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Steve Awodey [55]S. Awodey [11]Steven Awodey [3]Steven M. Awodey [1]
  1. An Answer to Hellman's Question: ‘Does Category Theory Provide a Framework for Mathematical Structuralism?’.Steve Awodey - 2004 - Philosophia Mathematica 12 (1):54-64.
    An affirmative answer is given to the question quoted in the title.
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  2. Structuralism, Invariance, and Univalence†: Articles.Steve Awodey - 2014 - Philosophia Mathematica 22 (1):1-11.
    The recent discovery of an interpretation of constructive type theory into abstract homotopy theory suggests a new approach to the foundations of mathematics with intrinsic geometric content and a computational implementation. Voevodsky has proposed such a program, including a new axiom with both geometric and logical significance: the Univalence Axiom. It captures the familiar aspect of informal mathematical practice according to which one can identify isomorphic objects. While it is incompatible with conventional foundations, it is a powerful addition to homotopy (...)
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  3. Category Theory.Steve Awodey - 2007 - Studia Logica 86 (1):133-135.
  4. Category Theory.Steve Awodey - 2010 - Oxford University Press.
    A comprehensive reference to category theory for students and researchers in mathematics, computer science, logic, cognitive science, linguistics, and philosophy. Useful for self-study and as a course text, the book includes all basic definitions and theorems, as well as numerous examples and exercises.
     
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  5. Structure in Mathematics and Logic: A Categorical Perspective.S. Awodey - 1996 - Philosophia Mathematica 4 (3):209-237.
    A precise notion of ‘mathematical structure’ other than that given by model theory may prove fruitful in the philosophy of mathematics. It is shown how the language and methods of category theory provide such a notion, having developed out of a structural approach in modern mathematical practice. As an example, it is then shown how the categorical notion of a topos provides a characterization of ‘logical structure’, and an alternative to the Pregean approach to logic which is continuous with the (...)
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  6.  35
    Peter T. Johnstone. Sketches of an Elephant: A Topos Theory Compendium. Oxford Logic Guides, Vols. 43, 44. Oxford University Press, Oxford, 2002, Xxii + 1160 Pp. [REVIEW]Steve Awodey - 2005 - Bulletin of Symbolic Logic 11 (1):65-69.
  7. Completeness and Categoricity. Part I: Nineteenth-Century Axiomatics to Twentieth-Century Metalogic.Steve Awodey & Erich H. Reck - 2002 - History and Philosophy of Logic 23 (1):1-30.
    This paper is the first in a two-part series in which we discuss several notions of completeness for systems of mathematical axioms, with special focus on their interrelations and historical origins in the development of the axiomatic method. We argue that, both from historical and logical points of view, higher-order logic is an appropriate framework for considering such notions, and we consider some open questions in higher-order axiomatics. In addition, we indicate how one can fruitfully extend the usual set-theoretic semantics (...)
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  8.  68
    First-Order Logical Duality.Steve Awodey - 2013 - Annals of Pure and Applied Logic 164 (3):319-348.
    From a logical point of view, Stone duality for Boolean algebras relates theories in classical propositional logic and their collections of models. The theories can be seen as presentations of Boolean algebras, and the collections of models can be topologized in such a way that the theory can be recovered from its space of models. The situation can be cast as a formal duality relating two categories of syntax and semantics, mediated by homming into a common dualizing object, in this (...)
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  9. Carnap’s Dream: Gödel, Wittgenstein, and Logical, Syntax.S. Awodey & A. W. Carus - 2007 - Synthese 159 (1):23-45.
    In Carnap’s autobiography, he tells the story how one night in January 1931, “the whole theory of language structure” in all its ramifications “came to [him] like a vision”. The shorthand manuscript he produced immediately thereafter, he says, “was the first version” of Logical Syntax of Language. This document, which has never been examined since Carnap’s death, turns out not to resemble Logical Syntax at all, at least on the surface. Wherein, then, did the momentous insight of 21 January 1931 (...)
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  10.  67
    Carnap, Completeness, and Categoricity:The Gabelbarkeitssatz OF 1928. [REVIEW]S. Awodey & A. W. Carus - 2001 - Erkenntnis 54 (2):145-172.
