 
Summary: Aspects of Predicative Algebraic Set Theory III:
Sheaves
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 sheaftheoretic 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
