21. Categorical Logic Lambek, J. and Scott, P. Introduction to HigherOrder Categorical Logic. Borceux, F. Handbook of Categorical Algebra (Encyclopedia of Mathematics and http://www.andrew.cmu.edu/user/awodey/catlog/

Categorical Logic Fall 2009 Course Information Place: BH 150 Time: M 3:30 - 5:50 Instructor: Steve Awodey Office: Baker 152 (mail: Baker 135) Office Hour: Monday 1-2, or by appointment Phone: 8947 Email: awodey@andrew Secretary: Baker 135 Overview This course focuses on applications of category theory in logic and computer science. A leading idea is functorial semantics, according to which a model of a logical theory is a set-valued functor on a structured category determined by the theory. This gives rise to a syntax-invariant notion of a theory and introduces many algebraic methods into logic, leading naturally to the universal and other general models that distinguish functorial from classical semantics. Such categorical models occur, for example, in denotational semantics. In this connection the lambda-calculus is treated via the theory of cartesian closed categories. Similarly higher-order logic is modelled by the categorical notion of a topos. Using sheaves, topos theory also subsumes Kripke semantics for intuitionistic logic. Prerequisites 80-413/713 Category Theory, or equivalent.

22. Category Theory To be followed by a Fall course on categorical logic. Borceux Handbook http://www.andrew.cmu.edu/course/80-413-713/

Category Theory Fall 2010 Course Information Place: PH A22 Time: TR 1:30 - 2:50 Instructor: Steve Awodey Office: Baker 135F Office Hour: Monday 1-2, or by appointment Phone: x8947 Email: awodey@andrew Secretary: Baker 135 Webpage: www.andrew.cmu.edu/course/80-413-713 Overview Category theory, a branch of abstract algebra, has found many applications in mathematics, logic, and computer science. Like such fields as elementary logic and set theory, category theory provides a basic conceptual apparatus and a collection of formal methods useful for addressing certain kinds of commonly occurring formal and informal problems, particularly those involving structural and functional considerations. This course is intended to acquaint students with these methods, and also to encourage them to reflect on the interrelations between category theory and the other basic formal disciplines. To be followed by a Fall course on categorical logic. Prerequisites Some familiarity with abstract algebra or logic. Texts Course notes will be provided.

23. Categorical Logic - Wikipedia, The Free Encyclopedia This has enabled proofs of metatheoretical properties of some logics by means of an appropriate categorical algebra. For instance, Freyd gave a proof of http://en.wikipedia.org/wiki/Categorical_logic

Categorical logic From Wikipedia, the free encyclopedia Jump to: navigation search This article is about mathematical logic in the context of category theory. For Aristotle's system of logic, see Term logic Categorical logic is a branch of category theory within mathematics , adjacent to mathematical logic but more notable for its connections to theoretical computer science . In broad terms, categorical logic represents both syntax and semantics by a category , and an interpretation by a functor . The categorical framework provides a rich conceptual background for logical and type-theoretic constructions. The subject has been recognisable in these terms since around 1970. edit Overview There are three important themes in the categorical approach to logic: Categorical semantics . Categorical logic introduces the notion of structure valued in a category C with the classical model theoretic notion of a structure appearing in the particular case where C is the category of sets and functions . This notion has proven useful when the set-theoretic notion of a model lacks generality and/or is inconvenient. R.A.G. Seely

24. Category Theory - Wikipedia, The Free Encyclopedia Categorical logic is now a welldefined field based on type theory for http://en.wikipedia.org/wiki/Category_theory

Category theory From Wikipedia, the free encyclopedia Jump to: navigation search A category with objects X, Y, Z and morphisms f g This article includes a list of references , related reading or external links , but its sources remain unclear because it lacks inline citations Please improve this article by introducing more precise citations where appropriate (November 2009) Category theory is an area of study in mathematics that deals in an abstract way with mathematical structures and relationships between them: it abstracts from sets and functions respectively to objects linked in diagrams by morphisms or arrows One of the simplest examples of a category (which is a very important concept in topology ) is that of groupoid , defined as a category whose arrows or morphisms are all invertible. Categories now appear in most branches of mathematics, some areas of theoretical computer science where they correspond to types , and mathematical physics where they can be used to describe vector spaces . Categories were first introduced by Samuel Eilenberg and Saunders Mac Lane in 1942–45, in connection with

25. Algebra And Logic File Format PDF/Adobe Acrobat Quick View http://www.dpmms.cam.ac.uk/~martin/Research/Slides/algebralogic.pdf

26. Springer Online Reference Works A branch of mathematics dealing with the interaction between logic (cf. also Mathematical .. Appl. Categorical Algebra , Amer. Math. Soc. (1970) pp. 1–14 http://eom.springer.de/c/c120060.htm

27. IngentaConnect Categorical Abstract Algebraic Logic Algebraizable by G Voutsadakis 2002 - Cited by 29 - Related articles http://www.ingentaconnect.com/content/klu/apcs/2002/00000010/00000006/05090913

28. CiteSeerX — Categorical Abstract Algebraic Logic Gentzen by G Voutsadakis 2005 - Cited by 1 - Related articles http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.85.4261

29. CiteSeerX — Categorical Semantics Of Linear Logic Paul-André by E Preuves 2007 http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.62.5117

30. Axiomatic And Categorical Foundations Of Mathematics I I 30 Mac Lane, S., 1971, Categorical algebra and SetTheoretic Sets, Topoi, and Internal Logic in Categories, Studies in Logic and the Foundations of http://myyn.org/m/article/axiomatic-and-categorical-foundations-of-mathematics-i