    In 1929 Carnap gave a paper in Prague on Investigations in General Axiomatics; a briefsummary was published soon after. Its subject lookssomething like early model theory, and the mainresult, called the Gabelbarkeitssatz, appears toclaim that a consistent set of axioms is complete justif it is categorical. This of course casts doubt onthe entire project. Though there is no furthermention of this theorem in Carnap''s publishedwritings, his Nachlass includes a largetypescript on the subject, Investigations inGeneral Axiomatics. We examine this work here,showing (...)
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  11.  57
    Homotopy Theoretic Models of Identity Types.Steve Awodey & Michael A. Warren - unknown
    Quillen [17] introduced model categories as an abstract framework for homotopy theory which would apply to a wide range of mathematical settings. By all accounts this program has been a success and—as, e.g., the work of Voevodsky on the homotopy theory of schemes [15] or the work of Joyal [11, 12] and Lurie [13] on quasicategories seem to indicate—it will likely continue to facilitate mathematical advances. In this paper we present a novel connection between model categories and mathematical logic, inspired (...)
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  12. Topology and Modality: The Topological Interpretation of First-Order Modal Logic: Topology and Modality.Steve Awodey - 2008 - Review of Symbolic Logic 1 (2):146-166.
    As McKinsey and Tarski showed, the Stone representation theorem for Boolean algebras extends to algebras with operators to give topological semantics for propositional modal logic, in which the “necessity” operation is modeled by taking the interior of an arbitrary subset of a topological space. In this article, the topological interpretation is extended in a natural way to arbitrary theories of full first-order logic. The resulting system of S4 first-order modal logic is complete with respect to such topological semantics.
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  13. 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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  14. From Wittgenstein's Prison to the Boundless Ocean : Carnap's Dream of Logical Syntax.Steve Awodey & A. W. Carus - 2009 - In Pierre Wagner (ed.), Carnap's Logical Syntax of Language. Palgrave-Macmillan.
  15.  53
    Completeness and Categoricity, Part II: Twentieth-Century Metalogic to Twenty-First-Century Semantics.Steve Awodey & Erich H. Reck - 2002 - History and Philosophy of Logic 23 (2):77-94.
    This paper is the second in a two-part series in which we discuss several notions of completeness for systems of mathematical axioms, with special focus on their interrelations and historical origins in the development of the axiomatic method. We argue that, both from historical and logical points of view, higher-order logic is an appropriate framework for considering such notions, and we consider some open questions in higher-order axiomatics. In addition, we indicate how one can fruitfully extend the usual set-theoretic semantics (...)
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  16.  82
    Type Theory and Homotopy.Steve Awodey - unknown
    of type theory has been used successfully to formalize large parts of constructive mathematics, such as the theory of generalized recursive definitions [NPS90, ML79]. Moreover, it is also employed extensively as a framework for the development of high-level programming languages, in virtue of its combination of expressive strength and desirable proof-theoretic properties [NPS90, Str91]. In addition to simple types A, B, . . . and their terms x : A b(x) : B, the theory also has dependent types x : (...)
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  17.  12
    Predicative Algebraic Set Theory.Steve Awodey & Michael A. Warren - unknown
    In this paper the machinery and results developed in [Awodey et al, 2004] are extended to the study of constructive set theories. Specifically, we introduce two constructive set theories BCST and CST and prove that they are sound and complete with respect to models in categories with certain structure. Specifically, basic categories of classes and categories of classes are axiomatized and shown to provide models of the aforementioned set theories. Finally, models of these theories are constructed in the category of (...)
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  18. Category Theory.S. Awodey - 2007 - Bulletin of Symbolic Logic 13 (3):371-372.
  19.  8
    Relating First-Order Set Theories and Elementary Toposes.Steve Awodey & Thomas Streicher - 2007 - Bulletin of Symbolic Logic 13 (3):340-358.
