About this topic
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 1995. 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. 
Related categories
Siblings:History/traditions: Category Theory

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  1. Logic and Categories as Tools for Building Theories.Samson Abramsky - 2010 - Journal of the Indian Council of Philosophical Research 27 (1).
  2. On Transferring Model Theoretic Theorems of $${\Mathcal{L}_{{\Infty},\Omega}}$$ L ∞, Ω in the Category of Sets to a Fixed Grothendieck Topos.Nathanael Leedom Ackerman - 2014 - 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 (...)
  3. The Modesty Topos and John of Damascus as a Not-so-Modest Author.Alexander Alexakis - 2005 - Byzantinische Zeitschrift 97 (2):521-530.
    Byzantine authors frequently used the well-known topos of modesty in the opening lines of their literary works. This common introduction, usually served two purposes: The authors expressed a genuine, or, perhaps, feigned concern about their ability to deal adequately with their subject both in terms of form and substance, and they preemptively tried to thwart any possible criticism on the part of the audience for any shortcomings in their work by beginning with this sort of captatio benevolentiae. Typical examples of (...)
  4. Axiomatic Method and Category Theory.Rodin Andrei - unknown
    Lawvere’s axiomatization of topos theory and Voevodsky’s axiomatization of heigher homotopy theory exemplify a new way of axiomatic theory-building, which goes beyond the classical Hibert-style Axiomatic Method. The new notion of Axiomatic Method that emerges in Categorical logic opens new possibilities for using this method in physics and other natural sciences.
  5. Transfer Principles in Nonstandard Intuitionistic Arithmetic.Jeremy Avigad & Jeffrey Helzner - 2002 - 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 (...)
  6. Category Theory.S. Awodey - 2007 - Bulletin of Symbolic Logic 13 (3):371-372.
  7. 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 (...)
  8. 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.
  9. 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.
  10. REVIEWS-Category Theory.S. Awodey & Jiri Rosicky - 2007 - Bulletin of Symbolic Logic 13 (3).
  11. 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 (...)
  12. 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 (...)
  13. 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 (...)
  14. 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.
  15. 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 (...)
  16. 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 (...)
  17. Category Theory.Steve Awodey - 2007 - Studia Logica 86 (1):133-135.
  18. Continuity and Logical Completeness: An Application of Sheaf Theory and Topoi.Steve Awodey - 2006 - In Johan van Benthem, Gerhard Heinzman, M. Rebushi & H. Visser (eds.), The Age of Alternative Logics. Springer. pp. 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 (...)
  19. Notes on Algebraic Set Theory.Steve Awodey - unknown
    Steve Awodey. Notes on Algebraic Set Theory.
  20. 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]Steve Awodey - 2005 - Bulletin of Symbolic Logic 11 (1):65-69.
  21. 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.
  22. Sheaf Representation for Topoi.Steve Awodey - unknown
    Steve Awodey. Sheaf Representation for Topoi.
  23. Elementary Axioms for Local Maps of Toposes.Steve Awodey & Lars Birkedal - unknown
    We present a complete elementary axiomatization of local maps of toposes.
  24. 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 (...)
  25. 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.
  26. 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 (...)
  27. 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.
  28. 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 (...)
  29. 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.
  30. 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 (...)
  31. Axiom of Choice and Excluded Middle in Categorical Logic.Steven Awodey - 1995 - Bulletin of Symbolic Logic 1:344.
  32. Logic in Topoi: Functorial Semantics for High-Order Logic.Steven M. Awodey - 1997 - 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 (...)
  33. An Outline of Algebraic Set Theory.Steve Awody - manuscript
  34. Extended Memory Evolutive Systems in a Hyperstructure Context.Nils A. Baas - 2009 - 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 (...)
  35. The Completeness of a Predicate-Functor Logic.John Bacon - 1985 - Journal of Symbolic Logic 50 (4):903-926.
  36. Complex Non-Linear Biodynamics in Categories, Higher Dimensional Algebra and ŁUkasiewicz''“Moisil Topos: Transformations of Neuronal, Genetic and Neoplastic Networks.I. C. Baianu, R. Brown, G. Georgescu & J. F. Glazebrook - 2006 - Axiomathes 16 (1):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 (...)
  37. Category-Theoretic Structure and Radical Ontic Structural Realism.Jonathan Bain - 2013 - 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 (...)
  38. Excluded Middle Versus Choice in a Topos.B. Banaschewski - 2005 - 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.
  39. A Globalisation of the Gelfand Duality Theorem.Bernhard Banaschewski & Christopher J. Mulvey - 2006 - 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.
  40. Symmetry, Compact Closure and Dagger Compactness for Categories of Convex Operational Models.Howard Barnum, Ross Duncan & Alexander Wilce - 2013 - 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 (...)
  41. Ringing in the Sheaves. [REVIEW]A. Bartlett - 2006 - Cosmos and History 2 (1-2):339-344.
  42. Categorical Modelling of Husserl's Intentionality.Imants Barušs - 1989 - Husserl Studies 6 (1):25-41.
    This paper is concerned with the application of constructions from category theory to Smith and McIntyre's interpretation of Husserl's intentionality. 1 Not only did Hussefl's own ideas change in the course of his lifetime 2 but there are a number of interpretations of Husserl's work 3 so that the line of philosophical investigation that Husserl strongly influenced is still in the process of development. In this vein, Smith and McIntyre have recognized the potential for a possible worlds interpretation of intentionality (...)
  43. Theories, Theorizers and the World: A Category-Theoretic Approach.Vadim Batitsky - 1996 - 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 (...)
  44. Review: J. Lambek, P. J. Scott, Introduction to Higher Order Categorical Logic. [REVIEW]J. L. Bell - 1989 - Journal of Symbolic Logic 54 (3):1113-1114.
  45. From Absolute to Local Mathematics.J. L. Bell - 1986 - 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.
  46. Categories, Toposes and Sets.J. L. Bell - 1982 - 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.
  47. Category Theory and the Foundations of Mathematics.J. L. Bell - 1981 - British Journal for the Philosophy of Science 32 (4):349-358.
  48. Two Approaches to Modelling the Universe: Synthetic Differential Geometry and Frame-Valued Sets.John L. Bell - unknown
    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 (...)
  49. Types, Sets and Categories.John L. Bell - unknown
    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, (...)
  50. Notes on Toposes and Local Set Theories.John L. Bell - unknown
    This book is written for those who are in sympathy with its spirit. This spirit is different from the one which informs the vast stream of European and American civilization in which all of us stand. That spirit expresses itself in an onwards movement, in building ever larger and more complicated structures; the other in striving in clarity and perspicuity in no matter what structure. The first tries to grasp the world by way of its periphery—in its variety; the second (...)
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