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Aspects of Predicative Algebraic Set Theory III: Benno van den Berg & Ieke Moerdijk

Summary: Aspects of Predicative Algebraic Set Theory III:
Benno van den Berg & Ieke Moerdijk
7 Dec, 2009
1 Introduction
This is the third in a series of papers on algebraic set theory, the aim of which is
to develop a categorical semantics for constructive set theories, including pred-
icative ones, based on the notion of a "predicative category with small maps".1
In the first paper in this series [8] we discussed how these predicative categories
with small maps provide a sound and complete semantics for constructive set
theory. In the second one [10], we explained how realizability extensions of such
predicative categories with small maps can be constructed. The purpose of the
present paper is to do the same for sheaf-theoretic extensions. This program
was summarised in [11], where we announced the results that we will present
and prove here.
For the convenience of the reader, and also to allow a comparison with the
work by other researchers, we outline the main features of our approach. As
said, the central concept in our theory is that of a predicative category with
small maps. It axiomatises the idea of a category whose objects are classes and
whose morphisms are functions between classes, and which is moreover equipped


Source: Andrews, Peter B. - Department of Mathematical Sciences, Carnegie Mellon University


Collections: Mathematics