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Sheaves for predicative toposes Benno van den Berg

Summary: Sheaves for predicative toposes
Benno van den Berg
12 July, 2005
Abstract: In this paper, we discuss the stability of predicative toposes
under taking sheaves for an internal site. What is the most useful no-
tion of a predicative topos has not been established yet, but we discuss
sheafification for some candidates for that title. The results obtained
here sharpen the results published by Moerdijk and Palmgren in [10]
and [11]. We also propose a new axiom and show its usefulness for the
issue at hand.
1 Introduction
The usefulness of toposes for the study of metamathematical properties of intu-
itionistic formal systems is well established. The crucial properties of toposes for
deriving various metamathematical results, like consistency and independence re-
sults and derived rules, are closure under certain 2-categorical limits, sheaves and
When it comes to the metamathematics over predicative intuitionistic systems,
however, this approach has clear limitations. Due to the presence of a subobject
classifier, the internal logic of toposes is impredicative.
To overcome this obstacle, a notion of predicative topos ought to be developed


Source: Andrews, Peter B. - Department of Mathematical Sciences, Carnegie Mellon University


Collections: Mathematics