    We show how to interpret the language of first-order set theory in an elementary topos endowed with, as extra structure, a directed structural system of inclusions . As our main result, we obtain a complete axiomatization of the intuitionistic set theory validated by all such interpretations. Since every elementary topos is equivalent to one carrying a dssi, we thus obtain a first-order set theory whose associated categories of sets are exactly the elementary toposes. In addition, we show that the full (...)
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  20.  28
    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.
  21.  49
    Relating First-Order Set Theories and Elementary Toposes.Steve Awodey, Carsten Butz & Alex Simpson - 2007 - Bulletin of Symbolic Logic 13 (3):340-358.
    We show how to interpret the language of first-order set theory in an elementary topos endowed with, as extra structure, a directed structural system of inclusions (dssi). As our main result, we obtain a complete axiomatization of the intuitionistic set theory validated by all such interpretations. Since every elementary topos is equivalent to one carrying a dssi, we thus obtain a first-order set theory whose associated categories of sets are exactly the elementary toposes. In addition, we show that the full (...)
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  22. Topological Completeness for Higher-Order Logic.S. Awodey & C. Butz - 2000 - Journal of Symbolic Logic 65 (3):1168-1182.
    Using recent results in topos theory, two systems of higher-order logic are shown to be complete with respect to sheaf models over topological spaces- so -called "topological semantics." The first is classical higher-order logic, with relational quantification of finitely high type; the second system is a predicative fragment thereof with quantification over functions between types, but not over arbitrary relations. The second theorem applies to intuitionistic as well as classical logic.
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  23.  74
    How Carnap Could Have Replied to Gödel.Steve Awodey & A. W. Carus - unknown
    Steve Awodey and A. W. Carus. How Carnap Could Have Replied to Gödel.
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  24. Explicating "Analytic".Steve Awodey - 2012 - In Pierre Wagner (ed.), Carnap's Ideal of Explication and Naturalism. Palgrave-Macmillan.
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  25. Frege's Lectures on Logic: Carnap's Student Notes, 1910-1914.Erich H. Reck & Steve Awodey - 2005 - Bulletin of Symbolic Logic 11 (3):445-447.
  26. Gödel and Carnap.Steve Awodey & A. W. Carus - 2010 - In Kurt Gödel, Solomon Feferman, Charles Parsons & Stephen G. Simpson (eds.), Kurt Gödel: Essays for His Centennial. Association for Symbolic Logic.
  27.  65
    Completeness and Categoricity, Part I: 19th Century Axiomatics to 20th Century Metalogic.Steve Awodey & Erich H. Reck - unknown
    This paper is the first in a two-part series in which we discuss several notions of completeness for systems of mathematical axioms, with special focus on their interrelations and historical origins in the development of the axiomatic method. We argue that, both from historical and logical points of view, higher-order logic is an appropriate framework for considering such notions, and we consider some open questions in higher-order axiomatics. In addition, we indicate how one can fruitfully extend the usual set-theoretic semantics (...)
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  28.  15
    A Cubical Model of Homotopy Type Theory.Steve Awodey - 2018 - Annals of Pure and Applied Logic 169 (12):1270-1294.
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  29. Carnap Versus Godel: On Syntax and Tolerance.S. Awodey & A. W. Carus - unknown
    One thing we have found out about logical empiricism, now that people are examining it more closely again, is that it was more a framework for a number of related views than a single doctrine. The pluralism of different approaches among various adherents to the Vienna and Berlin groups has been much emphasized. Some have gone so far as to suggest that the kind of speculative philosophy now often called "continental" (including, say, phenomenology) can be seen as falling within the (...)
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  30.  5
    Topos Semantics for Higher-Order Modal Logic.Steve Awodey, Kohei Kishida & Hans-Cristoph Kotzsch - 2014 - Logique Et Analyse 228:591-636.
    We define the notion of a model of higher-order modal logic in an arbitrary elementary topos E. In contrast to the well-known interpretation of higher-order logic, the type of propositions is not interpreted by the subobject classifier ΩE, but rather by a suitable complete Heyting algebra H. The canonical map relating H and ΩE both serves to interpret equality and provides a modal operator on H in the form of a comonad. Examples of such structures arise from surjective geometric morphisms (...)
