 
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 di#erent 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
