# Search Results for "topology-and-category-theory-in-computer-science"

## Topology and Category Theory in Computer Science

**Author**: George M. Reed,A. W. Roscoe**Publisher:**Oxford University Press**ISBN:**0198537603**Category:**Computers**Page:**390**View:**1604

This work consists of a selection of papers from the proceedings of a special session on topology and category theory in computer science, held at The Oxford Topology Symposium in June 1989. The session achieved a mixing of ideas between the two communities - giving one a course of new problems with a more practical flavour, and the other a source of solutions and ideas.

## Categorical Methods in Computer Science

*With Aspects from Topology*

**Author**: Hartmut Ehrig,Horst Herrlich,Hans-Jörg Kreowski,Gerhard Preuß**Publisher:**Springer Science & Business Media**ISBN:**9783540517221**Category:**Computers**Page:**354**View:**706

This volume contains selected papers of the International Workshop on "Categorical Methods in Computer Science - with Aspects from Topology" and of the "6th International Data Type Workshop" held in August/September 1988 in Berlin. The 23 papers of this volume are grouped into three parts: Part 1 includes papers on categorical foundations and fundamental concepts from category theory in computer science. Part 2 presents applications of categorical methods to algebraic specification languages and techniques, data types, data bases, programming, and process specifications. Part 3 comprises papers on categorial aspects from topology which mainly concentrate on special adjoint situations like cartesian closeness, Galois connections, reflections, and coreflections which are of growing interest in categorical topology and computer science.

## Tool and Object

*A History and Philosophy of Category Theory*

**Author**: Ralph Krömer**Publisher:**Springer Science & Business Media**ISBN:**3764375248**Category:**Mathematics**Page:**367**View:**2701

Category theory is a general mathematical theory of structures and of structures of structures. It occupied a central position in contemporary mathematics as well as computer science. This book describes the history of category theory whereby illuminating its symbiotic relationship to algebraic topology, homological algebra, algebraic geometry and mathematical logic and elaboratively develops the connections with the epistemological significance.

## An Introduction to Category Theory

**Author**: Harold Simmons**Publisher:**Cambridge University Press**ISBN:**1139503324**Category:**Mathematics**Page:**N.A**View:**2495

Category theory provides a general conceptual framework that has proved fruitful in subjects as diverse as geometry, topology, theoretical computer science and foundational mathematics. Here is a friendly, easy-to-read textbook that explains the fundamentals at a level suitable for newcomers to the subject. Beginning postgraduate mathematicians will find this book an excellent introduction to all of the basics of category theory. It gives the basic definitions; goes through the various associated gadgetry, such as functors, natural transformations, limits and colimits; and then explains adjunctions. The material is slowly developed using many examples and illustrations to illuminate the concepts explained. Over 200 exercises, with solutions available online, help the reader to access the subject and make the book ideal for self-study. It can also be used as a recommended text for a taught introductory course.

## Category Theory

*Proceedings of the International Conference held in Como, Italy, July 22-28, 1990*

**Author**: Aurelio Carboni,Maria C. Pedicchio,Giuseppe Rosolini**Publisher:**Springer**ISBN:**3540464352**Category:**Mathematics**Page:**496**View:**5110

With one exception, these papers are original and fully refereed research articles on various applications of Category Theory to Algebraic Topology, Logic and Computer Science. The exception is an outstanding and lengthy survey paper by Joyal/Street (80 pp) on a growing subject: it gives an account of classical Tannaka duality in such a way as to be accessible to the general mathematical reader, and to provide a key for entry to more recent developments and quantum groups. No expertise in either representation theory or category theory is assumed. Topics such as the Fourier cotransform, Tannaka duality for homogeneous spaces, braided tensor categories, Yang-Baxter operators, Knot invariants and quantum groups are introduced and studies. From the Contents: P.J. Freyd: Algebraically complete categories.- J.M.E. Hyland: First steps in synthetic domain theory.- G. Janelidze, W. Tholen: How algebraic is the change-of-base functor?.- A. Joyal, R. Street: An introduction to Tannaka duality and quantum groups.- A. Joyal, M. Tierney: Strong stacks andclassifying spaces.- A. Kock: Algebras for the partial map classifier monad.- F.W. Lawvere: Intrinsic co-Heyting boundaries and the Leibniz rule in certain toposes.- S.H. Schanuel: Negative sets have Euler characteristic and dimension.-

