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         Categorical Algebra And Logic:     more books (15)
  1. Handbook of Categorical Algebra: Volume 1, Basic Category Theory (Encyclopedia of Mathematics and its Applications) (v. 1) by Francis Borceux, 2008-04-24
  2. Algebraic Theories: A Categorical Introduction to General Algebra (Cambridge Tracts in Mathematics) by J. Adámek, J. Rosický, et all 2010-12-31
  3. Realizability, Volume 152: An Introduction to its Categorical Side (Studies in Logic and the Foundations of Mathematics) by Jaap van Oosten, 2008-04-24
  4. Sheaves, Games, and Model Completions: A Categorical Approach to Nonclassical Propositional Logics (Trends in Logic) by Silvio Ghilardi, M. Zawadowski, 2002-07-31
  5. Categorical Logic and Type Theory, Volume 141 (Studies in Logic and the Foundations of Mathematics) by B. Jacobs, 2001-05-24
  6. Goguen Categories: A Categorical Approach to L-fuzzy Relations (Trends in Logic) by Michael Winter, 2010-11-02
  7. Categorical Foundations: Special Topics in Order, Topology, Algebra, and Sheaf Theory (Encyclopedia of Mathematics and its Applications)
  8. Sheaves, Games, and Model Completions: A Categorical Approach to Nonclassical Propositional Logics (Trends in Logic) by Silvio Ghilardi, M. Zawadowski, 2010-11-02
  9. Categorical Closure Operators by Gabriele Castellini, 2003-05-15
  10. Categorical Topology
  11. Categorical Methods in Computer Science: With Aspects from Topology (Lecture Notes in Computer Science)
  12. From a Geometrical Point of View: A Study of the History and Philosophy of Category Theory (Logic, Epistemology, and the Unity of Science) by Jean-Pierre Marquis, 2008-12-05
  13. Introduction to Higher-Order Categorical Logic (Cambridge Studies in Advanced Mathematics) by J. Lambek, P. J. Scott, 1986-07-25
  14. Categorical Structure of Closure Operators: With Applications to Topology, Algebra and Discrete Mathematics (Mathematics and Its Applications) by D. Dikranjan, Walter Tholen, 1995-10-31

1. Categorical Logic
Instructor Steve Awodey Office Baker 152 (mail Baker 135) Office Hour Thursday 12, or by appointment Secretary Baker 135 This course focuses on applications of category theory in logic and computer science. categorical logic. 80-520/820 Borceux, F. Handbook of categorical algebra (Encyclopedia of Mathematics and its Applications).
Categorical Logic
Fall 2002
Course Information
Instructor: Steve Awodey
Office: Baker 152 (mail: Baker 135)
Office Hour: Thursday 1-2, or by appointment
Phone: 8947
Email: awodey@andrew
Secretary: Baker 135
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.
80-413/713 Category Theory, or equivalent.

Research topics include mathematical models and theories in the empirical sciences, models and theories Category Society Philosophy Philosophy of Mathematics...... Handbook of categorical algebra 3Categories of Sheaves. Locales, Sheaves, GrothendieckToposes, The Classifying Topos,Elementary Toposes, Internal logic of a
Research topics include mathematical models and theories in the empirical sciences, models and theories in mathematics, category theory, and the use of mathematical structures in theoretical computer science.
Research Bibliography
Mathematical Theories and Models
Scientific Theories and Models
Category Theory
Theoretical Computer Science ... WWW Research Sites
Mathematical Theories and Models
  • Agazzi and Darvas. Philosophy of Mathematics Today. Kluwer Academic Publishers, 1997
  • Anglin and Lambek. The Heritage of Thales. Springer-Verlag, 1995
  • Akin, Ethan. The General Topology of Dynamical Systems. American Mathematical Society, 1993
  • Barwise, Jon. (ed) Handbook of Mathematical Logic. North-Holland,1977
  • Barwise, Jon. "Axioms for Abstract Model Theory" ,Annals of Mathematical Logic 7(1974) 221-265.
  • Bell, John and Machover,Moshe. A Course in Mathematical Logic. North-Holland, 1977
  • Bridge, Jane. Beginning Model Theory. Clarendon Press, 1977
  • Burgess, John and Rosen, Gifeon. A Subject with No Object Oxford Press, 1997