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  31.  12
    Algebraic Models of Intuitionistic Theories of Sets and Classes.Steve Awodey & Henrik Forssell - unknown
    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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  32.  51
    Martin-Löf Complexes.S. Awodey & M. A. Warren - 2013 - Annals of Pure and Applied Logic 164 (10):928-956.
    In this paper we define Martin-L¨of complexes to be algebras for monads on the category of (reflexive) globular sets which freely add cells in accordance with the rules of intensional Martin-L¨of type theory. We then study the resulting categories of algebras for several theories. Our principal result is that there exists a cofibrantly generated Quillen model structure on the category of 1-truncated Martin-L¨of complexes and that this category is Quillen equivalent to the category of groupoids. In particular, 1-truncated Martin-L¨of complexes (...)
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  33.  9
    Voevodsky’s Univalence Axiom in Homotopy Type Theory.Steve Awodey, Alvaro Pelayo & Michael A. Warren - unknown
    In this short note we give a glimpse of homotopy type theory, a new field of mathematics at the intersection of algebraic topology and mathematical logic, and we explain Vladimir Voevodsky’s univalent interpretation of it. This interpretation has given rise to the univalent foundations program, which is the topic of the current special year at the Institute for Advanced Study.
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  34. In Memoriam: Saunders Mac Lane, 1909-2005.Steve Awodey - 2007 - Bulletin of Symbolic Logic 13 (1):115-119.
  35.  36
    A Study of Categorres of Algebras and Coalgebras.Jesse Hughes, Steve Awodey, Dana Scott, Jeremy Avigad & Lawrence Moss - unknown
    This thesis is intended t0 help develop the theory 0f coalgebras by, Hrst, taking classic theorems in the theory 0f universal algebras amd dualizing them and, second, developing an interna] 10gic for categories 0f coalgebras. We begin with an introduction t0 the categorical approach t0 algebras and the dual 110tion 0f coalgebras. Following this, we discuss (c0)a,lg€bra.s for 2. (c0)monad and develop 2. theory 0f regular subcoalgebras which will be used in the interna] logic. We also prove that categories 0f (...)
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  36.  51
    Lawvere-Tierney Sheaves in Algebraic Set Theory.S. Awodey, N. Gambino & M. A. Warren - 2009 - Journal of Symbolic Logic 74 (3):861 - 890.
    We present a solution to the problem of defining a counterpart in Algebraic Set Theory of the construction of internal sheaves in Topos Theory. Our approach is general in that we consider sheaves as determined by Lawvere-Tierney coverages, rather than by Grothendieck coverages, and assume only a weakening of the axioms for small maps originally introduced by Joyal and Moerdijk, thus subsuming the existing topos-theoretic results.
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  37.  34
    Completeness and Categoricty, Part II: 20th Century Metalogic to 21st Century Semantics.Steve Awodey & Erich H. Reck - unknown
    This paper is the second in a two-part series in which we discuss several notions of completeness for systems of mathematical axioms, with special focus on their interrelations and historical origins in the development of the axiomatic method. We argue that, both from historical and logical points of view, higher-order logic is an appropriate framework for considering such notions, and we consider some open questions in higher-order axiomatics. In addition, we indicate how one can fruitfully extend the usual set-theoretic semantics (...)
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  38.  59
    Carnap and the Invariance of Logical Truth.Steve Awodey - 2017 - Synthese 194 (1):67-78.
    The failed criterion of logical truth proposed by Carnap in the Logical Syntax of Language was based on the determinateness of all logical and mathematical statements. It is related to a conception which is independent of the specifics of the system of the Syntax, hints of which occur elsewhere in Carnap’s writings, and those of others. What is essential is the idea that the logical terms are invariant under reinterpretation of the empirical terms, and are therefore semantically determinate. A certain (...)
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  39.  66
    Ultrasheaves and Double Negation.Jonas Eliasson & Steve Awodey - 2004 - Notre Dame Journal of Formal Logic 45 (4):235-245.
    Moerdijk has introduced a topos of sheaves on a category of filters. Following his suggestion, we prove that its double negation subtopos is the topos of sheaves on the subcategory of ultrafilters - the ultrasheaves. We then use this result to establish a double negation translation of results between the topos of ultrasheaves and the topos on filters.
