# Category Theory

Edited by Shay Logan (Kansas State University)
 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 1971. Textbooks aimed more at philosophical audiences include Goldblatt 2006, Awodey 2006, and McLarty 1991. 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. On Adjoint and Brain Functors.David Ellerman - 2016 - Axiomathes 26 (1):41-61.
There is some consensus among orthodox category theorists that the concept of adjoint functors is the most important concept contributed to mathematics by category theory. We give a heterodox treatment of adjoints using heteromorphisms that parses an adjunction into two separate parts. Then these separate parts can be recombined in a new way to define a cognate concept, the brain functor, to abstractly model the functions of perception and action of a brain. The treatment uses relatively simple category theory and (...)

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2. In this paper, we develop a mathematical model of awareness based on the idea of plurality. Instead of positing a singular principle, telos, or essence as noumenon, we model it as plurality accessible through multiple forms of awareness (“n-awareness”). In contrast to many other approaches, our model is committed to pluralist thinking. The noumenon is plural, and reality is neither reducible nor irreducible. Nothing dies out in meaning making. We begin by mathematizing the concept of awareness by appealing to the (...)

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3. How Category Theory Works.David Ellerman - manuscript
The purpose of this paper is to show that the dual notions of elements & distinctions are the basic analytical concepts needed to unpack and analyze morphisms, duality, and universal constructions in the Sets, the category of sets and functions. The analysis extends directly to other concrete categories (groups, rings, vector spaces, etc.) where the objects are sets with a certain type of structure and the morphisms are functions that preserve that structure. Then the elements & distinctions-based definitions can be (...)

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4. Saunders Mac Lane famously remarked that "Bourbaki just missed" formulating adjoints in a 1948 appendix (written no doubt by Pierre Samuel) to an early draft of Algebre--which then had to wait until Daniel Kan's 1958 paper on adjoint functors. But Mac Lane was using the orthodox treatment of adjoints that only contemplates the object-to-object morphisms within a category, i.e., homomorphisms. When Samuel's treatment is reconsidered in view of the treatment of adjoints using heteromorphisms or hets (object-to-object morphisms between objects in (...)

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5. Recent developments in pure mathematics and in mathematical logic have uncovered a fundamental duality between "existence" and "information." In logic, the duality is between the Boolean logic of subsets and the logic of quotient sets, equivalence relations, or partitions. The analogue to an element of a subset is the notion of a distinction of a partition, and that leads to a whole stream of dualities or analogies--including the development of new logical foundations for information theory parallel to Boole's development of (...)

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6. This paper shows how the universals of category theory in mathematics provide a model (in the Platonic Heaven of mathematics) for the self-predicative strand of Plato's Theory of Forms as well as for the idea of a "concrete universal" in Hegel and similar ideas of paradigmatic exemplars in ordinary thought. The paper also shows how the always-self-predicative universals of category theory provide the "opposite bookend" to the never-self-predicative universals of iterative set theory and thus that the paradoxes arose from having (...)

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7. This essay endeavors to define the concept of indefinite extensibility in the setting of category theory. I argue that the generative property of indefinite extensibility for set-theoretic truths in the category of sets is identifiable with the elementary embeddings of large cardinal axioms. A modal coalgebraic automata's mappings are further argued to account for both reinterpretations of quantifier domains as well as the ontological expansion effected by the elementary embeddings in the category of sets. The interaction between the interpretational and (...)
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8. This sketch of a perhaps future 'Elementary Theory of the Category of Mereological Sums (including Mereological Wholes and Parts)' relates to my previous papers "The Topos of Emergence" and "Intelligible Gunk". I assert that for successfully categorizing Mereology one has to start with a specific setting of gunk. In this paper we will give a sketch of a categorically version of particular mereological structures. I.e. we will follow the example of F.W.Lawvere’s “An elementary theory of the category of sets” -/- (...)

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9. The Elementary Process Theory (EPT) is a collection of seven elementary process-physical principles that describe the individual processes by which interactions have to take place for repulsive gravity to exist. One of the two main problems of the EPT is that there is no proof that the four fundamental interactions (gravitational, electromagnetic, strong, and weak) as we know them can take place in the elementary processes described by the EPT. This paper sets forth the method by which it can be (...)