3. MAA-Amazon Math Book List: Logic
Modern algebra and the Rise of Mathematical A logical Approach (Oxford logic Guides,No 16 Uncountably categorical Theories (Translations of Mathematical
Category Theory
Abstract and Concrete Categories : The Joy of Cats Ships in 2-3 days Jiri Adamek, et al / Hardcover / Published 1990
Our Price: $113.00
Read more about this title...
The Theory of Quantaloids (Pitman Research Notes in Mathematics, No 348) Kimmo I. Rosenthal / Paperback / Published 1996
Our Price: $40.33 Algebra Vol 190 Carl Clifton, Faith / Hardcover / Published 1981
Our Price: $54.00 (Special Order) Algebra in a Localic Topos With Application to Ring Theory Francis Borceux / Paperback / Published 1983
Our Price: $13.00 (Special Order) Applications of Category Theory to Fuzzy Subsets (Theory and Decision Library. Series B, Mathematical and Statistical Methods, Vol. 14) Stephen Ernest Rodabaugh, et al / Hardcover / Published 1992
Our Price: $69.30 ~ You Save: $29.70 (30%) (Special Order) Approach Spaces : The Missing Link in the Topology-Uniformity-Metric Triad (Oxford Mathematical Monographs) Robert Lowen / Hardcover / Published 1997
Our Price: $115.00 (Not Yet Published) Automata and Algebras in Categories (Mathematics and Its Applications; East European Series, No 37)

4. Handbook Of Categorical Algebra - Cambridge University Press
This document provides an introduction to the interaction between category theory and mathematical logic which is slanted towards computer scientists. categorical and algebraic aspects of martinlof type theory (context) - Obtu - 1989. 1 with English summary (context) - Minc, the et al. - 1977. 1 the unity of algebra and logic
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Handbook of Categorical Algebra
Volume 3 , Sheaf Theory
Francis Borceux
In stock
Preface; Introduction to the handbook; 1. Locales; 2. Sheaves; 3. Grothendieck toposes; 4. The classifying topos; 5. Elementary toposes; 6. Internal logic of a topos; 7. The law of excluded middle; 8. The axiom of infinity; 9. Sheaves in a topos; Index.
Cambridge University Press 2001. Security
Order by phone (+44 (0)1223 326050) or fax (+44 (0)1223 326111).

5. Categorical Geometry Homepage
This site contains online books and research papers on the subjects of categorical algebra, categorical logic, categorical geometry, lattice theory, universal algebra, algebraic geometry. categorical Geometry. categorical algebra. 1 0. categorical logic. 0 1. categorical Geometry

6. Springer NY Categorical Algebra And Its Applications
categorical algebra and its applications contain several fundamental papers on general category theory, by the top specialists in the field, and many interesting papers on the applications of category theory in functional analysis, algebraic Go to the Mathematics home page. categorical algebra and its Applications ring theory, cohomology, differential geometry, group theory, mathematical logic and computer sciences.

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Go to the Mathematics home page Categorical Algebra and its Applications Proceedings of a Conference, Held in Louvain-la-Neuve, Belgium, July 26 - August 1, 1987 Series: Lecture Notes in Mathematics , Vol. 1348 Edited by Francis Borceux , Catholique University of Louvain, Louvain-la-Neuve, Belgium Price: $52.95 375 pages hardcover ISBN: 0-387-50362-5, published 1988 Availability: Out of stock. Order now, and we will ship when available (your card will not be charged until we ship). (Why you should order now) Error: Error: expected Error: Error: expected [/BROWSER], but found [/SHOWIF] instead[/SHOWIF], but found Error: Error: expected [/BROWSER], but found [/!] instead[/!] insteadError: Error: expected [/BROWSER], but found [/!] instead[/!] EMAIL THIS PAGE ABOUT THIS BOOK Categorical algebra and its applications contain several fundamental papers on general category theory, by the top specialists in the field, and many interesting papers on the applications of category theory in functional analysis, algebraic topology, algebraic geometry, general topology, ring theory, cohomology, differential geometry, group theory, mathematical logic and computer sciences. The volume contains 28 carefully selected and refereed papers, out of 96 talks delivered, and illustrates the usefulness of category theory today as a powerful tool of investigation in many other areas.

