 
Summary: Algebraic models of sets and classes in categories
of ideals
Steve Awodey and Henrik Forssell and Michael A. Warren
May 31, 2006
Revised: November 2006
Abstract
We introduce a new sheaftheoretic construction called the ideal comple
tion of a category and investigate its logical properties. We show that
it satisfies the axioms for a category of classes in the sense of Joyal and
Moerdijk [17], so that the tools of algebraic set theory can be applied
to produce models of various elementary set theories. These results are
then used to prove the conservativity of different set theories over various
classical and constructive type theories.
1 Introduction
It is well known that various type theories may be modelled in certain kinds of
categories (cf. [15]). For instance, cartesian closed categories are models of the
typed lambda calculus and toposes are models of intuitionistic higher order logic
(IHOL). Similarly, Joyal and Moerdijk [17] showed that one can axiomatize a
notion of small map in a category in such a way that the resulting category will
contain an algebraic model of elementary set theory. In this paper we employ
