Geodesic
Algebraic models of intuitionistic theories of sets and classes
Theory and applications of categories, CT2004, Tome 15 (2005), pp. 147-163.

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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.
Classification : 18B05, 18B25, 18C10, 03G30, 03E70, 03F60
Keywords: algebraic set theory, topos theory, sheaf theory 
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     title = {Algebraic models of intuitionistic theories of sets and classes},
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     year = {2005},
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     url = {http://geodesic.mathdoc.fr/item/TAC_2005_15_a4/}
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S. Awodey; H. Forssell. Algebraic models of intuitionistic theories of sets and classes. Theory and applications of categories, CT2004, Tome 15 (2005), pp. 147-163. http://geodesic.mathdoc.fr/item/TAC_2005_15_a4/