7. Introduction
Lambek continued his work in ring theory and categorical algebra (often in Therethen followed a fruitful collaboration on categorical logic with P. Scott
Introduction to MSCS volume
This volume is dedicated to our friend, colleague, and teacher Joachim (Jim) Lambek. On December 5,1997, a small conference was held at McGill on the occasion of Jim Lambek's 75th birthday. Subsequently it was decided to publish two volumes of papers contributed in his honour to mark this occasion; this issue of Mathematical Structures in Computer Science is one of the volumes, the other is Volume 6 of the journal Theory and Applications of Categories . At the December 1997 conference, a brief biographical essay was presented by Michael Barr; that essay appears in the TAC volume. But we wish to make some further remarks here. Jim completed his Ph.D. at McGill under Hans Zassenhaus in 1950, and has remained at McGill since then. But it is of interest to note that Jim wrote two theses: the second involved biquaternions in mathematical physics, and so forshadows a significant feature of his career: Jim has consistently shown a remarkable range of interests, from physics to linguistics, from algebra to logic, from the history and philosophy of mathematics to the theory of computing science (although he never touches a computer, to this day!). Let us just review a small sample of his more than 100 published papers. In the mid 1960's Lambek became increasingly interested in category theory. His first monograph, "Completions of Categories" (Springer LNM

8. Logic In New York City
foundations of logic programming. Semantics Universal algebra; category theory; categorical model theory; classical
Logic at CUNY
and the New York Metropolis to the Logic Workshop
Seminars and Conferences:
Logic Worldwide
Logic in the City
There is an enormous variety of logic going on at the City University of New York, from large cardinals and forcing in set theory to models of arithmetic to philosophical logic and theoretical computer science. In all this diversity of interests and expertise, the CUNY logic faculty combines the resources of experienced, established researchers with those of younger up-and-coming logicians to create a productive, active research atmosphere. The CUNY Logic Workshop, for example, is a weekly research seminar here at the CUNY Graduate Center that hosts distinguished speakers from around the New York area and around the world. Additional logic talks occasionally supplement the seminar. The CUNY-administered Mid-Atlantic Mathematical Logic Seminar (MAMLS), a traveling conference meeting up to four times yearly at various east-coast universities, has recently been held once each year at CUNY. In addition, CUNY logicians are active in the Rutgers Logic Seminar. Plans are underway for the New York Logic Colloquium, a monthly lecture series that will bring distinguished speakers in logic to New York City.

9. WebGuest - Open Directory : Science : Math : Algebra : Category Theory
Homepage This site contains online books and research papers on the subjects ofcategorical algebra, categorical logic, categorical geometry, lattice theory
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10. Categorical Logic - Pitts (ResearchIndex)
1988 1 categorical logic (context) Crole, Types et al. - 1993 1 categoricaland algebraic 1977 1 the unity of algebra and logic (context) - Lambek
Categorical Logic (2001) (Make Corrections) (26 citations)
Andrew M. Pitts Handbook of Logic in Computer Science, Volume 5: Algebraic and Logical Structures
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Abstract: This document provides an introduction to the interaction between category theory and mathematical logic which is slanted towards computer scientists. (Update) Context of citations to this paper: More ...notion of internal language theorems. 4. 1 Internal languages Introductions to categorical logic such as Lambek and Scott [17] and Pitts , give examples of internal languages for speci c categories, but no generic de nition of what it is to provide the internal language of a Alternative approaches to the semantics of (intuitionistic) dependent types are presented in, for example, Cartmell [4] Pitts These presentations lack the conceptual distinction provided by indexed categories. We need the base category C to account for the structural...