31. Borceux F Handbook Of Categorical Algebra Vol 2 - File Search (788145 Results) Files borceux f handbook of categorical algebra vol 3 categories sheaves cup . djvu, /handbookof-philosophical-logic-volume-1-7-2nd-edition_3468.html http://rapiddigger.com/borceux-f-handbook-of-categorical-algebra-vol-2/

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32. The Temporal Logic Of Coalgebras Via Galois Algebras by B Jacobs 2002 - Cited by 56 - Related articles http://portal.acm.org/citation.cfm?id=966840

33. Category Theory (Stanford Encyclopedia Of Philosophy) by JP Marquis 2010 - Cited by 17 - Related articles http://plato.stanford.edu/entries/category-theory/

Cite this entry Search the SEP Advanced Search Tools ... Please Read How You Can Help Keep the Encyclopedia Free Category Theory First published Fri Dec 6, 1996; substantive revision Thu Feb 25, 2010 1. General Definitions, Examples and Applications 2. Brief Historical Sketch 3. Philosophical Significance Bibliography ... Related Entries 1. General Definitions, Examples and Applications 1.1 Definitions et al . 2000, 2001, 2002). The very definition of a category is not without philosophical importance, since one of the objections to category theory as a foundational framework is the claim that since categories are defined Definition : A mapping e will be called an identity if and only if the existence of any product e e implies that e e Definition C is an aggregate Ob of abstract elements, called the objects of C , and abstract elements Map , called mappings of the category. The mappings are subject to the following five axioms: is defined. When either is defined, the associative law are defined. e e is defined, and at least one identity e such that e (C4) The mapping e X corresponding to each object X is an identity.

34. Information-based Complexity, By J. F. Traub, G. W. Wasilkowski Your browser may not have a PDF reader available. Google recommends visiting our text version of this document. http://www.ams.org/bull/1989-21-02/S0273-0979-1989-15851-5/S0273-0979-1989-15851

35. Categorical Logic : Definition Of Categorical Logic And Synonym Of Categorical L This has enabled proofs of metatheoretical properties of some logics by means of an appropriate categorical algebra. For instance, Freyd gave a proof of http://dictionary.sensagent.com/categorical logic/en-en/

36. Categorical Abstract Algebraic Logic: Subdirect Representation Of Pofunctors - C asi nacute ska and Pigozzi developed a theory of partially ordered varieties and quasivarieties of algebras with the goal of addressing issues pertaining http://www.informaworld.com/smpp/56763741-7167005/content~db=all~content=a770777

37. Chung-Kil Hur (허충길) My specific interests include category theory, especially categorical algebra; equational logic and term rewriting; stepindexing, logical relation and http://www.mpi-sws.org/~gil/

Chung-Kil Hur (허충길) MPI-SWS Campus E1.4 Germany gil#mpi-sws:org where # is @ and : is . Tel: +49 (0)681 9303 8711 From Oct. 2010, I am a postdoc researcher in Max Planck Institute for Software Systems From Nov. 2009 to Sep. 2010, I was a postdoc researcher in CNRS I received my phd degree from Computer Laboratory at University of Cambridge under the supervison of Marcelo Fiore My specific interests include: category theory, especially categorical algebra; equational logic and term rewriting; step-indexing, logical relation and compositional compiler correctness; Coq proof assistant. I won a bronze medal in the International Mathematical Olympiad in 1994. Tools Heq A Coq library to support reasoning about Heterogeneous Equality. Thesis Categorical equational systems: algebraic models and equational reasoning Computer Laboratory, University of Cambridge, UK, 2010 [summary: pdf ] [thesis: pdf Publications A kripke logical relation between ML and assembly (with Derek Dreyer To appear in the 38th ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages ( POPL 2011 [preprint: pdf ] [appendix: pdf Mathematical synthesis of equational logic (with Marcelo Fiore To appear in Logical Method in Computer Science.

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39. George VOUTSADAKIS CATEGORICAL ABSTRACT ALGEBRAIC LOGIC File Format PDF/Adobe Acrobat Quick View http://www.iphils.uj.edu.pl/rml/rml-44/07-voutsadakis.pdf

40. Notes On: The Logical Foundations Of Mathematics Mar 16, 1995 Chapter 1 First Order Logic A presentation of first order logic including a general treatment of Chapter 8 - Categorial Algebra http://www.rbjones.com/rbjpub/philos/bibliog/hatch82.htm

by on The Logical Foundations of Mathematics by William S. Hatcher Chapter 1 - First Order Logic A presentation of first order logic including a general treatment of "variable binding term operators", such as set comprehension, which are often required in foundation systems. Chapter 2 - The Origin of Modern Foundational Studies Chapter 3 - Frege's System and the Paradoxes Chapter 4 - The Theory of Types Chapter 5 - Zermelo Fraenkel Set Theory Chapter 7 - The Foundational Systems of W.V.Quine Chapter 8 - Categorical Algebra Chapter 2 - The Origin of Modern Foundational Studies Section 1 - Mathematics as an Independent Science Section 2 - The Arithmetisation of Analysis Section 3 - Constructivism Section 4 - Frege and the notion of a Formal System Section 5 - Criteria of Foundations "What must a foundation be, and what must it do?" Chapter 8 - Categorial Algebra Introduction In the introduction, Hatcher describes the relevance of category theory for foundations. Section 8.1 The notion of a category An informal introduction to category theory. Section 8.2

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