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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 & Jiri Rosicky (2007). REVIEWS-Category Theory. Bulletin of Symbolic Logic 13 (3).
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  5. 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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  6. 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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  7. 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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  8. 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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  9. 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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  10. 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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  11. 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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  12. 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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  13. 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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  14. 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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  15. John L. Bell, The Development of Categorical Logic.
    5.5. Every topos is linguistic: the equivalence theorem.
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  16. Jean Bénabou (1985). Fibered Categories and the Foundations of Naive Category Theory. Journal of Symbolic Logic 50 (1):10-37.
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  17. 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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  18. 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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  19. Richard Blute & Philip Scott (2004). Category Theory for Linear Logicians. In Thomas Ehrhard (ed.), Linear Logic in Computer Science. Cambridge University Press. 316--3.
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  20. Izabela Bondecka-Krzykowska & Roman Murawski (2008). Structuralism and Category Theory in the Contemporary Philosophy of Mathematics. Logique Et Analyse 51 (204):365.
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  21. Anna Bucalo & Giuseppe Rosolini (2013). Topologies and Free Constructions. Logic and Logical Philosophy 22 (3):327-346.
    The standard presentation of topological spaces relies heavily on (naïve) set theory: a topology consists of a set of subsets of a set (of points). And many of the high-level tools of set theory are required to achieve just the basic results about topological spaces. Concentrating on the mathematical structures, category theory offers the possibility to look synthetically at the structure of continuous transformations between topological spaces addressing specifically how the fundamental notions of point and open come about. As a (...)
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  22. Otavio Bueno, Outline of a Paraconsistent Category Theory.
    The aim of this paper is two-fold: (1) To contribute to a better knowledge of the method of the Argentinean mathematicians Lia Oubifia and Jorge Bosch to formulate category theory independently of set theory. This method suggests a new ontology of mathematical objects, and has a profound philosophical significance (the underlying logic of the resulting category theory is classical iirst—order predicate calculus with equality). (2) To show in outline how the Oubina-Bosch theory can be modified to give rise to a (...)
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  23. M. W. Bunder (1984). Category Theory Based on Combinatory Logic. Archive for Mathematical Logic 24 (1):1-16.
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  24. Marta Bunge (1984). Toposes in Logic and Logic in Toposes. Topoi 3 (1):13-22.
    The purpose of this paper is to justify the claim that Topos theory and Logic (the latter interpreted in a wide enough sense to include Model theory and Set theory) may interact to the advantage of both fields. Once the necessity of utilizing toposes (other than the topos of Sets) becomes apparent, workers in Topos theory try to make this task as easy as possible by employing a variety of methods which, in the last instance, find their justification in metatheorems (...)
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  25. Olivia Caramello (2011). A Characterization Theorem for Geometric Logic. Annals of Pure and Applied Logic 162 (4):318-321.
    We establish a criterion for deciding whether a class of structures is the class of models of a geometric theory inside Grothendieck toposes; then we specialize this result to obtain a characterization of the infinitary first-order theories which are geometric in terms of their models in Grothendieck toposes, solving a problem posed by Ieke Moerdijk in 1989.
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  26. Jessica Carter (2008). Categories for the Working Mathematician: Making the Impossible Possible. Synthese 162 (1):1 - 13.
    This paper discusses the notion of necessity in the light of results from contemporary mathematical practice. Two descriptions of necessity are considered. According to the first, necessarily true statements are true because they describe ‘unchangeable properties of unchangeable objects’. The result that I present is argued to provide a counterexample to this description, as it concerns a case where objects are moved from one category to another in order to change the properties of these objects. The second description concerns necessary (...)
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  27. J. L. Castiglioni, M. Menni & M. Sagastume (2008). On Some Categories of Involutive Centered Residuated Lattices. Studia Logica 90 (1):93 - 124.
    Motivated by an old construction due to J. Kalman that relates distributive lattices and centered Kleene algebras we define the functor K • relating integral residuated lattices with 0 (IRL0) with certain involutive residuated lattices. Our work is also based on the results obtained by Cignoli about an adjunction between Heyting and Nelson algebras, which is an enrichment of the basic adjunction between lattices and Kleene algebras. The lifting of the functor to the (...)
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  28. Roberto Cignoli (1979). Coproducts in the Categories of Kleene and Three-Valued Łukasiewicz Algebras. Studia Logica 38 (3):237 - 245.
    It is given an explicit description of coproducts in the category of Kleene algebras in terms of the dual topological spaces. As an application, a description of dual spaces of free Kleene algebras is given. It is also shown that the coproduct of a family of three-valued ukasiewicz algebras in the category of Kleene algebras is the same as the coproduct in the subcategory of three-valued ukasiewicz algebras.