11. Rewriting Properties Of Combinators For Intuitionistic Linear Logic - Nesi, De P
lambda calculus Maria Emilia Maietti (Correct) 0.3 Relating categorical Semanticsfor In Proceedings of the Workshop on Higher Order algebra, logic and Term
Rewriting Properties of Combinators for Intuitionistic Linear Logic (1994) (Make Corrections) (1 citation)
Monica Nesi, Valeria de Paiva, Eike Ritter
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(Enter summary) Abstract: this paper we investigate the possibility of developing a (semi-)automatic rewriting tool for manipulating and reasoning about combinators for Intuitionistic Linear Logic. In particular, we develop a canonical (i.e. confluent and terminating) term rewriting system associated to a theory of categorical combinators for (rudimentary) Linear Logic. In order to do that, we make use of the Knuth-Bendix completion algorithm [3] to transform the equational theory for the combinators into an equivalent... (Update) Context of citations to this paper: More The calculus appears in a companion paper [GdPR98] and some of the rewriting properties of (a fragment of) the system are described in In this paper we concentrate on the more refined categorical models for the linear calculus extended with explicit substitutions....

12. KLUWER Academic Publishers | Foundations, Sets And Categories
algebra and logic Yuri L Applied categorical Structures A Journal Devoted to Applicationsof categorical Methods in algebra, Analysis, Order, Topology and
Title Authors Affiliation ISBN ISSN advanced search search tips Home Browse by Subject ... Mathematics Foundations, Sets and Categories
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Publication Date

A Primer on Determinism

John Earman
November 1986, ISBN 90-277-2241-2, Paperback
Out of Print
A Primer on Determinism

John Earman
August 1986, ISBN 90-277-2240-4, Hardbound Printing on Demand Price: 166.50 EUR / 210.00 USD / 126.75 GBP Add to cart A Short Introduction to Intuitionistic Logic Grigori Mints June 2002, ISBN 0-306-46975-8, eBook Price: 87.00 EUR / 82.50 USD A Short Introduction to Intuitionistic Logic Grigori Mints October 2000, ISBN 0-306-46394-6, Hardbound Price: 64.00 EUR / 60.50 USD / 41.75 GBP Add to cart Algebra and Logic Yuri L. Ershov, Sergei S. Goncharov 2003, Volume 42 (6 issues) Price: 1960.00 EUR / 1950.00 USD ISSN: 0002-5232 Add to cart Applied Categorical Structures A Journal Devoted to Applications of Categorical Methods in Algebra, Analysis, Order, Topology and Computer Science R. Lowen 2003, Volume 11 (6 issues) Price: 363.00 EUR / 363.00 USD ISSN: 0927-2852 Add to cart Architecture of Systems Problem Solving, Second Edition

13. KLUWER Academic Publishers | Category Theory, Homological Algebra
Universal algebra, algebraic logic, and Databases B Theory of Topological StructuresAn Approach to categorical Topology Gerhard Preuss December 1987

14. Research
FE J. Linton, Ph.D. Columbia categorical algebra, functorial semantics, topoi.James Lipton, Ph.D. Cornell logic and computation, logic programming, type

Faculty, Staff, and Graduate Students
Undergrad Math Program Calendar of Events Graduate Program ... Contact Us
Some of the areas of research currently active in the department are:
  • Algebra and number theory Algebraic topology and category theory Bioinformatics Combinatorics and complexity Discrete groups, complex and geometric analysis Ergodic theory and topological dynamics General and low dimensional topology Logic and theoretical computer science
There are weekly seminars in most of the above areas as well as a regular departmental colloquium series and a weekly graduate-student-run lunchtime seminar. The following list describes the research interests of individual faculty members and current visitors. A list of faculty webpages is located here. Petra Bonfert-Taylor , Ph.D. Technical University of Berlin

15. Graduate Mathematics And Computer Science Program
algebraic topology, analysis of algorithms, categorical algebra, combinatorics,complex analysis, computational logic, data mining, ergodic theory, geometric