## Category Theory and Applications

*A Textbook for Beginners*

**Author**: Marco Grandis**Publisher:**World Scientific**ISBN:**9813231084**Category:****Page:**304**View:**4060

Category Theory now permeates most of Mathematics, large parts of theoretical Computer Science and parts of theoretical Physics. Its unifying power brings together different branches, and leads to a deeper understanding of their roots. This book is addressed to students and researchers of these fields and can be used as a text for a first course in Category Theory. It covers its basic tools, like universal properties, limits, adjoint functors and monads. These are presented in a concrete way, starting from examples and exercises taken from elementary Algebra, Lattice Theory and Topology, then developing the theory together with new exercises and applications. Applications of Category Theory form a vast and differentiated domain. This book wants to present the basic applications and a choice of more advanced ones, based on the interests of the author. References are given for applications in many other fields. Contents: Introduction Categories, Functors and Natural Transformations Limits and Colimits Adjunctions and Monads Applications in Algebra Applications in Topology and Algebraic Topology Applications in Homological Algebra Hints at Higher Dimensional Category Theory References Indices Readership: Graduate students and researchers of mathematics, computer science, physics. Keywords: Category TheoryReview: Key Features: The main notions of Category Theory are presented in a concrete way, starting from examples taken from the elementary part of well-known disciplines: Algebra, Lattice Theory and Topology The theory is developed presenting other examples and some 300 exercises; the latter are endowed with a solution, or a partial solution, or adequate hints Three chapters and some extra sections are devoted to applications

## Topology Via Logic

**Author**: Steven Vickers**Publisher:**Cambridge University Press**ISBN:**9780521576512**Category:**Computers**Page:**200**View:**1071

This is an advanced textbook on topology for computer scientists. It is based on a course given by the author to postgraduate students of computer science at Imperial College.

## Category Theory

**Author**: Steve Awodey**Publisher:**Oxford University Press**ISBN:**0191513822**Category:**Mathematics**Page:**256**View:**1968

This text and reference book on Category Theory, a branch of abstract algebra, is aimed not only at students of Mathematics, but also researchers and students of Computer Science, Logic, Linguistics, Cognitive Science, Philosophy, and any of the other fields that now make use of it. Containing clear definitions of the essential concepts, illuminated with numerous accessible examples, and providing full proofs of all important propositions and theorems, this book aims to make the basic ideas, theorems, and methods of Category Theory understandable to this broad readership. Although it assumes few mathematical pre-requisites, the standard of mathematical rigour is not compromised. The material covered includes the standard core of categories; functors; natural transformations; equivalence; limits and colimits; functor categories; representables; Yoneda's lemma; adjoints; monads.

## Category Theory and Computer Science

*7th International Conference, CTCS'97, Santa Margherita Ligure Italy, September 4-6, 1997, Proceedings*

**Author**: Eugenio Moggi,Giuseppe Rosolini**Publisher:**Springer Science & Business Media**ISBN:**9783540634553**Category:**Computers**Page:**319**View:**5335

This book constitutes the refereed proceedings of the 7th International Conference on Category Theory and Computer Science, CTCS'97, held in Santa Margheria Ligure, Italy, in September 1997. Category theory attracts interest in the theoretical computer science community because of its ability to establish connections between different areas in computer science and mathematics and to provide a few generic principles for organizing mathematical theories. This book presents a selection of 15 revised full papers together with three invited contributions. The topics addressed include reasoning principles for types, rewriting, program semantics, and structuring of logical systems.

## Understanding Concurrent Systems

**Author**: A.W. Roscoe**Publisher:**Springer Science & Business Media**ISBN:**9781848822580**Category:**Computers**Page:**530**View:**7310

CSP notation has been used extensively for teaching and applying concurrency theory, ever since the publication of the text Communicating Sequential Processes by C.A.R. Hoare in 1985. Both a programming language and a specification language, the theory of CSP helps users to understand concurrent systems, and to decide whether a program meets its specification. As a member of the family of process algebras, the concepts of communication and interaction are presented in an algebraic style. An invaluable reference on the state of the art in CSP, Understanding Concurrent Systems also serves as a comprehensive introduction to the field, in addition to providing material for a number of more advanced courses. A first point of reference for anyone wanting to use CSP or learn about its theory, the book also introduces other views of concurrency, using CSP to model and explain these. The text is fully integrated with CSP-based tools such as FDR, and describes how to create new tools based on FDR. Most of the book relies on no theoretical background other than a basic knowledge of sets and sequences. Sophisticated mathematical arguments are avoided whenever possible. Topics and features: presents a comprehensive introduction to CSP; discusses the latest advances in CSP, covering topics of operational semantics, denotational models, finite observation models and infinite-behaviour models, and algebraic semantics; explores the practical application of CSP, including timed modelling, discrete modelling, parameterised verifications and the state explosion problem, and advanced topics in the use of FDR; examines the ability of CSP to describe and enable reasoning about parallel systems modelled in other paradigms; covers a broad variety of concurrent systems, including combinatorial, timed, priority-based, mobile, shared variable, statecharts, buffered and asynchronous systems; contains exercises and case studies to support the text; supplies further tools and information at the associated website: http://www.comlab.ox.ac.uk/ucs/. From undergraduate students of computer science in need of an introduction to the area, to researchers and practitioners desiring a more in-depth understanding of theory and practice of concurrent systems, this broad-ranging text/reference is essential reading for anyone interested in Hoare’s CSP.

