Presheaf Models for Constructive Set Theories

We introduce a new kind of models for constructive set theories based on categories of presheaves. These models are a counterpart of the presheaf models for intuitionistic set theories defined by Dana Scott in the ’80s. We also show how presheaf models fit into the framework of Algebraic Set Theory and sketch an application to an independence result. 1. Variable sets in foundations and practice Presheaves are of central importance both for the foundations and the practice of mathematics. The notion of a presheaf formalizes well the idea of a variable set, that is relevant in all the areas of mathematics concerned with the study of indexed families of objects [19]. One may then readily see how presheaves are of interest also in foundations: both Cohen’s forcing models for classical set theories and Kripke models for intuitionistic logic involve the idea of sets indexed by stages. Constructive aspects start to emerge when one considers the internal logic of categories of presheaves. This logic, which does not include classical principles such as the law of the excluded middle, provides a useful language to manipulate objects and arrows, and can be used as an alternative to diagrammatic reasoning [25]. Furthermore, it is sufficiently expressive to allow the definitions of complex mathematical constructions. This aspect has led to important developments in the study of elementary toposes [16]. The main purpose of this paper is to show how presheaves can be used to obtain models for constructive set theories [23, 5] analogous to the ones defined by Dana Scott for intuitionistic set theories [26]. In order to do so, we will have to overcome the challenges intrinsic to working with generalised predicative formal systems. By a generalised predicative formal system we mean here a system that is proof-theoretically reducible to Martin-Löf dependent type theories with W -types and universes [20, 12]. Generalised predicative systems typically contain axioms allowing generalized forms of inductive definitions [1] instead of proof-theoretically strong axioms such as Power Set. Our development will focus on categories of classes rather than categories of sets as the starting point to define presheaves, thus assuming the perspective of Algebraic Set Theory [15, 27, 7, 22, 6]. The main reason for this choice is that the properties of categories of sets do not always reflect directly the set-theoretical axioms adopted to define them. There are indeed axioms, such as Replacement, that do not express directly properties of sets, but regard the interaction between sets and classes. In categories of classes we can overcome this problem without loss, since sets can be isolated as special objects, those that are in some sense ‘small’. Date: December 10th, 2004. Research supported by an EPSRC Postdoctoral Fellowship in Mathematics (GR/R95975/01).