Faculty, Staff, and Graduate Students
Undergrad Math Program Calendar of Events Graduate Program ... Contact Us Graduate Program Programs of Study Courses Research Areas Facilities ... More Information The Department's graduate programs include a Ph. D. program in mathematics and M. A. programs in mathematics and in computer science. The research emphasis at Wesleyan is in pure mathematics and theoretical computer science. One of the distinctive features of our department is the close interaction between the computer science faculty and the mathematics faculty, particularly those in logic and discrete mathematics. Among possible fields of specialization for Ph.D. candidates are algebraic topology, analysis of algorithms, categorical algebra, combinatorics, complex analysis, computational logic, data mining, ergodic theory, geometric analysis, general topology, graph theory, homological algebra, Kleinian groups and discrete groups, lattice-ordered algebraic structures, logic programming, mathematical physics, model theory, model-theoretic algebra, number theory, operator algebras, probability theory, proof theory, topological dynamics, and topological groups.

16. Categorical Logic, Also Called “Aristotelian Logic” And “syllogistic Logic”
the algebra of logic how to handle proper names ( Socrates is mortal ), the oddityof 1+1=1, the existential quantifier, and the relationship of categorical of logic.html
A Brief Summary of the History and Philosophy of Logic: Categorical, Propositional, and Predicate Logic Categorical logic , also called "Aristotelian logic" and "syllogistic logic" has been around at least since Aristotle’s discovery of it (reported in Prior Analytics ) around 335 B.B. Propositional logic became an important part of Stoic philosophy, and was developed by the Stoics around 225 B.C. The Stoics were primarily concerned with conditionals—"if…then…" statements of various formats, because of their belief in determinism and cause-effect laws. Even Aristotle occasionally used letters to stand for terms, but neither the Aristotelian tradition nor the Stoic one extensively developed a systematic symbolization of logical form, nor diagrams (or tables). In the long period of time from the Aristotelians and Stoics into the 19 th A very abbreviated discussion of the history then jumps to predicate logic (properly "first order predicate logic" and sometimes "quantificational logic") which was due to notational work by the American C.S. Peirce (1870) and the more systematic efforts of the German Gottlob Frege (1879). This notation and conception of logic integrated propositional and categorical logic into one system, and made it vastly more powerful; it was popularized in the Principia Mathematica (1910-1914) of English philosopher-mathematicians Bertrand Russell and A.N. Whitehead. (Russell was one of the most famous and influential philosophers of the 20

Grammar, algebra and logic ***** Prof. Em. resulting in a monographwith Phil Scott Introduction to Higher order categorical logic .
The Leo Apostel Center invites everyone to the 47st of its interdisciplinary seminars in the Foundations series. In this series CLEA invites scholars that are actively engaged in the research on the foundations of a particular discipline. Their lectures will always be directed to an interdisciplinary audience, and the discussions aim at confronting the foundations of the different disciplines. Grammar, Algebra and Logic ************************** Prof. Em. Jim Lambek, McGill, Montreal ************************************** Monday, April 10, 17.00h, 10F734. About the lecture and speaker: Prof. Lambek is author of some monographs in mathematics, eg. "Completions of categories" (1966), "Torsion theories, additive semantics, and rings of quotients" (1970), but in particular a standard mathematical reference work "Lectures on Rings and Modules" (1966) of which there was a third edition in 1986. He obtained all his degrees at McGill, Montreal. In 1958, he published his first paper on the syntactic calculus, and for a while the "Lambek Grammars" were an essential opponent to "Chomskian Grammars", essentially supported in Europe - contrary to the North Americans who made Chomski win the day. Prof. Lambek then turned his thoughts for most of a decade to ring theory, particularly to rings of quotients, including the above mentioned monographs. Around 1965 he got interested in categories, resulting in a monograph with Phil Scott "Introduction to Higher order Categorical Logic". That same time he renewed his interest in mathematical linguistics, studying formally verb conjugations in French and Latin. Also to be mentioned is a paper "How to program the abacus" in which he invents independently and simultaneously with Marvin Minsky the Minsky machine, which is Turing complete, but conceptually much simpler than the Turing machine. Currently Prof. Lambek still publishes regularly on categorical logic, still producing highly valued papers, and on linguistics, returning to the grammars of syntactic types. In this lecture he will outline his ideas on the latter. (An appreciation of Prof. Lambek by Prof. Barr on the occasion of Jim Lambek's 75th birthday can be found on )

