Sheaves for predicative toposes

Heyting-valued interpretations for Constructive Set Theory

The theory of locales [23] has a twofold interplay with intuitionistic mathematics: first of all, the internal logic of toposes and intuitionistic set theories provide suitable settings for the development of the theory of locales [24], and secondly, the notion of a locale determines two important forms of toposes and of interpretations for intuitionistic set theories, namely localic toposes [2...

Aspects of predicative algebraic set theory I: Exact completion

Sheaf theory and realisability have been effective methods for constructing models of various constructive and intuitionistic type theories [22, 29, 15]. In particular, toposes constructed using sheaves or realisability provide models for intuitionistic higher order logic (HAH), and it was shown by Freyd, Fourman, Friedman respectively by McCarthy in the 1980s that from these toposes one can co...

Exact completions and toposes

Toposes and quasi-toposes have been shown to be useful in mathematics, logic and computer science. Because of this, it is important to understand the different ways in which they can be constructed. Realizability toposes and presheaf toposes are two important classes of toposes. All of the former and many of the latter arise by adding “good” quotients of equivalence relations to a simple catego...

Type theories, toposes and constructive set theory: predicative aspects of AST

Intuitionistic First-Order Logic: Categorical Semantics via the Curry-Howard Isomorphism

In this technical report, the achievements obtained by the author during his sabbatical period in the Department of Pure Mathematics, University of Leeds, UK, under the Marie Curie Intra-European Fellowship ‘Predicative Theories and Grothendieck Toposes’ are documented. Synthetically, this reports introduces a novel sound and complete semantics for firstorder intuitionistic logic, in the framew...