The standard presentation of topological spaces relies heavily on (naïve) set theory: a topology consists of a set of subsets of a set (of points). And many of the high-level tools of set theory are required to achieve just the basic results about topological spaces. Concentrating on the mathematical structures, category theory offers the possibility to look synthetically at the structure of continuous transformations between topological spaces addressing specifically how the fundamental notions of point and open come about. As a (...) byproduct of this, one may look at the different approaches to topology from an external perspective and compare them in a unified way. Technically, the category of sober topological spaces can be seen as consisting of (co)algebraic structures in the exact completion of the elementary category of sets and relations. Moreover, the same abstract construction of taking the exact completion, when applied to the category of topological spaces and continuous functions produces an extension of it which is cartesian closed. In other words, there is one general mathematical construction that, when applied to a very elementary category, generates the category of topological spaces and continuous functions, and when applied to that category produces a very suitable category where to deal with all sorts functions spaces. Yet, via such free constructions it is possible to give a new meaning to Marshall Stone’s dictum: “always topologize” as the category of sets and relations is the most natural way to give structure to logic and the category of topological spaces and continuous functions is obtained from it by a good mix of free – i.e. syntactic – constructions. (shrink)

Kleene's realizability interpretation for first-order arithmetic was shown by Hyland to fit into the internal logic of an elementary topos, the “Effective topos” . In this paper it is shown, that there is an internal realizability definition in , i.e. a syntactical translation of the internal language of into itself of form “n realizes ” , which extends Kleene's definition, and such that for sentences , the equivalence [harr]n is true in . The internal realizability definition depends on finding separated (...) covers for non-separated objects of . However, for the objects arising in higher-order arithmetic, canonical covers are available, which are definable in higher-order arithmetic. These canonical covers yield “covering axioms”. It is shown that these covering axioms, together with uniformity principles and Extended Church's Thesis, axiomatize a formalized extension of Kleene realizability to a constructive system of higher-order arithmetic. The details are worked out for second and third-order arithmetic, but the method can be extended to any order. As an application, it is shown in the final section that a certain completeness property of the category of “modest sets” can be derived in third-order arithmetic from the realizability axioms. (shrink)

We present a general notion of realizability encompassing both standard Kleene style realizability over partial combinatory algebras and Kleene style realizability over more general structures, including all partial cartesian closed categories. We shown how the general notion of realizability can be used to get models of dependent predicate logic, thus obtaining as a corollary (the known result) that the category Equ of equilogical spaces models dependent predicate logic. Moreover, we characterize when the general notion of realizability gives rise to a (...) topos, i.e., a model of impredicative intuitionistic higher-order logic. (shrink)

In the present paper we use the theory of exact completions to study categorical properties of small setoids in Martin-Löf type theory and, more generally, of models of the Constructive Elementary Theory of the Category of Sets, in terms of properties of their subcategories of choice objects. Because of these intended applications, we deal with categories that lack equalisers and just have weak ones, but whose objects can be regarded as collections of global elements. In this context, we study the (...) internal logic of the categories involved, and employ this analysis to give a sufficient condition for the local cartesian closure of an exact completion. Finally, we apply this result to show when an exact completion produces a model of CETCS. (shrink)

Two straightforward “extensionalisations” of Kleene's realizability are considered; denoted re and e. It is shown that these realizabilities are not equivalent. While the re-notion is a subset of Kleene's realizability, the e-notion is not. The problem of an axiomatization of e-realizability is attacked and one arrives at an axiomatization over a conservative extension of arithmetic, in a language with variables for finite sets. A derived rule for arithmetic is obtained by the use of a q-variant of e-realizability; this rule subsumes (...) the well-known Extended Church's Rule. The second part of the paper focuses on toposes for these realizabilities. By a relaxation of the notion of partial combinatory algebra, a new class of realizability toposes emerges. Relationships between the various realizability toposes are given, and results analogous to Robinson and Rosolini's characterization of the effective topos, are obtained for a topos generalizing e-realizability. (shrink)

5.5. Every topos is linguistic: the equivalence theorem.

We present a topos-theoretic interpretation of (a categorical generalization of) Fraïssé’s construction in Model Theory, with applications to homogeneous models and countably categorical theories.