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Summary Category theory is a branch of mathematics that has played a very important role in twentieth and twenty-first century mathematics. A category is a mathematical structure made up of objects -- which can be helpfully thought of as mathematical structures of some sort and morphisms, which can helpfully be thought of as abstract mappings connecting the objects. A canonical example of a category is the category with sets for objects and functions for morphisms. From the philosophical perspective category theory is important for a variety of reasons, including its role as an alternative foundation for mathematics, because of the development and growth of categorial logic, and for its role in providing a canonical codification of the notion of isomorphism.
Key works The definitions of categories, functors, and natural transformations all appeared for the first time in MacLane & Eilenberg 1945. This paper is difficult for a variety of both historical and mathematical reasons; the standard textbook on category theory is Maclane 1978. Textbooks aimed more at philosophical audiences include Goldblatt 2006, Awodey 2010, and McLarty 1996. For discussion on the role of category theory as an autonomous foundation of mathematics, the conversation contained in the following papers is helpful: Feferman 1977, Hellman 2003, Awodey 2004, Linnebo & Pettigrew 2011, and Logan 2015. The references in these papers will direct the reader in helpful directions for further research.
Introductions Landry & Marquis 2005 and Landry 1999 provide excellent overviews of the area. McLarty 1990 provides an overview of the history of philosophical uses of category theory focused on Topos theory. 
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  1. Nathanael Leedom Ackerman (2014). On Transferring Model Theoretic Theorems of {Mathcal{L}_{{Infty},Omega}} in the Category of Sets to a Fixed Grothendieck Topos. Logica Universalis 8 (3-4):345-391.
    Working in a fixed Grothendieck topos Sh(C, J C ) we generalize \({\mathcal{L}_{{\infty},\omega}}\) to allow our languages and formulas to make explicit reference to Sh(C, J C ). We likewise generalize the notion of model. We then show how to encode these generalized structures by models of a related sentence of \({\mathcal{L}_{{\infty},\omega}}\) in the category of sets and functions. Using this encoding we prove analogs of several results concerning \({\mathcal{L}_{{\infty},\omega}}\) , such as the downward Löwenheim–Skolem theorem, the completeness theorem and (...)
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  2. Jeremy Avigad & Jeffrey Helzner (2002). Transfer Principles in Nonstandard Intuitionistic Arithmetic. Archive for Mathematical Logic 41 (6):581-602.
    Using a slight generalization, due to Palmgren, of sheaf semantics, we present a term-model construction that assigns a model to any first-order intuitionistic theory. A modification of this construction then assigns a nonstandard model to any theory of arithmetic, enabling us to reproduce conservation results of Moerdijk and Palmgren for nonstandard Heyting arithmetic. Internalizing the construction allows us to strengthen these results with additional transfer rules; we then show that even trivial transfer axioms or minor strengthenings of these rules destroy (...)
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  3. S. Awodey (1996). Structure in Mathematics and Logic: A Categorical Perspective. 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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  4. S. Awodey & C. Butz (2000). Topological Completeness for Higher-Order Logic. 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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  5. S. Awodey, N. Gambino & M. A. Warren (2009). Lawvere-Tierney Sheaves in Algebraic Set Theory. 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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  6. S. Awodey & Jiri Rosicky (2007). REVIEWS-Category Theory. Bulletin of Symbolic Logic 13 (3).
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  7. S. Awodey & M. A. Warren (2013). Martin-Löf Complexes. 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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  8. Steve Awodey, Natural Models of Homotopy Type Theory.
    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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  9. Steve Awodey (2013). First-Order Logical Duality. 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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  10. Steve Awodey (2010). Category Theory. Oup Oxford.
    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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  11. Steve Awodey (2009). From Sets to Types to Categories to Sets. 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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  12. Steve Awodey (2008). A Brief Introduction to Algebraic Set Theory. 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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  13. Steve Awodey (2006). Continuity and Logical Completeness: An Application of Sheaf Theory and Topoi. In Johan van Benthem, Gerhard Heinzman, M. Rebushi & H. Visser (eds.), The Age of Alternative Logics. Springer. 139--149.