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10. This Gentle Introduction is very much still work in progress. Roughly aimed at those who want something a bit more discursive, slower-moving, than Awodey's or Leinster's excellent books. -/- The current [Jan 2018] version is 291pp.
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11. Scientists and philosophers frequently speak about levels of description, levels of explanation, and ontological levels. This paper presents a framework for studying levels. I give a general definition of a system of levels and discuss several applications, some of which refer to descriptive or explanatory levels while others refer to ontological levels. I illustrate the usefulness of this framework by bringing it to bear on some familiar philosophical questions. Is there a hierarchy of levels, with a fundamental level at the (...)

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13. Yale Gallery Talk, Language Perception and Representation.PhD Tanya Kelley - forthcoming - Https://Drive.Google.Com/File/D/1YHzX_YR_wOWC3JUvfBcW7KucWX0Wlr_o/View.
Yale Gallery Talk, Language Perception and Representation Tanya Kelley and James Prosek Linguist and artist Tanya Kelley, Ph.D., and artist, writer, and naturalist James Prosek, B.A. 1997, discuss color manuals used by artist-naturalists and biologists and lead visitors in close looking and drawing. Presented in conjunction with the exhibition James Prosek: Art, Artifact, Artifice. Space is limited. Open to: General Public .

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14. Categories and the Foundations of Classical Field Theories.James Owen Weatherall - forthcoming - In Elaine Landry (ed.), Categories for the Working Philosopher. Oxford, UK: Oxford University Press.
I review some recent work on applications of category theory to questions concerning theoretical structure and theoretical equivalence of classical field theories, including Newtonian gravitation, general relativity, and Yang-Mills theories.

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15. Coalgebra And Abstraction.Graham Leach-Krouse - 2021 - Notre Dame Journal of Formal Logic 62 (1):33-66.
Frege’s Basic Law V and its successor, Boolos’s New V, are axioms postulating abstraction operators: mappings from the power set of the domain into the domain. Basic Law V proved inconsistent. New V, however, naturally interprets large parts of second-order ZFC via a construction discovered by Boolos in 1989. This paper situates these classic findings about abstraction operators within the general theory of F-algebras and coalgebras. In particular, we show how Boolos’s construction amounts to identifying an initial F-algebra in a (...)
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16. Impurity in Contemporary Mathematics.Ellen Lehet - 2021 - Notre Dame Journal of Formal Logic 62 (1):67-82.
Purity has been recognized as an ideal of proof. In this paper, I consider whether purity continues to have value in contemporary mathematics. The topics (e.g., algebraic topology, algebraic geometry, category theory) and methods of contemporary mathematics often favor unification and generality, values that are more often associated with impurity rather than purity. I will demonstrate this by discussing several examples of methods and proofs that highlight the epistemic significance of unification and generality. First, I discuss the examples of algebraic (...)

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17. Enriched Meanings: Natural Language Semantics with Category Theory.Ash Asudeh & Gianluca Giorgolo - 2020 - Oxford University Press.
This book develops a theory of enriched meanings for natural language interpretation that uses the concept of monads and related ideas from category theory. The volume is interdisciplinary in nature, and will appeal to graduate students and researchers from a range of disciplines interested in natural language understanding and representation.

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18. Choice-Free Stone Duality.Nick Bezhanishvili & Wesley H. Holliday - 2020 - Journal of Symbolic Logic 85 (1):109-148.
The standard topological representation of a Boolean algebra via the clopen sets of a Stone space requires a nonconstructive choice principle, equivalent to the Boolean Prime Ideal Theorem. In this article, we describe a choice-free topological representation of Boolean algebras. This representation uses a subclass of the spectral spaces that Stone used in his representation of distributive lattices via compact open sets. It also takes advantage of Tarski’s observation that the regular open sets of any topological space form a Boolean (...)

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19. Logical Rules as Fractions and Logics as Sketches.Dominique Duval - 2020 - Logica Universalis 14 (3):395-405.
In this short paper, using category theory, we argue that logical rules can be seen as fractions and logics as limit sketches.
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20. Review of Alain Badiou, The Pornographic Age. [REVIEW]Ekin Erkan - 2020 - Theory, Culture and Society 37.
This review of Alain Badiou’s The Pornographic Age—as well of the essays included in the book by William Watkin, A.J. Bartlett and Justin Clemens—illuminates that this is one of the few, if not only, texts where Badiou reverses the operational directionality of the event qua category theory, so as to “dis-image” power. In doing so, Badiou provides a theory of power based on intentionality and relation, rather than the more common Foucauldian genealogic-historical methodologies so often co-opted by contemporary thinkers of (...)