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  29. D. Corfield (2002). Review of F. W. Lawvere and S. H. Schanuel, Conceptual Mathematics: A First Introduction to Categories; and J. L. Bell, A Primer of Infinitesimal Analysis. [REVIEW] Studies in History and Philosophy of Science Part B 33 (2):359-366.
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  30. David Corfield, Some Implications of the Adoption of Category Theory for Philosophy.
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  31. David Corfield (2002). Conceptual Mathematics: A First Introduction to Categories. Studies in History and Philosophy of Science Part B 33 (2):359-366.
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  32. Calvin C. Elgot (1974). Review: F. William Lawvere, S. Eilenberg, D. K. Harrison, S. MacLane, H. Rohrl, The Category of Categories as a Foundation for Mathematics. [REVIEW] Journal of Symbolic Logic 39 (2):341-341.
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  33. Jonas Eliasson (2004). Ultrapowers as Sheaves on a Category of Ultrafilters. Archive for Mathematical Logic 43 (7):825-843.
    In the paper we investigate the topos of sheaves on a category of ultrafilters. The category is described with the help of the Rudin-Keisler ordering of ultrafilters. It is shown that the topos is Boolean and two-valued and that the axiom of choice does not hold in it. We prove that the internal logic in the topos does not coincide with that in any of the ultrapowers. We also show that internal set theory, an axiomatic nonstandard set theory, can be (...)
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  34. David Ellerman, Category Theory and Universal Models: Adjoints and Brain Functors.
    Since its formal definition over sixty years ago, category theory has been increasingly recognized as having a foundational role in mathematics. It provides the conceptual lens to isolate and characterize the structures with importance and universality in mathematics. The notion of an adjunction (a pair of adjoint functors) has moved to center-stage as the principal lens. The central feature of an adjunction is what might be called "internalization through a universal" based on universal mapping properties. A recently developed "heteromorphic" theory (...)
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  35. David Ellerman (forthcoming). On Concrete Universals: A Modern Treatment Using Category Theory. AL-MUKHATABAT.
    Today it would be considered "bad Platonic metaphysics" to think that among all the concrete instances of a property there could be a universal instance so that all instances had the property by virtue of participating in that concrete universal. Yet there is a mathematical theory, category theory, dating from the mid-20th century that shows how to precisely model concrete universals within the "Platonic Heaven" of mathematics. This paper, written for the philosophical logician, develops this category-theoretic treatment of concrete universals (...)
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  36. David Ellerman (2007). Adjoints and Emergence: Applications of a New Theory of Adjoint Functors. [REVIEW] Axiomathes 17 (1):19-39.
    Since its formal definition over sixty years ago, category theory has been increasingly recognized as having a foundational role in mathematics. It provides the conceptual lens to isolate and characterize the structures with importance and universality in mathematics. The notion of an adjunction (a pair of adjoint functors) has moved to center-stage as the principal lens. The central feature of an adjunction is what might be called “determination through universals” based on universal mapping properties. A recently developed “heteromorphic” theory about (...)
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  37. David P. Ellerman (1988). Category Theory and Concrete Universals. Erkenntnis 28 (3):409 - 429.
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  38. Erwin Engeler & Helmut Röhrl (1969). On the Problem of Foundations of Category Theory. Dialectica 23 (1):58-66.
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  39. Solomon Feferman, Foundations of Category Theory: What Remains to Be Done.
    • Session on CF&FCT proposed by E. Landry; participants: G. Hellman, E. Landry, J.-P. Marquis and C. McLarty..
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  40. Solomon Feferman, Enriched Stratified Systems for the Foundations of Category Theory.
    Four requirements are suggested for an axiomatic system S to provide the foundations of category theory: (R1) S should allow us to construct the category of all structures of a given kind (without restriction), such as the category of all groups and the category of all categories; (R2) It should also allow us to construct the category of all functors between any two given categories including the ones constructed under (R1); (R3) In addition, S should allow us to establish the (...)
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  41. Solomon Feferman (2013). Foundations of Unlimited Category Theory: What Remains to Be Done. Review of Symbolic Logic 6 (1):6-15.
    Following a discussion of various forms of set-theoretical foundations of category theory and the controversial question of whether category theory does or can provide an autonomous foundation of mathematics, this article concentrates on the question whether there is a foundation for or category theory. The author proposed four criteria for such some years ago. The article describes how much had previously been accomplished on one approach to meeting those criteria, then takes care of one important obstacle that had been met (...)
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  42. Siegfried Gottwald (2006). Universes of Fuzzy Sets and Axiomatizations of Fuzzy Set Theory. Part II: Category Theoretic Approaches. Studia Logica 84 (1):23 - 50.