    The notion of a continuously variable quantity can be regarded as a generalization of that of a particular quantity, and the properties of such quantities are then akin to, and derived from, the properties of constants. For example, the continuous, real-valued functions on a topological space behave like the field of real numbers in many ways, but instead form a ring. Topos theory permits one to apply this same idea to logic, and to consider continuously variable sets . In this (...)
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  14. Steve Awodey, Notes on Algebraic Set Theory.
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  15. Steve Awodey (2005). Johnstone Peter T.. Sketches of an Elephant: A Topos Theory Compendium. Oxford Logic Guides, Vols. 43, 44. Oxford University Press, Oxford, 2002, Xxii+ 1160 Pp. [REVIEW] Bulletin of Symbolic Logic 11 (1):65-69.
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  16. Steve Awodey (2004). An Answer to Hellman's Question: ‘Does Category Theory Provide a Framework for Mathematical Structuralism?’. Philosophia Mathematica 12 (1):54-64.
    An affirmative answer is given to the question quoted in the title.
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  17. Steve Awodey, Sheaf Representation for Topoi.
    Steve Awodey. Sheaf Representation for Topoi.
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  18. Steve Awodey & Lars Birkedal, Elementary Axioms for Local Maps of Toposes.
    We present a complete elementary axiomatization of local maps of toposes.
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  19. Steve Awodey, Carsten Butz & Alex Simpson (2007). Relating First-Order Set Theories and Elementary Toposes. 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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  20. Steve Awodey, Carston Butz, Alex Simpson & Thomas Streicher, Relating First-Order Set Theories, Toposes and Categories of Classes.
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  21. Steve Awodey & Jonas Eliasson (2004). Ultrasheaves and Double Negation. 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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  22. 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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  23. 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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  24. Steve Awodey & Michael A. Warren, Predicative Algebraic Set Theory.
    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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  25. Steven Awodey (1995). Axiom of Choice and Excluded Middle in Categorical Logic. Bulletin of Symbolic Logic 1:344.
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  26. Steven M. Awodey (1997). Logic in Topoi: Functorial Semantics for High-Order Logic. Dissertation, The University of Chicago
    The category-theoretic notion of a topos is called upon to study the syntax and semantics of higher-order logic. Syntactical systems of logic are replaced by topoi, and models by functors on those topoi, as per the general scheme of functorial semantics. Each logical theory T gives rise to a syntactic topos ${\cal I}\lbrack U\sb{T}\rbrack$ of polynomial-like objects. The chief result is the universal characterization of ${\cal I}\lbrack U\sb{T}\rbrack$ as a so-called classifying topos: for any topos ${\cal E},$ the category ${\bf (...)
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  27. Steve Awody, An Outline of Algebraic Set Theory.
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  28. Nils A. Baas (2009). Extended Memory Evolutive Systems in a Hyperstructure Context. Axiomathes 19 (2):215-221.
    This paper is just a comment to the impressive work by A. C. Ehresmann and J.-P. Vanbremeersch on the theory of Memory Evolutive Systems (MES). MES are truly higher order systems. Hyperstructures represent a new concept which I introduced in order to capture the essence of what a higher order structure is—encompassing hierarchies and emergence. Hyperstructures are motivated by cobordism theory in topology and higher category theory. The morphism concept is replaced by the concept of a bond. In the paper (...)
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  29. I. C. Baianu, R. Brown, G. Georgescu & J. F. Glazebrook (2006). Complex Non-Linear Biodynamics in Categories, Higher Dimensional Algebra and Łukasiewicz–Moisil Topos: Transformations of Neuronal, Genetic and Neoplastic Networks. [REVIEW] Axiomathes 16 (1-2):65-122.
    A categorical, higher dimensional algebra and generalized topos framework for Łukasiewicz–Moisil Algebraic–Logic models of non-linear dynamics in complex functional genomes and cell interactomes is proposed. Łukasiewicz–Moisil Algebraic–Logic models of neural, genetic and neoplastic cell networks, as well as signaling pathways in cells are formulated in terms of non-linear dynamic systems with n-state components that allow for the generalization of previous logical models of both genetic activities and neural networks. An algebraic formulation of variable ‘next-state functions’ is extended to a Łukasiewicz–Moisil (...)
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  30. Jonathan Bain (2013). Category-Theoretic Structure and Radical Ontic Structural Realism. Synthese 190 (9):1621-1635.
