Type theories, toposes and constructive set theory: predicative aspects of AST

Abstract We introduce a predicative version of topos (stratified pseudotopos) based on the notion of small maps in algebraic set theory, developed by Joyal and one of the authors. Examples of stratified pseudotoposes can be constructed in Martin-Lof type theory, which is a predicative theory. A stratified pseudotopos admits construction of the internal category of sheaves, which is again a stratified pseudotopos. We also show how to build models of Aczel-Myhill constructive set theory using this categorical structure.

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