## Practical Foundations of Mathematics

**Author**: Paul Taylor**Publisher:**Cambridge University Press**ISBN:**9780521631075**Category:**Mathematics**Page:**572**View:**2666

Practical Foundations collects the methods of construction of the objects of twentieth-century mathematics. Although it is mainly concerned with a framework essentially equivalent to intuitionistic Zermelo-Fraenkel logic, the book looks forward to more subtle bases in categorical type theory and the machine representation of mathematics. Each idea is illustrated by wide-ranging examples, and followed critically along its natural path, transcending disciplinary boundaries between universal algebra, type theory, category theory, set theory, sheaf theory, topology and programming. Students and teachers of computing, mathematics and philosophy will find this book both readable and of lasting value as a reference work.

## Categorical Perspectives

**Author**: Jürgen Koslowski,Austin Melton**Publisher:**Springer Science & Business Media**ISBN:**1461213703**Category:**Mathematics**Page:**281**View:**7848

"Categorical Perspectives" consists of introductory surveys as well as articles containing original research and complete proofs devoted mainly to the theoretical and foundational developments of category theory and its applications to other fields. A number of articles in the areas of topology, algebra and computer science reflect the varied interests of George Strecker to whom this work is dedicated. Notable also are an exposition of the contributions and importance of George Strecker's research and a survey chapter on general category theory. This work is an excellent reference text for researchers and graduate students in category theory and related areas. Contributors: H.L. Bentley * G. Castellini * R. El Bashir * H. Herrlich * M. Husek * L. Janos * J. Koslowski * V.A. Lemin * A. Melton * G. Preuá * Y.T. Rhineghost * B.S.W. Schroeder * L. Schr"der * G.E. Strecker * A. Zmrzlina

## Sheaves in Geometry and Logic

*A First Introduction to Topos Theory*

**Author**: Saunders MacLane,Ieke Moerdijk**Publisher:**Springer Science & Business Media**ISBN:**9780387977102**Category:**Mathematics**Page:**630**View:**2575

Sheaves arose in geometry as coefficients for cohomology and as descriptions of the functions appropriate to various kinds of manifolds. Sheaves also appear in logic as carriers for models of set theory. This text presents topos theory as it has developed from the study of sheaves. Beginning with several examples, it explains the underlying ideas of topology and sheaf theory as well as the general theory of elementary toposes and geometric morphisms and their relation to logic.

## Associahedra, Tamari Lattices and Related Structures

*Tamari Memorial Festschrift*

**Author**: Folkert Müller-Hoissen,Jean Marcel Pallo,Jim Stasheff**Publisher:**Springer Science & Business Media**ISBN:**3034804059**Category:**Mathematics**Page:**436**View:**9066

Tamari lattices originated from weakenings or reinterpretations of the familar associativity law. This has been the subject of Dov Tamari's thesis at the Sorbonne in Paris in 1951 and the central theme of his subsequent mathematical work. Tamari lattices can be realized in terms of polytopes called associahedra, which in fact also appeared first in Tamari's thesis. By now these beautiful structures have made their appearance in many different areas of pure and applied mathematics, such as algebra, combinatorics, computer science, category theory, geometry, topology, and also in physics. Their interdisciplinary nature provides much fascination and value. On the occasion of Dov Tamari's centennial birthday, this book provides an introduction to topical research related to Tamari's work and ideas. Most of the articles collected in it are written in a way accessible to a wide audience of students and researchers in mathematics and mathematical physics and are accompanied by high quality illustrations.