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21. A Generic Figures Reconstruction of Peirce’s Existential Graphs.Rocco Gangle, Gianluca Caterina & Fernando Tohme - 2020 - Erkenntnis 85:1-34.
We present a category-theoretical analysis, based on the concept of generic figures, of a diagrammatic system for propositional logic ). The straightforward construction of a presheaf category \ of cuts-only Existential Graphs provides a basis for the further construction of the category \ which introduces variables in a reconstructedly generic, or label-free, mode. Morphisms in these categories represent syntactical embeddings or, equivalently but dually, extensions. Through the example of Peirce’s system, it is shown how the generic figures approach facilitates the (...)

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22. Foundations of Mathematics: From Hilbert and Wittgenstein to the Categorical Unity of Science.Yoshihiro Maruyama - 2020 - In Shyam Wuppuluri & Newton da Costa (eds.), Wittgensteinian : Looking at the World From the Viewpoint of Wittgenstein's Philosophy. Springer Verlag. pp. 245-274.
Wittgenstein’s philosophy of mathematics is often devalued due to its peculiar features, especially its radical departure from any of standard positions in foundations of mathematics, such as logicism, intuitionism, and formalism. We first contrast Wittgenstein’s finitism with Hilbert’s finitism, arguing that Wittgenstein’s is perspicuous or surveyable finitism whereas Hilbert’s is transcendental finitism. We then further elucidate Wittgenstein’s philosophy by explicating his natural history view of logic and mathematics, which is tightly linked with the so-called rule-following problem and Kripkenstein’s paradox, yielding (...)

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23. Foundations of Mathematics: From Hilbert and Wittgenstein to the Categorical Unity of Science.Yoshihiro Maruyama - 2020 - In A. C. Grayling, Shyam Wuppuluri, Christopher Norris, Nikolay Milkov, Oskari Kuusela, Danièle Moyal-Sharrock, Beth Savickey, Jonathan Beale, Duncan Pritchard, Annalisa Coliva, Jakub Mácha, David R. Cerbone, Paul Horwich, Michael Nedo, Gregory Landini, Pascal Zambito, Yoshihiro Maruyama, Chon Tejedor, Susan G. Sterrett, Carlo Penco, Susan Edwards-Mckie, Lars Hertzberg, Edward Witherspoon, Michel ter Hark, Paul F. Snowdon, Rupert Read, Nana Last, Ilse Somavilla & Freeman Dyson (eds.), Wittgensteinian : Looking at the World From the Viewpoint of Wittgenstein’s Philosophy. Springer Verlag. pp. 245-274.
Wittgenstein’s philosophy of mathematics is often devalued due to its peculiar features, especially its radical departure from any of standard positions in foundations of mathematics, such as logicism, intuitionism, and formalism. We first contrast Wittgenstein’s finitism with Hilbert’s finitism, arguing that Wittgenstein’s is perspicuous or surveyable finitism whereas Hilbert’s is transcendental finitism. We then further elucidate Wittgenstein’s philosophy by explicating his natural history view of logic and mathematics, which is tightly linked with the so-called rule-following problem and Kripkenstein’s paradox, yielding (...)

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24. Metalogic is an open-ended cognitive, formal methodology pertaining to semantics and information processing. The language that mathematizes metalogic is known as metalanguage and deals with metafunctions purely by extension on patterns. A metalogical process involves an effective enrichment in knowledge as logical statements, and, since human cognition is an inherently logic–based representation of knowledge, a metalogical process will always be aimed at developing the scope of cognition by exploring possible cognitive implications reflected on successive levels of abstraction. Indeed, it is (...)

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25. Composition of Deductions Within the Propositions-As-Types Paradigm.Ivo Pezlar - 2020 - Logica Universalis (4):1-13.
Kosta Došen argued in his papers Inferential Semantics (in Wansing, H. (ed.) Dag Prawitz on Proofs and Meaning, pp. 147–162. Springer, Berlin 2015) and On the Paths of Categories (in Piecha, T., Schroeder-Heister, P. (eds.) Advances in Proof-Theoretic Semantics, pp. 65–77. Springer, Cham 2016) that the propositions-as-types paradigm is less suited for general proof theory because—unlike proof theory based on category theory—it emphasizes categorical proofs over hypothetical inferences. One specific instance of this, Došen points out, is that the Curry–Howard isomorphism (...)

