionistic logic. Heyting categories where defined in [58]. They also have an internal language, but this internal language is a many sorted predicate logic that does not always have higher order quantification. An early reference of regular and exact categories is [3]. First Mac Lane developed Abelian categories (see [49]) for algebraic topology. Subsequent authors looked at categories that omitted parts of the algebraic structure of Abelian categories, while retaining the non algebraic properties, until Barr settled on the regular and exact categories we use in this thesis. In [11] we find a construction of exact categories out of categories with finite limits – the ex/lex completion – and subsequently many similar constructs have been defined: [10], [12]. Menni worked out under which conditions these completion constructions result in toposes [52], [53]. Lawvere introduced hyperdoctrines in [43]. Both fibred locales and complete fibred Heyting algebras are – up to a 2-equivalence of 2-categories – examples of hyperdoc- vi INTRODUCTION trines and we could have called them regular and first order hyperdoctrines. We de- cided to work with the fibred categories instead of category valued (pseudo)functors, in order to make our work less dependent on set theory, therefore new names seemed appropriate. Grothendieck introduced fibred categories (see [23]) for algebraic geom- etry. B´enabou started applying them to logic [4], [5]. A tripos is a special type of complete fibred Heyting algebra. In [56] and [29], one can find a construction of toposes out of triposes. The tripos-to-topos construction was soon applied to realizability, resulting in Hyland’s effective topos, see [27]. We give a definition of this category here. Definition 0.2.1. The effective topos Eff is the category whose objects are pairs (X, E ⊆ N × X × X) for which s, t ∈ N exists such that for all (m, x, y) ∈ E and (n, z, x) ∈ E, sm and tmn are defined and (sm, y, x) and (tmn, z, y) ∈ E. Morphisms are defined as follows. For all X and all U, V ⊆ N × X we let U |=X V if there is an m ∈ N such that for all (n, x) ∈ U, mn↓ and (mn, x) ∈ V . Then U ⇐⇒ X V if 0 0 U |=X V and V |=X U. A morphism (X, E) → (X ,E ) is an ⇐⇒X -equivalence class 0 φ of F ⊆ N × X × X for which e, r, s0, s1, u ∈ N exists such that • for all (m, x, x) ∈ E, there is a z ∈ X0 such that em↓ and (em, x, z) ∈ F ; • for all (m, x, y) ∈ F ,(n, x, x0) ∈ E and (p, y, y0) ∈ E0, ((rm)n)p↓ and (((rm)n)p, x0, y0) ∈ F ; • for all (m, x, y) ∈ F , s0m and s1m are defined, (s0m, x, x) ∈ E and 0 (s1m, y, y) ∈ E ; • for all (m, x, y), (n, x, z) ∈ F ,(um)n↓ and ((um)n, y, z) ∈ E0. The composition of two morphisms φ :(X, E) → (X0,E0) and χ :(X00,E00) → 0 0 (X ,E ) is the ⇐⇒X -equivalence class χ ◦ φ that for all F ∈ φ and G ∈ χ contains 00 G ◦ F = {(hn, mi, x, y) ∈ N × X × X |∃z.(n, x, z) ∈ F, (m, z, y) ∈ G} Here, h−, −i is the pairing combinator from the definition at the beginning of this introduction.
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