    Radical Ontic Structural Realism (ROSR) claims that structure exists independently of objects that may instantiate it. Critics of ROSR contend that this claim is conceptually incoherent, insofar as, (i) it entails there can be relations without relata, and (ii) there is a conceptual dependence between relations and relata. In this essay I suggest that (ii) is motivated by a set-theoretic formulation of structure, and that adopting a category-theoretic formulation may provide ROSR with more support. In particular, I consider how a (...)
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  31. B. Banaschewski (2005). Excluded Middle Versus Choice in a Topos. Mathematical Logic Quarterly 51 (3):282.
    It is shown for an arbitrary topos that the Law of the Excluded Middle holds in its propositional logic iff it satisfies the limited choice principle that every epimorphism from 2 = 1 ⊕ 1 splits.
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  32. Bernhard Banaschewski & Christopher J. Mulvey (2006). A Globalisation of the Gelfand Duality Theorem. Annals of Pure and Applied Logic 137 (1):62-103.
    In this paper we bring together results from a series of previous papers to prove the constructive version of the Gelfand duality theorem in any Grothendieck topos , obtaining a dual equivalence between the category of commutative C*-algebras and the category of compact, completely regular locales in the topos.
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  33. Howard Barnum, Ross Duncan & Alexander Wilce (2013). Symmetry, Compact Closure and Dagger Compactness for Categories of Convex Operational Models. Journal of Philosophical Logic 42 (3):501-523.
    In the categorical approach to the foundations of quantum theory, one begins with a symmetric monoidal category, the objects of which represent physical systems, and the morphisms of which represent physical processes. Usually, this category is taken to be at least compact closed, and more often, dagger compact, enforcing a certain self-duality, whereby preparation processes (roughly, states) are interconvertible with processes of registration (roughly, measurement outcomes). This is in contrast to the more concrete “operational” approach, in which the states and (...)
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  34. A. Bartlett (2006). Ringing in the Sheaves. [REVIEW] Cosmos and History: The Journal of Natural and Social Philosophy 2 (1-2):339-344.
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  35. Vadim Batitsky (1996). Theories, Theorizers and the World: A Category-Theoretic Approach. Dissertation, University of Pennsylvania
    In today's philosophy of science, scientific theories are construed as abstract mathematical objects: formal axiomatic systems or classes of set-theoretic models. By focusing exclusively on the logico-mathematical structure of theories, however, this approach ignores their essentially cognitive nature: that theories are conceptualizations of the world produced by some cognitive agents. As a result, traditional philosophical analyses of scientific theories are incapable of coherently accounting for the relevant relations between highly abstract and idealized models in science and concrete empirical phenomena in (...)
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  36. J. L. Bell (1989). Review: J. Lambek, P. J. Scott, Introduction to Higher Order Categorical Logic. [REVIEW] Journal of Symbolic Logic 54 (3):1113-1114.
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  37. J. L. Bell (1986). From Absolute to Local Mathematics. Synthese 69 (3):409 - 426.
    In this paper (a sequel to [4]) I put forward a "local" interpretation of mathematical concepts based on notions derived from category theory. The fundamental idea is to abandon the unique absolute universe of sets central to the orthodox set-theoretic account of the foundations of mathematics, replacing it by a plurality of local mathematical frameworks - elementary toposes - defined in category-theoretic terms.
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  38. J. L. Bell (1982). Categories, Toposes and Sets. Synthese 51 (3):293 - 337.
    This paper is an introduction to topos theory which assumes no prior knowledge of category theory. It includes a discussion of internal logic in a topos, A characterization of the category of sets, And an investigation of the notions of topology and sheaf in a topos.
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  39. J. L. Bell (1981). Category Theory and the Foundations of Mathematics. British Journal for the Philosophy of Science 32 (4):349-358.
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  40. John L. Bell, Two Approaches to Modelling the Universe: Synthetic Differential Geometry and Frame-Valued Sets.
    I describe two approaches to modelling the universe, the one having its origin in topos theory and differential geometry, the other in set theory. The first is synthetic differential geometry. Traditionally, there have been two methods of deriving the theorems of geometry: the analytic and the synthetic. While the analytical method is based on the introduction of numerical coordinates, and so on the theory of real numbers, the idea behind the synthetic approach is to furnish the subject of geometry with (...)