    For classical sets one has with the cumulative hierarchy of sets, with axiomatizations like the system ZF, and with the category SET of all sets and mappings standard approaches toward global universes of all sets.We discuss here the corresponding situation for fuzzy set theory. Our emphasis will be on various approaches toward (more or less naively formed) universes of fuzzy sets as well as on axiomatizations, and on categories of fuzzy sets.
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  43. Brice Halimi (2012). Diagrams as Sketches. Synthese 186 (1):387-409.
    This article puts forward the notion of “evolving diagram” as an important case of mathematical diagram. An evolving diagram combines, through a dynamic graphical enrichment, the representation of an object and the representation of a piece of reasoning based on the representation of that object. Evolving diagrams can be illustrated in particular with category-theoretic diagrams (hereafter “diagrams*”) in the context of “sketch theory,” a branch of modern category theory. It is argued that sketch theory provides a diagrammatic* theory of diagrams*, (...)
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  44. Michael John Healy & Thomas Preston Caudell (2006). Ontologies and Worlds in Category Theory: Implications for Neural Systems. [REVIEW] Axiomathes 16 (1-2):165-214.
    We propose category theory, the mathematical theory of structure, as a vehicle for defining ontologies in an unambiguous language with analytical and constructive features. Specifically, we apply categorical logic and model theory, based upon viewing an ontology as a sub-category of a category of theories expressed in a formal logic. In addition to providing mathematical rigor, this approach has several advantages. It allows the incremental analysis of ontologies by basing them in an interconnected hierarchy of theories, with an operation on (...)
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  45. Geoffrey Hellman (2003). Does Category Theory Provide a Framework for Mathematical Structuralism? Philosophia Mathematica 11 (2):129-157.
    Category theory and topos theory have been seen as providing a structuralist framework for mathematics autonomous vis-a-vis set theory. It is argued here that these theories require a background logic of relations and substantive assumptions addressing mathematical existence of categories themselves. We propose a synthesis of Bell's many-topoi view and modal-structuralism. Surprisingly, a combination of mereology and plural quantification suffices to describe hypothetical large domains, recovering the Grothendieck method of universes. Both topos theory and set theory can be carried out (...)
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  46. Chris Heunen, Klaas Landsman & Bas Spitters, The Principle of General Tovariance.
    We tentatively propose two guiding principles for the construction of theories of physics, which should be satisfied by a possible future theory of quantum gravity. These principles are inspired by those that led Einstein to his theory of general relativity, viz. his principle of general covariance and his equivalence principle, as well as by the two mysterious dogmas of Bohr's interpretation of quantum mechanics, i.e. his doctrine of classical concepts and his principle of complementarity. An appropriate mathematical language for combining (...)
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  47. Martin Hofmann (1997). An Application of Category-Theoretic Semantics to the Characterisation of Complexity Classes Using Higher-Order Function Algebras. Bulletin of Symbolic Logic 3 (4):469-486.
    We use the category of presheaves over PTIME-functions in order to show that Cook and Urquhart's higher-order function algebra PV ω defines exactly the PTIME-functions. As a byproduct we obtain a syntax-free generalisation of PTIME-computability to higher types. By restricting to sheaves for a suitable topology we obtain a model for intuitionistic predicate logic with ∑ 1 b -induction over PV ω and use this to re-establish that the provably total functions in this system are polynomial time computable. Finally, we (...)
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  48. David G. Holdsworth (1977). Category Theory and Quantum Mechanics (Kinematics). Journal of Philosophical Logic 6 (1):441 - 453.
  49. C. J. Isham (2005). Quantising on a Category. Foundations of Physics 35 (2):271-297.
    We review the problem of finding a general framework within which one can construct quantum theories of non-standard models for space, or space-time. The starting point is the observation that entities of this type can typically be regarded as objects in a category whose arrows are structure-preserving maps. This motivates investigating the general problem of quantising a system whose ‘configuration space’ (or history-theory analogue) is the set of objects Ob(Q) in a category Q. We develop a scheme based on constructing (...)
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  50. C. J. Isham & J. Butterfield (2000). Some Possible Roles for Topos Theory in Quantum Theory and Quantum Gravity. Foundations of Physics 30 (10):1707-1735.
    We discuss some ways in which topos theory (a branch of category theory) can be applied to interpretative problems in quantum theory and quantum gravity. In Sec.1, we introduce these problems. In Sec.2, we introduce topos theory, especially the idea of a topos of presheaves. In Sec.3, we discuss several possible applications of topos theory to the problems in Sec.1. In Sec.4, we draw some conclusions.
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