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26. Creating New Concepts in Mathematics: Freedom and Limitations. The Case of Category Theory.Zbigniew Semadeni - 2020 - Philosophical Problems in Science 69:33-65.
In the paper we discuss the problem of limitations of freedom in mathematics and search for criteria which would differentiate the new concepts stemming from the historical ones from the new concepts that have opened unexpected ways of thinking and reasoning. We also investigate the emergence of category theory and its origins. In particular we explore the origins of the term functor and present the strong evidence that Eilenberg and Carnap could have learned the term from Kotarbiński and Tarski.

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27. Category Theory in the Hands of Physicists, Mathematicians, and Philosophers.Mariusz Stopa - 2020 - Philosophical Problems in Science 69:283-293.
Book review: Category Theory in Physics, Mathematics, and Philosophy, Kuś M., Skowron B., Springer Proc. Phys. 235, 2019, pp.xii+134.

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28. A Model-Theoretic Analysis of Fidel-Structures for mbC.Marcelo E. Coniglio & Aldo Figallo-Orellano - 2019 - In Can Başkent & Thomas Macaulay Ferguson (eds.), Graham Priest on Dialetheism and Paraconsistency. Springer Verlag. pp. 189-216.
In this paper, the class of Fidel-structures for the paraconsistent logic mbC is studied from the point of view of Model Theory and Category Theory. The basic point is that Fidel-structures for mbC can be seen as first-order structures over the signature of Boolean algebras expanded by two binary predicate symbols N and O satisfying certain Horn sentences. This perspective allows us to consider notions and results from Model Theory in order to analyze the class of mbC-structures. Thus, substructures, union (...)

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29. MES: A Mathematical Model for the Revival of Natural Philosophy.Andrée Ehresmann & Jean-Paul Vanbremeersch - 2019 - Philosophies 4 (1):9-0.
The different kinds of knowledge which were connected in Natural Philosophy have been later separated. The real separation came when Physics took its individuality and developed specific mathematical models, such as dynamic systems. These models are not adapted to an integral study of _living systems_, by which we mean evolutionary multi-level, multi-agent, and multi-temporality self-organized systems, such as biological, social, or cognitive systems. For them, the physical models can only be applied to the local dynamic of each co-regulator agent, but (...)

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30. The Logic in Philosophy of Science.Hans Halvorson - 2019 - Cambridge and New York: Cambridge University Press.
Major figures of twentieth-century philosophy were enthralled by the revolution in formal logic, and many of their arguments are based on novel mathematical discoveries. Hilary Putnam claimed that the Löwenheim-Skølem theorem refutes the existence of an objective, observer-independent world; Bas van Fraassen claimed that arguments against empiricism in philosophy of science are ineffective against a semantic approach to scientific theories; W. V. O. Quine claimed that the distinction between analytic and synthetic truths is trivialized by the fact that any theory (...)

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31. Eta-Rules in Martin-Löf Type Theory.Ansten Klev - 2019 - Bulletin of Symbolic Logic 25 (3):333-359.
The eta rule for a set A says that an arbitrary element of A is judgementally identical to an element of constructor form. Eta rules are not part of what may be called canonical Martin-Löf type theory. They are, however, justified by the meaning explanations, and a higher-order eta rule is part of that type theory. The main aim of this paper is to clarify this somewhat puzzling situation. It will be argued that lower-order eta rules do not, whereas the (...)

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32. Category Theory in Physics, Mathematics, and Philosophy.Marek Kuś & Bartłomiej Skowron (eds.) - 2019 - Springer Verlag.
The contributions gathered here demonstrate how categorical ontology can provide a basis for linking three important basic sciences: mathematics, physics, and philosophy. Category theory is a new formal ontology that shifts the main focus from objects to processes. The book approaches formal ontology in the original sense put forward by the philosopher Edmund Husserl, namely as a science that deals with entities that can be exemplified in all spheres and domains of reality. It is a dynamic, processual, and non-substantial ontology (...)

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33. Categories with Families and First-Order Logic with Dependent Sorts.Erik Palmgren - 2019 - Annals of Pure and Applied Logic 170 (12):102715.
First-order logic with dependent sorts, such as Makkai's first-order logic with dependent sorts (FOLDS), or Aczel's and Belo's dependently typed (intuitionistic) first-order logic (DFOL), may be regarded as logic enriched dependent type theories. Categories with families (cwfs) is an established semantical structure for dependent type theories, such as Martin-Löf type theory. We introduce in this article a notion of hyperdoctrine over a cwf, and show how FOLDS and DFOL fit in this semantical framework. A soundness and completeness theorem is proved (...)

