Aspects of predicative algebraic set theory I: Exact completion

This is the first in a series of papers on Predicative Algebraic Set Theory, where we lay the necessary groundwork for the subsequent parts, one on realizability [B. van den Berg, I. Moerdijk, Aspects of predicative algebraic set theory II: Realizability, Theoret. Comput. Sci. (in press). Available from: arXiv:0801.2305, 2008], and the other on sheaves [B. van den Berg, I. Moerdijk, Aspects of predicative algebraic set theory III: Sheaf models, 2008 (in preparation)]. We introduce the notion of a predicative category with small maps and show that it provides a sound and complete semantics for constructive set theories like IZF and CZF. The main technical contribution of this paper is that it shows in detail that such categories can always be conservatively embedded in categories that are exact. These exactness properties play a crucial rôle in showing that predicative categories with small maps contain models of set theory and that they are closed under sheaves and realizability. We will prove the former statement in this paper as well, leaving a proof of the closure properties to the papers on realizability and sheaves as mentioned above.