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  41. John L. Bell, Types, Sets and Categories.
    This essay is an attempt to sketch the evolution of type theory from its beginnings early in the last century to the present day. Central to the development of the type concept has been its close relationship with set theory to begin with and later its even more intimate relationship with category theory. Since it is effectively impossible to describe these relationships (especially in regard to the latter) with any pretensions to completeness within the space of a comparatively short article, (...)
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  42. John L. Bell (2001). Observations on Category Theory. Axiomathes 12 (1-2):151-155.
    is a presentation of mathematics in terms of the fundamental concepts of transformation, and composition of transformations. While the importance of these concepts had long been recognized in algebra (for example, by Galois through the idea of a group of permutations) and in geometry (for example, by Klein in his Erlanger Programm), the truly universal role they play in mathematics did not really begin to be appreciated until the rise of abstract algebra in the 1930s. In abstract algebra the idea (...)
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  43. John L. Bell, The Development of Categorical Logic.
    5.5. Every topos is linguistic: the equivalence theorem.
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  44. John L. Bell & Silvia Gebellato (1996). Precovers, Modalities and Universal Closure Operators in a Topos. Mathematical Logic Quarterly 42 (1):289-299.
    In this paper we develop the notion of formal precover in a topos by defining a relation between elements and sets in a local set theory. We show that such relations are equivalent to modalities and to universal closure operators. Finally we prove that these relations are well characterized by a convenient restriction to a particular set.
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  45. Jean Bénabou (1985). Fibered Categories and the Foundations of Naive Category Theory. Journal of Symbolic Logic 50 (1):10-37.
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  46. Bodil Biering (2008). Cartesian Closed Dialectica Categories. Annals of Pure and Applied Logic 156 (2):290-307.
    When Gödel developed his functional interpretation, also known as the Dialectica interpretation, his aim was to prove consistency of first order arithmetic by reducing it to a quantifier-free theory with finite types. Like other functional interpretations Gödel’s Dialectica interpretation gives rise to category theoretic constructions that serve both as new models for logic and semantics and as tools for analysing and understanding various aspects of the Dialectica interpretation itself. Gödel’s Dialectica interpretation gives rise to the Dialectica categories , in: Contemp. (...)
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  47. Lars Birkedal (2002). A General Notion of Realizability. Bulletin of Symbolic Logic 8 (2):266-282.
    We present a general notion of realizability encompassing both standard Kleene style realizability over partial combinatory algebras and Kleene style realizability over more general structures, including all partial cartesian closed categories. We shown how the general notion of realizability can be used to get models of dependent predicate logic, thus obtaining as a corollary (the known result) that the category Equ of equilogical spaces models dependent predicate logic. Moreover, we characterize when the general notion of realizability gives rise to a (...)
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  48. Lars Birkedal & Jaap van Oosten (2002). Relative and Modified Relative Realizability. Annals of Pure and Applied Logic 118 (1-2):115-132.
    The classical forms of both modified realizability and relative realizability are naturally described in terms of the Sierpinski topos. The paper puts these two observations together and explains abstractly the existence of the geometric morphisms and logical functors connecting the various toposes at issue. This is done by advancing the theory of triposes over internal partial combinatory algebras and by employing a novel notion of elementary map.
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  49. Georges Blanc & Anne Preller (1975). Lawvere's Basic Theory of the Category of Categories. Journal of Symbolic Logic 40 (1):14-18.
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  50. Andreas Blass & Andre Scedrov (1992). Complete Topoi Representing Models of Set Theory. Annals of Pure and Applied Logic 57 (1):1-26.
    By a model of set theory we mean a Boolean-valued model of Zermelo-Fraenkel set theory allowing atoms (ZFA), which contains a copy of the ordinary universe of (two-valued,pure) sets as a transitive subclass; examples include Scott-Solovay Boolean-valued models and their symmetric submodels, as well as Fraenkel-Mostowski permutation models. Any such model M can be regarded as a topos. A logical subtopos E of M is said to represent M if it is complete and its cumulative hierarchy, as defined by Fourman (...)
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