Product Description
This book develops aspects of category theory fundamental to the study of algebraic K-theory. Starting with categories in general, the text then examines categories of K-theory and moves on to tensor products and the Morita theory. The categorical approach to localizations and completions of modules is formulated in terms of direct and inverse limits. The authors consider local-global techniques that supply information about modules from their localizations and completions and underlie some interesting applications ofK-theory to number theory and geometry. Many useful exercises, concrete illustrations of abstract concepts, and an extensive list of references are included. ... Read more

Customer Reviews (1)

A good introduction to category theory
I am the author's grad student so my rating may be biased. Nonetheless I would recommend the book as a comprehensive and well-written intro to category theory. There is a wealth of examples and exercises to motivate discussion and add depth to the material. ... Read more

Customer Reviews (2)

Good classical book
This is a book first published in the 70's and it has the advantage ofbeing easy reading and short. It presents to the student with littlebackground (a course of real analysis or calculus is all that is required,together with some familiarity with set theoretic reasoning) some of theclassical and powerful results of measure and set theory and analysis in anelegant and modern way. It explores in a diversity of ways the analogies ofmeasure and category and the uses of Baire and Borel's theorems. I believeit is one of the best introductions of measure theory that can be found inthe literature.

Interesting monograph on measure theory
This short book is an interesting account of measure theory and how itrelates to topology. The topics are the standard ones that one would findin a book on the subject, and the book is a real pleasure to read. Itslength motivates one to finish the book and the book is very applicable tothe theory of dynamical systems and the theory of large deviations. Itcould be used as a textbook on measure theory or foundations of analysis ifone supplemented it with problem sets and some outside reading. A goodbook. ... Read more

Product Description
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. ... Read more

Product Description
Category theory is a mathematical subject whose importance in several areas of computer science, most notably the semantics of programming languages and the design of programmes using abstract data types, is widely acknowledged. This book introduces category theory at a level appropriate for computer scientists and provides practical examples in the context of programming language design. "Categories, Types and structures" provides a self-contained introduction to general category theory and explains the mathematical structures that have been the foundation of language design for the past two decades. The authors observe that the language of categories could provide a powerful means of standardizing of methods and language, and offer examples ranging from the early dialects of LISP, to Edinburgh ML, to work in polymorphisms and modularity. The book familiarizes readers with categorical concepts through examples based on elementary mathematical notions such as monoids, groups and toplogical spaces, as well as elementary notions from programming-language semantics such as partial orders and categories of domains in denotational semantics.It then pursues the more complex mathematical semantics of data types and programs as objects and morphisms of categories. ... Read more

Customer Reviews (1)

Out of print, but available via ftp
You can get PostScript files for all chapters of the book at ftp.ens.fr/pub/dmi/users/longo/CategTypesStructures ... Read more

Product Description

Demonstrates how category theory can be used for formal software development.

The mathematical toolbox for the Software Engineering in the new age of complex interactive systems.

Customer Reviews (1)

Don't Waste You're Money or Time
I had high hopes for this book, but I was greatly disappointed.The publisher misrepresents the content of this book and hence has incurred my wrath which you will read below.

This book purports to give a "gentle, software engineering oriented introduction to category theory" (see back cover).This is false.There is nothing gentle about this book.I think Springer more than any other publisher attempts to publish books buy academic 'experts'.The usual result, and what we have here, is a book written by an expert that appears to be a cut and paste job of the text and Latex files from their academic papers with very little value added in the way of explanations for the non peer, non expert.

Certainly this author gives lip service to 'gently introducing' the material in the first chapter, and the first section of the second chapter seems reasonable.The author then dives into throwing truly dense mathematical notation at the reader for the remainder of the book.I have a BS degree in Math so I'm used to dense notation, but with no or confusing explanations it can be very rough going indeed.

Also, the prose of the text is littered with references to the literature which the author uses as a 'copout'; ie the author repeatedly tells the read to consult reference [x] for more information.This is typical, the reader is expected to do the work; both in hunting down the actual references, and putting the peices together because this author provides very little assistance.One may ask why the author even bothered?

The book is divided into 3 sections:

Section 1: This is the introduction to Category Theory.The author states that Categories are different but equivalent formulation of mathematics than Set theory.Then proceeds to define Categories thru set theory.Okay, I've studied a lot of math, I can handle that.The author then goes on to define Categories as being compositions of nodes and edges over graphs.At this point the notation becomes so dense and the prose so excruciating I couldn't follow it anymore.At no point past the first section of chapter 2 does the author let up and give the reader an overview of what he's getting at.

Section 2: Advanced Topics.I didn't even try to read this.

