This paper constructs models of intuitionistic set theory in suitable categories. First, a Basic Intuitionistic Set Theory (BIST) is stated, and the categorical semantics are given. Second, we give a notion of an ideal over a category, using which one can build a model of BIST in which a given topos occurs as the sets. And third, a sheaf model is given of a Basic Intuitionistic Class Theory conservatively extending BIST. The paper extends the results in Awodey, Butz, Simpson and Streicher (2003) by introducing a new and perhaps more natural notion of ideal, and in the class theory of part three.
Keywords: algebraic set theory, topos theory, sheaf theory
2000 MSC: 18B05, 18B25, 18C10, 03G30, 03E70, 03F60
Theory and Applications of Categories,
Vol. 15, CT2004,
No. 5, pp 147-163.