 
Summary: Sheaves for predicative toposes
Benno van den Berg
22 July, 2005
Abstract: In this paper, we identify some categorical structures in
which one can model predicative formal systems: in other words, pred
icative analogues of the notion of a topos, with the aim of using sheaf
models to interprete predicative formal systems. Among our techni
cal results, we prove that all the notions of a ``predicative topos'' that
we consider, are stable under presheaves, while most are stable under
sheaves.
1 Introduction
The importance for logic of toposes is due to the fact that they are categorical models
of a impredicative constructive theories, like higherorder type theoretic logics or
the set theory IZF. The theory of toposes provides a large stock of examples of such
models in the form of toposes of sheaves for a site, thereby also incorporating the
settheoretic method of forcing. Toposes of sheaves are an especially fruitful source
of examples, because the construction can be iterated: i.e., the notion of a sheaf for
a site can be formulated internally in a topos and these sheaves for an internal site
again form a topos. Using these topostheoretic models, one can derive consistency
and independence results and derived rules (good sources for logical applications of