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  40.  15
    Sheaf Representation for Topoi.Steve Awodey - unknown
    Steve Awodey. Sheaf Representation for Topoi.
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  41. Methodology.Peter T. Johnstone & Steve Awodey - unknown
    Notices Amer. Math. Sac. 51, 2004). Logically, such a "Grothendieck topos" is something like a universe of continuously variable sets. Before long, however, F.W. Lawvere and M. Tierney provided an elementary axiomatization..
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  42.  40
    Local Realizability Toposes and a Modal Logic for Computability.Steve Awodey, Lars Birkedal & Dana Scott - unknown
    This work is a step toward the development of a logic for types and computation that includes not only the usual spaces of mathematics and constructions, but also spaces from logic and domain theory. Using realizability, we investigate a configuration of three toposes that we regard as describing a notion of relative computability. Attention is focussed on a certain local map of toposes, which we first study axiomatically, and then by deriving a modal calculus as its internal logic. The resulting (...)
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  43.  9
    Algebraic Models of Sets and Classes in Categories of Ideals.Steve Awodey, Henrik Forssell & Michael A. Warren - unknown
    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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  44.  23
    Continuity and Logical Completeness.Steve Awodey - unknown
    The notion of a continuously variable quantity can be regarded as a generalization of that of a particular (constant) quantity, and the properties of such quantities are then akin to, and derived from, the..
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  45.  99
    From Sets to Types to Categories to Sets.Steve Awodey - 2009 - Philosophical Explorations.
    Three different styles of foundations of mathematics are now commonplace: set theory, type theory, and category theory. How do they relate, and how do they differ? What advantages and disadvantages does each one have over the others? We pursue these questions by considering interpretations of each system into the others and examining the preservation and loss of mathematical content thereby. In order to stay focused on the “big picture”, we merely sketch the overall form of each construction, referring to the (...)
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  46.  44
    Sheaf Toposes for Realizability.Steven Awodey & Andrej Bauer - 2008 - Archive for Mathematical Logic 47 (5):465-478.
    Steve Awodey and Audrej Bauer. Sheaf Toposes for Realizability.
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  47.  18
    Propositions as [Types].Steve Awodey & Andrej Bauer - unknown
    Image factorizations in regular categories are stable under pullbacks, so they model a natural modal operator in dependent type theory. This unary type constructor [A] has turned up previously in a syntactic form as a way of erasing computational content, and formalizing a notion of proof irrelevance. Indeed, semantically, the notion of a support is sometimes used as surrogate proposition asserting inhabitation of an indexed family. We give rules for bracket types in dependent type theory and provide complete semantics using (...)
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  48.  5
    Topological Completeness for Higher-Order Logic.S. Awodey & C. Butz - 2000 - Journal of Symbolic Logic 65 (3):1168-1182.
    Using recent results in topos theory, two systems of higher-order logic are shown to be complete with respect to sheaf models over topological spaces-so-called "topological semantics". The first is classical higher-order logic, with relational quantification of finitely high type; the second system is a predicative fragment thereof with quantification over functions between types, but not over arbitrary relations. The second theorem applies to intuitionistic as well as classical logic.
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  49.  41
    Natural Models of Homotopy Type Theory.Steve Awodey - unknown
    The notion of a natural model of type theory is defined in terms of that of a representable natural transfomation of presheaves. It is shown that such models agree exactly with the concept of a category with families in the sense of Dybjer, which can be regarded as an algebraic formulation of type theory. We determine conditions for such models to satisfy the inference rules for dependent sums Σ, dependent products Π, and intensional identity types Id, as used in (...)
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  50.  2
    Relating Topos Theory and Set Theory Via Categories of Classes.Steve Awodey, Alex Simpson & Thomas Streicher - unknown
    We investigate a certain system of intuitionistic set theory from three points of view: an elementary set theory with bounded separation, a topos with distinguished inclusions, and a category of classes with a system of small maps. The three presentations are shown to be equivalent in a strong sense.
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