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34. Categories of Scientific Theories.Hans Halvorson & Dimitris Tsementzis - 2018 - In Elaine Landry (ed.), Categories for the Working Philosopher. Oxford University Press.
We discuss ways in which category theory might be useful in philosophy of science, in particular for articulating the structure of scientific theories. We argue, moreover, that a categorical approach transcends the syntax-semantics dichotomy in 20th century analytic philosophy of science.

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35. Syntax-Semantics Interaction in Mathematics.Michael Heller - 2018 - Studia Semiotyczne 32 (2):87-105.
Mathematical tools of category theory are employed to study the syntax-semantics problem in the philosophy of mathematics. Every category has its internal logic, and if this logic is sufficiently rich, a given category provides semantics for a certain formal theory and, vice versa, for each formal theory one can construct a category, providing a semantics for it. There exists a pair of adjoint functors, Lang and Syn, between a category and a category of theories. These functors describe, in a formal (...)

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36. Filozoficznie prowokująca teoria kategorii.Michał Heller - 2018 - Philosophical Problems in Science 65:232-241.
Recenzja książki: Elaine Landry, _Categories for the Working Philosopher_, Oxford University Press, Oxford, 2017, ss. xiv+471.
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37. On Kalman’s Functor for Bounded Hemi-Implicative Semilattices and Hemi-Implicative Lattices.Ramon Jansana & Hernán Javier San Martín - 2018 - Logic Journal of the IGPL 26 (1):47-82.

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38. Canonical Maps.Jean-Pierre Marquis - 2018 - In Elaine Landry (ed.), Categories for the Working Philosophers. Oxford, UK: pp. 90-112.
Categorical foundations and set-theoretical foundations are sometimes presented as alternative foundational schemes. So far, the literature has mostly focused on the weaknesses of the categorical foundations. We want here to concentrate on what we take to be one of its strengths: the explicit identification of so-called canonical maps and their role in mathematics. Canonical maps play a central role in contemporary mathematics and although some are easily defined by set-theoretical tools, they all appear systematically in a categorical framework. The key (...)

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39. A Computable Functor From Graphs to Fields.Russell Miller, Bjorn Poonen, Hans Schoutens & Alexandra Shlapentokh - 2018 - Journal of Symbolic Logic 83 (1):326-348.
Fried and Kollár constructed a fully faithful functor from the category of graphs to the category of fields. We give a new construction of such a functor and use it to resolve a longstanding open problem in computable model theory, by showing that for every nontrivial countable structure${\cal S}$, there exists a countable field${\cal F}$of arbitrary characteristic with the same essential computable-model-theoretic properties as${\cal S}$. Along the way, we develop a new “computable category theory”, and prove that our functor and (...)

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40. Negating as Turning Upside Down.Bartłomiej Skowron & Wiesław Kubiś - 2018 - Studies in Logic, Grammar and Rhetoric 54 (1):115-129.
In order to understand negation as such, at least since Aristotle’s time, there have been many ways of conceptually modelling it. In particular, negation has been studied as inconsistency, contradictoriness, falsity, cancellation, an inversion of arrangements of truth values, etc. In this paper, making substantial use of category theory, we present three more conceptual and abstract models of negation. All of them capture negation as turning upside down the entire structure under consideration. The first proposal turns upside down the structure (...)

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41. The Modernity of Dedekind’s Anticipations Contained in What Are Numbers and What Are They Good For?J. Soliveres Tur & J. Climent Vidal - 2018 - Archive for History of Exact Sciences 72 (2):99-141.
We show that Dedekind, in his proof of the principle of definition by mathematical recursion, used implicitly both the concept of an inductive cone from an inductive system of sets and that of the inductive limit of an inductive system of sets. Moreover, we show that in Dedekind’s work on the foundations of mathematics one can also find specific occurrences of various profound mathematical ideas in the fields of universal algebra, category theory, the theory of primitive recursive mappings, and set (...)