## Frobenius Algebras and 2-D Topological Quantum Field Theories

**Author**: Joachim Kock**Publisher:**Cambridge University Press**ISBN:**9780521540315**Category:**Mathematics**Page:**240**View:**9606

This 2003 book describes a striking connection between topology and algebra, namely that 2D topological quantum field theories are equivalent to commutative Frobenius algebras. The precise formulation of the theorem and its proof is given in terms of monoidal categories, and the main purpose of the book is to develop these concepts from an elementary level, and more generally serve as an introduction to categorical viewpoints in mathematics. Rather than just proving the theorem, it is shown how the result fits into a more general pattern concerning universal monoidal categories for algebraic structures. Throughout, the emphasis is on the interplay between algebra and topology, with graphical interpretation of algebraic operations, and topological structures described algebraically in terms of generators and relations. The book will prove valuable to students or researchers entering this field who will learn a host of modern techniques that will prove useful for future work.

## Basic Category Theory for Computer Scientists

**Author**: Benjamin C. Pierce,Benjamin C.. Pierce,Ierce Benjamin**Publisher:**MIT Press**ISBN:**9780262660716**Category:**Computers**Page:**100**View:**402

Basic Category Theory for Computer Scientists provides a straightforward presentation of the basic constructions and terminology of category theory, including limits, functors, natural transformations, adjoints, and cartesian closed categories. Category theory is a branch of pure mathematics that is becoming an increasingly important tool in theoretical computer science, especially in programming language semantics, domain theory, and concurrency, where it is already a standard language of discourse. Assuming a minimum of mathematical preparation, Basic Category Theory for Computer Scientists provides a straightforward presentation of the basic constructions and terminology of category theory, including limits, functors, natural transformations, adjoints, and cartesian closed categories. Four case studies illustrate applications of category theory to programming language design, semantics, and the solution of recursive domain equations. A brief literature survey offers suggestions for further study in more advanced texts. Contents Tutorial * Applications * Further Reading

## Handbook of Mathematics

**Author**: Thierry Vialar**Publisher:**BoD - Books on Demand**ISBN:**2955199001**Category:****Page:**1132**View:**7825

The book consists of XI Parts and 28 Chapters covering all areas of mathematics. It is a tool for students, scientists, engineers, students of many disciplines, teachers, professionals, writers and also for a general reader with an interest in mathematics and in science. It provides a wide range of mathematical concepts, definitions, propositions, theorems, proofs, examples, and numerous illustrations. The difficulty level can vary depending on chapters, and sustained attention will be required for some. The structure and list of Parts are quite classical: I. Foundations of Mathematics, II. Algebra, III. Number Theory, IV. Geometry, V. Analytic Geometry, VI. Topology, VII .Algebraic Topology, VIII. Analysis, IX. Category Theory, X. Probability and Statistics, XI. Applied Mathematics. Appendices provide useful lists of symbols and tables for ready reference. The publisher’s hope is that this book, slightly revised and in a convenient format, will serve the needs of readers, be it for study, teaching, exploration, work, or research.

## Category Theory and Computer Science

*Paris, France, September 3-6, 1991. Proceedings*

**Author**: David H. Pitt,Pierre-Louis Curien,Samson Abramsky,Andrew Pitts,Axel Poigne,David E. Rydeheard**Publisher:**Springer Science & Business Media**ISBN:**9783540544951**Category:**Mathematics**Page:**304**View:**2629

The papers in this volume were presented at the fourth biennial Summer Conference on Category Theory and Computer Science, held in Paris, September3-6, 1991. Category theory continues to be an important tool in foundationalstudies in computer science. It has been widely applied by logicians to get concise interpretations of many logical concepts. Links between logic and computer science have been developed now for over twenty years, notably via the Curry-Howard isomorphism which identifies programs with proofs and types with propositions. The triangle category theory - logic - programming presents a rich world of interconnections. Topics covered in this volume include the following. Type theory: stratification of types and propositions can be discussed in a categorical setting. Domain theory: synthetic domain theory develops domain theory internally in the constructive universe of the effective topos. Linear logic: the reconstruction of logic based on propositions as resources leads to alternatives to traditional syntaxes. The proceedings of the previous three category theory conferences appear as Lecture Notes in Computer Science Volumes 240, 283 and 389.

## Between Mind and Computer

*Fuzzy Science and Engineering*

**Author**: P.-Z. Wang,K.F. Loe**Publisher:**World Scientific**ISBN:**9789810236632**Category:**Mathematics**Page:**387**View:**5739

The ?Fuzzy Explosion? emanating from Japan has compelled more people than ever to ponder the meaning and potential of fuzzy engineering. Scientists all over are now beginning to harness the power of fuzzy recognition and decision-making ? reminescent of the way the human mind works ? in computer applications.In this book a blue-ribbon list of contributors discusses the latest developments in topics such as possibility logic programming, truth-valued flow inference, fuzzy neural-logic networks and default knowledge representation. This volume is the first in a series aiming to document advances in fuzzy set theory and its applications.