Sheaves for predicative toposes

In this paper, we identify some categorical structures in which one can model predicative formal systems: in other words, predicative analogues of the notion of a topos, with the aim of using sheaf models to interprete predicative formal systems. Among our technical results, we prove that all the notions of a ``predicative topos'' that we consider, are stable under presheaves, while most are stable under sheaves.