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42. Teoria kategorii i niektóre jej logiczne aspekty.Mariusz Stopa - 2018 - Philosophical Problems in Science 64:7-58.
This article is intended for philosophers and logicians as a short partial introduction to category theory and its peculiar connection with logic. First, we consider CT itself. We give a brief insight into its history, introduce some basic definitions and present examples. In the second part, we focus on categorical topos semantics for propositional logic. We give some properties of logic in toposes, which, in general, is an intuitionistic logic. We next present two families of toposes whose tautologies are identical (...)
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43. Approximating Cartesian Closed Categories in NF-Style Set Theories.Morgan Thomas - 2018 - Journal of Philosophical Logic 47 (1):143-160.
I criticize, but uphold the conclusion of, an argument by McLarty to the effect that New Foundations style set theories don’t form a suitable foundation for category theory. McLarty’s argument is from the fact that Set and Cat are not Cartesian closed in NF-style set theories. I point out that these categories do still have a property approximating Cartesian closure, making McLarty’s argument not conclusive. After considering and attempting to address other problems with developing category theory in NF-style set theories, (...)

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44. The Quantum Field Theory (QFT) Dual Paradigm in Fundamental Physics and the Semantic Information Content and Measure in Cognitive Sciences.Gianfranco Basti - 2017 - In Gordana Dodig-Crnkovic & Raffaela Giovagnoli (eds.), Representation of Reality: Humans, Other Living Organism and Intelligent Machines. Springer.
In this paper we explore the possibility of giving a justification of the “semantic information” content and measure, in the framework of the recent coalgebraic approach to quantum systems and quantum computation, extended to QFT systems. In QFT, indeed, any quantum system has to be considered as an “open” system, because it is always interacting with the background fluctuations of the quantum vacuum. Namely, the Hamiltonian in QFT always includes the quantum system and its inseparable thermal bath, formally “entangled” like (...)

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45. Theories, Sites, Toposes: Relating and Studying Mathematical Theories Through Topos-Theoretic 'Bridges'.Olivia Caramello - 2017 - Oxford, England: Oxford University Press UK.
This book introduces a set of methods and techniques for studying mathematical theories and relating them to each other through the use of Grothendieck toposes.

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46. Duality as a Category-Theoretic Concept.David Corfield - 2017 - Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 59:55-61.
In a paper published in 1939, Ernest Nagel described the role that projective duality had played in the reformulation of mathematical understanding through the turn of the nineteenth century, claiming that the discovery of the principle of duality had freed mathematicians from the belief that their task was to describe intuitive elements. While instances of duality in mathematics have increased enormously through the twentieth century, philosophers since Nagel have paid little attention to the phenomenon. In this paper I will argue (...)

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47. The Future of Mathematics in Economics: A Philosophically Grounded Proposal.Ricardo Crespo & Fernando Tohmé - 2017 - Foundations of Science 22 (4):677-693.
The use of mathematics in economics has been widely discussed. The philosophical discussion on what mathematics is remains unsettled on why it can be applied to the study of the real world. We propose to get back to some philosophical conceptions that lead to a language-like role for the mathematical analysis of economic phenomena and present some problems of interest that can be better examined in this light. Category theory provides the appropriate tools for these analytical approach.

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48. Category Theory and Set Theory as Theories About Complementary Types of Universals.David P. Ellerman - 2017 - Logic and Logical Philosophy 26 (2):1-18.
Instead of the half-century old foundational feud between set theory and category theory, this paper argues that they are theories about two different complementary types of universals. The set-theoretic antinomies forced naïve set theory to be reformulated using some iterative notion of a set so that a set would always have higher type or rank than its members. Then the universal u_{F}={x|F(x)} for a property F() could never be self-predicative in the sense of u_{F}∈u_{F}. But the mathematical theory of categories, (...)

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49. "Quantum Field Theory and Coalgebraic Logic in Theoretical Computer Science.Giuseppe Vitiello Gianfanco Basti, Antonio Capolupo - 2017 - Progress in Biophysics and Molecular Biology 130:39-52.
In this paper we suggest that in the framework of the Category Theory it is possible to demonstrate the mathematical and logical dual equivalence between the category of the q-deformed Hopf Coalgebras and the category of the q-deformed Hopf Algebras in QFT, interpreted as a thermal field theory. Each pair algebra-coalgebra characterizes, indeed, a QFT system and its mirroring thermal bath, respectively, so to model dissipative quantum systems persistently in far-from-equilibrium conditions, with an evident significance also for biological sciences. The (...)
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50. The category-theoretic representation of quantum event structures provides a canonical setting for confronting the fundamental problem of truth valuation in quantum mechanics as exemplified, in particular, by Kochen–Specker’s theorem. In the present study, this is realized on the basis of the existence of a categorical adjunction between the category of sheaves of variable local Boolean frames, constituting a topos, and the category of quantum event algebras. We show explicitly that the latter category is equipped with an object of truth values, (...)