1. Henrik Forssell (2012). Topological Representation of Geometric Theories. Mathematical Logic Quarterly 58 (6):380-393.
    Using Butz and Moerdijk's topological groupoid representation of a topos with enough points, a ‘syntax-semantics’ duality for geometric theories is constructed. The emphasis is on a logical presentation, starting with a description of the semantic topological groupoid of models and isomorphisms of a theory. It is then shown how to extract a theory from equivariant sheaves on a topological groupoid in such a way that the result is a contravariant adjunction between theories and groupoids, the restriction of which is a (...)
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  2. Steve Awodey, Henrik Forssell & Michael A. Warren, Algebraic Models of Sets and Classes in Categories of Ideals.
    We introduce a new sheaf-theoretic construction called the ideal completion of a category and investigate its logical properties. We show that it satisfies the axioms for a category of classes in the sense of Joyal and Moerdijk [17], so that the tools of algebraic set theory can be applied to produce models of various elementary set theories. These results are then used to prove the conservativity of different set theories over various classical and constructive type theories.
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  3. Steve Awodey & Henrik Forssell, Algebraic Models of Intuitionistic Theories of Sets and Classes.
    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 [2] by introducing a (...)
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