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  1. A Brief Introduction to Algebraic Set Theory.Steve Awodey - 2008 - Bulletin of Symbolic Logic 14 (3):281-298.
    This brief article is intended to introduce the reader to the field of algebraic set theory, in which models of set theory of a new and fascinating kind are determined algebraically. The method is quite robust, applying to various classical, intuitionistic, and constructive set theories. Under this scheme some familiar set theoretic properties are related to algebraic ones, while others result from logical constraints. Conventional elementary set theories are complete with respect to algebraic models, which arise in a variety of (...)
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  • Models of Intuitionistic Set Theory in Subtoposes of Nested Realizability Toposes.Samuele Maschio & Thomas Streicher - 2015 - Annals of Pure and Applied Logic 166 (6):729-739.
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  • The Associated Sheaf Functor Theorem in Algebraic Set Theory.Nicola Gambino - 2008 - Annals of Pure and Applied Logic 156 (1):68-77.
    We prove a version of the associated sheaf functor theorem in Algebraic Set Theory. The proof is established working within a Heyting pretopos equipped with a system of small maps satisfying the axioms originally introduced by Joyal and Moerdijk. This result improves on the existing developments by avoiding the assumption of additional axioms for small maps and the use of collection sites.
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  • Well-Foundedness in Realizability.M. Hofmann, J. van Oosten & T. Streicher - 2006 - Archive for Mathematical Logic 45 (7):795-805.
  • Aspects of Predicative Algebraic Set Theory I: Exact Completion.Benno van den Berg & Ieke Moerdijk - 2008 - Annals of Pure and Applied Logic 156 (1):123-159.
    This is the first in a series of papers on Predicative Algebraic Set Theory, where we lay the necessary groundwork for the subsequent parts, one on realizability [B. van den Berg, I. Moerdijk, Aspects of predicative algebraic set theory II: Realizability, Theoret. Comput. Sci. . Available from: arXiv:0801.2305, 2008], and the other on sheaves [B. van den Berg, I. Moerdijk, Aspects of predicative algebraic set theory III: Sheaf models, 2008 ]. We introduce the notion of a predicative category with small (...)
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  • Relating First-Order Set Theories, Toposes and Categories of Classes.Steve Awodey, Carsten Butz, Alex Simpson & Thomas Streicher - 2014 - Annals of Pure and Applied Logic 165 (2):428-502.