Aspects of Predicative Algebraic Set Theory III: Sheaves

This is the third installment in a series of papers on algebraic set theory. In it, we develop a uniform approach to sheaf models of constructive set theories based on ideas from categorical logic. The key notion is that of a “predicative category with small maps” which axiomatises the idea of a category of classes and class morphisms, together with a selected class of maps whose fibres are sets (in some axiomatic set theory). The main result of the present paper is that such predicative categories with small maps are stable under internal sheaves. We discuss the sheaf models of constructive set theory this leads to, as well as ideas for future work.