18. Hidden Algebra
algebraic exposition of first order logic, proof planning and a systematic expositionof universal algebra and its in August 1999.); A categorical Manifesto, in
Hidden Algebra Homepage Contents A Brief Overview of Hidden Algebra Hidden algebra aims to give a semantics for software engineering, and in particular for concurrent distributed object systems, supporting correctness proofs that are as simple and mechanized as possible. This emphasis on effective proofs rather than semantic models supports taking a calculational approach based on equations , rather than one based on, for example, higher order logic, type theory, denotational semantics, or any particular kind of model or set theory, because equational proofs achieve maximal simplicity and mechanization, while still allowing adequate expressiveness. It is also convenient that the models of a hidden algebraic specification are precisely its possible implementations. Hidden algebra effectively handles the most troubling features of large systems, including concurrency, distribution, nondeterminism, and local states, as well as the usual features of the object paradigm, including classes, subclasses (inheritance), attributes and methods, in addition to logical variables (as in logic programming), abstract data types, generic modules and more generally, the very powerful module system of prameterized programming. Hidden algebra generalizes the process algebra and transition system approaches to include non-monadic operations, so that it can take advantage of equations involving data, parameterized methods and attributes; this extra power can dramatically simplify proofs. Coinduction proof methods appear to be more effective for behavioral properties (including behavioral refinement) than any alternative of which we are aware, and moreover, they can be automated to a very significant degree.

19. Guests Of The Algebra And Logic Group At The University Of Saskatchewan
the strict refinement property) in terms of certain formulas (hformulas) definedby E. A. Palyutin in categorical Horn classes, I. algebra and logic 19(1980
of the
Mathematical Sciences Group

University of Saskatchewan

106 Wiggins Road
Saskatoon, SK, S7N 5E6, Canada
Phone: (306) 966-6081 - Fax: (306) 966-6086 Past Talks of Our Guests Friday, November 19, 1997, 4:00 p.m. Professor Sibylla Priess-Crampe
gave a talk in the Department Colloquium on
Fixed Point and Coincidence Theorems for Ultrametric Spaces
An ultrametric space (X,d,G) is a set X with an ultrametric distance functions d from X to G , where G is a partially ordered set with a smallest element 0. d has the same properties as a metric but instead of the triangle inequality the following one: For all g of G , if d(x,y) and d(y,z) are at most g then also d(x,z) is at most g . A special role for ultrametric spaces play spherically complete ultrametric spaces. "Sperically complete" corresponds to the property "maximal valued" for valued fields. For spherically complete ultrametric spaces there holds a fixed point theorem which looks like Banach's fixed point theorem for metric spaces. One has furthermore a generalization of this singlevalued fixed point theorem to multivalued mappings (again as it is the case in the metric situation). Some hints to applications of the theorems will be given. Friday, February 13, 1998, 4:00 p.m.

20. Algebra And Logic In Computer Science Group - Department Of Computing Science
algebra and logic in Computer Science Group Nonclassical logic, generalised terms compositionsare used to provide a basis for categorical unification in
Algebra and Logic in Computer Science Group
Non-classical logic, generalised terms, and unification
Unification in non-classical logics, with various approaches to handling generalised terms, has drawn more and more attention in recent years. It turns out to be among the most promising areas of research on non-classical logics and its applications. Composition of monads provides a method for extending the notion of terms. Variable substitutions, viewed as morphisms in the corresponding Kleisli categories over composed monads, correspond to variables being assigned to various generalised power sets of terms. These techniques for monad compositions are used to provide a basis for categorical unification in the framework of generalised terms. Monads have shown to be useful in different fields related to computer science. In functional programming monad compositions are applied to structuring of functional programs. In particular, in functional programs like parsers or type checkers the monad needed is often a composed monad.
Patrik Eklund
Selected papers:
Electronic Notes in Theoretical Computer Science 40 (2001)

Department of Computing Science
Umeå university
The information on this page was last checked 2002-03-20

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