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  1. 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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  2. 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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  3. 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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  4. 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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  5. 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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  6. 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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  7. 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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  8. 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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  9. 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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  10. 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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  11. 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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  12. John L. Bell, The Development of Categorical Logic.
    5.5. Every topos is linguistic: the equivalence theorem.
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  13. Jean Bénabou (1985). Fibered Categories and the Foundations of Naive Category Theory. Journal of Symbolic Logic 50 (1):10-37.
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  14. 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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  15. 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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  16. 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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  17. 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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  18. 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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  19. 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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  20. 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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  21. 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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  22. 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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  23. 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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  24. David Corfield, Some Implications of the Adoption of Category Theory for Philosophy.
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  25. 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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  26. 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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  27. David P. Ellerman (1988). Category Theory and Concrete Universals. Erkenntnis 28 (3):409 - 429.
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  28. 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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  29. 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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  30. 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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  31. 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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  32. 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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  33. 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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  34. 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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  35. David G. Holdsworth (1977). Category Theory and Quantum Mechanics (Kinematics). Journal of Philosophical Logic 6 (1):441 - 453.
  36. 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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  37. 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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  38. Chris Isham & Jeremy Butterfield, A Topos Perspective on the Kochen-Specker Theorem: I. Quantum States as Generalised Valuations.
    Any attempt to construct a realist interpretation of quantum theory founders on the Kochen-Specker theorem, which asserts the impossibility of assigning values to quantum quantities in a way that preserves functional relations between them. We construct a new type of valuation which is defined on all operators, and which respects an appropriate version of the functional composition principle. The truth-values assigned to propositions are (i) contextual; and (ii) multi-valued, where the space of contexts and the multi-valued logic for each context (...)
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  39. Bart Jacobs (2013). Dagger Categories of Tame Relations. Logica Universalis 7 (3):341-370.
    Within the context of an involutive monoidal category the notion of a comparison relation ${\mathsf{cp} : \overline{X} \otimes X \rightarrow \Omega}$ is identified. Instances are equality = on sets, inequality ${\leq}$ on posets, orthogonality ${\perp}$ on orthomodular lattices, non-empty intersection on powersets, and inner product ${\langle {-}|{-} \rangle}$ on vector or Hilbert spaces. Associated with a collection of such (symmetric) comparison relations a dagger category is defined with “tame” relations as morphisms. Examples include familiar categories in the foundations of quantum (...)
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  40. C. Barry Jay (1991). Coherence in Category Theory and the Church-Rosser Property. Notre Dame Journal of Formal Logic 33 (1):140-143.
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  41. C. Barry Jay (1989). A Note on Natural Numbers Objects in Monoidal Categories. Studia Logica 48 (3):389 - 393.
    The internal language of a monoidal category yields simple proofs of results about a natural numbers object therein.
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  42. Paul C. Kainen (2009). On the Ehresmann–Vanbremeersch Theory and Mathematical Biology. Axiomathes 19 (3):225-244.
    Category theory has been proposed as the ultimate algebraic model for biology. We review the Ehresmann–Vanbremeersch theory in the context of other mathematical approaches.
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  43. Molly Kao, Nicolas Fillion & John Bell (2010). J Ean -P Ierre M Arquis . From a Geometrical Point of View: A Study of the History and Philosophy of Category Theory. Philosophia Mathematica 18 (2):227-234.
    (No abstract is available for this citation).
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  44. M. Kary (2009). (Math, Science, ?). Axiomathes 19 (3):61-86.
    In science as in mathematics, it is popular to know little and resent much about category theory. Less well known is how common it is to know little and like much about set theory. The set theory of almost all scientists, and even the average mathematician, is fundamentally different from the formal set theory that is contrasted against category theory. The latter two are often opposed by saying one emphasizes Substance, the other Form. However, in all known systems of mathematics (...)
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  45. Goro Kato & D. Struppa (2002). Category Theory and Consciousness. In Kunio Yasue, Marj Jibu & Tarcisio Della Senta (eds.), No Matter, Never Mind. John Benjamins.
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  46. Goro Kato & Tsunefumi Tanaka (2006). Double-Slit Interference and Temporal Topos. Foundations of Physics 36 (11):1681-1700.
    The electron double-slit interference is re-examined from the point of view of temporal topos. Temporal topos (or t-topos) is an abstract algebraic (categorical) method using the theory of sheaves. A brief introduction to t-topos is given. When the structural foundation for describing particles is based on t-topos, the particle-wave duality of electron is a natural consequence. A presheaf associated with the electron represents both particle-like and wave-like properties depending upon whether an object in the site (t-site) is specified (particle-like) or (...)
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  47. Jerzy Król (2006). A Model for Spacetime: The Role of Interpretation in Some Grothendieck Topoi. [REVIEW] Foundations of Physics 36 (7):1070-1098.
    We analyse the proposition that the spacetime structure is modified at short distances or at high energies due to weakening of classical logic. The logic assigned to the regions of spacetime is intuitionistic logic of some topoi. Several cases of special topoi are considered. The quantum mechanical effects can be generated by such semi-classical spacetimes. The issues of: background independence and general relativity covariance, field theoretic renormalization of divergent expressions, the existence and definition of path integral measures, are briefly discussed (...)
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  48. Luis M. Laita (1976). A Study of Algebraic Logic From the Point of View of Category Theory. Notre Dame Journal of Formal Logic 17 (1):89-118.
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  49. J. Lambek (1989). On Some Connections Between Logic and Category Theory. Studia Logica 48 (3):269 - 278.
    Categories may be viewed as deductive systems or as algebraic theories. We are primarily interested in the interplay between these two views and trace it through a number of structured categories and their internal languages, bearing in mind their relevance to the foundations of mathematics. We see this as a common thread running through the six contributions to this issue of Studia Logica.
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  50. Seremeti Lambrini & Kameas Achilles (forthcoming). Composable Relations Induced in Networks of Aligned Ontologies: A Category Theoretic Approach. Axiomathes:1-27.
    A network of aligned ontologies is a distributed system, whose components (constituent ontologies) are interacting and interoperating, the result of this interaction being, either the extension of local assertions, which are valid within each individual ontology, to global assertions holding between remote ontology syntactic entities (concepts, individuals) through a network path, or to local assertions holding between local entities of an ontology, but induced by remote ontologies, through a cycle in the network. The mechanism for achieving this interaction is the (...)
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