Section 3: Applications.The author presents work he was involved with concerning an 'architectural' description language, CommUnity.Apparently CommUnity is related to other concurrency oriented languages in the CSP/CCP style.Supposedly, the constructs of the language can be modeled via some Categories the author develops in the preceding chapters.The examples are so simple (as are typical for these types of academic languages) to be completely meaningless.Again, the author goes into excruciating minute detail about obscure topics without ever giving the reader a hint why anything is relevant.

With this type of book the writer is writing for his peers and to have a book on his CV.Where's the value for you and I, the non experts?I found very little value in this book.I recommend checking it out at your library (if possible) before purchasing to see what you're buying.

... Read more

Product Description
This book presents the proceedings of the Sixth International Conference on Category Theory and Computer Science, CTCS '95, held in Cambridge, UK in August 1995. The 15 revised full papers included in the volume document the exploitation of links between logic and category theory leading to a solid basis for much of the understanding of the semantics of computation. Notable amongst other advances is the introduction of linear logic and other substructural logics, providing a new approach to proof theory. Further aspects covered are semantics of lambda calculi and type theories, program specification and development, and domain theory. ... Read more

Product Description
The technique of diagrammatic morphisms is an important ingredient in comprehending and visualizing certain types of categories with structure. It was widely used in this capacity in many areas of algebra, low-dimensional topology and physics. It was also applied to problems in classical and quantum information processing and logic.

This volume contains articles based on talks at the Special Session, "Diagrammatic Morphisms in Algebra, Category Theory, and Topology", at the AMS Sectional Meeting in San Francisco. The articles describe recent achievements in several aspects of diagrammatic morphisms and their applications. Some of them contain detailed expositions on various diagrammatic techniques. The introductory article by D. Yetter is a thorough account of the subject in a historical perspective. ... Read more

Product Description
The papers in this volume were presented at the fourthbiennial SummerConference on Category Theory and ComputerScience, held in Paris, September3-6, 1991. Category theorycontinues to be an important tool in foundationalstudies incomputer science. It has been widely applied by logicianstoget concise interpretations of many logical concepts. Linksbetweenlogic and computer science have been developed nowfor over twenty years, notably via the Curry-Howardisomorphism which identifies programs with proofs and typeswith propositions. The triangle category theory - logic-programming presents a rich world of interconnections.Topics covered in this volume include the following. Typetheory: stratification of typesand propositions can bediscussed in a categorical setting. Domain theory:syntheticdomain theory develops domain theory internally intheconstructive universe of the effective topos. Linear logic:the reconstruction of logic based on propositions asresources leads to alternatives to traditional syntaxes. Theproceedings of the previous three category theoryconferences appear as Lecture Notes in Computer ScienceVolumes 240, 283 and 389. ... Read more

Product Description
High Quality Content by WIKIPEDIA articles! It is a basic fact that every small category has a skeleton; more generally, every accessible category has a skeleton. Also, although a category may have many distinct skeletons, any two skeletons are isomorphic as categories, so up to isomorphism of categories, the skeleton of a category is unique.The importance of skeletons comes from the fact that they are (up to isomorphism of categories), canonical representatives of the equivalence classes of categories under the equivalence relation of equivalence of categories. This follows from the fact that any skeleton of a category C is equivalent to C, and that two categories are equivalent if and only if they have isomorphic skeletons. ... Read more

Product Description
"Applications of Category Theory to Fuzzy Subsets" is the first major work to comprehensively describe the deeper mathematical aspects of fuzzy sets, particularly those aspects which are category-theoretic in nature, and is intimately related to the first eleven years of the renowned International Seminar on Fuzzy Set Theory. Though it brings the reader to the very frontier of the mathematics of fuzzy set theory, its extensive bibliography, indices, and the tutorial nature of its longer chapters also make it suitable as a text for advanced graduate students. Part I develops model-theoretic foundations for fuzzy set theory, and in doing so, comprises an extensive study of monoid-valued sets, sheaves over commutative cl-monoids, weak and quasi topoi, local existence in such settings, and categories with two closed structures, including the logic and inference rules in these latter categories for the unbalanced subobjects modeling fuzzy subsets.Part II refines and works within non-model-theoretic approaches to fuzzy sets, giving a full account of the use of categorical methods to describe fuzzy topology from the structure-theoretic, category-theoretic, and point-set lattice-theoretic viewpoints. Explored in detail are set functors, topological constructs, convergence, and the relationship between locales, fuzzy topologies, and functor categories. Part III addresses issues related to Part I and II, including pointless geometry, fuzzy paths and relational databases, fuzzy points and locales, sobriety and Urysohn Lemmas, and the relationship between the Helly space and the fuzzy unit interval. The fourth part consists of several appendices: round tables and open questions, a comprehensive bibliography, and two extensive indices. ... Read more

Customer Reviews (1)

Category
Fuzzy topology, Quasi-uniforms spaces, ordered spaces ... Read more

Product Description
This volume reflects the growing use of techniques from topology and category theory in the field of theoretical computer science.In so doing it offers a source of new problems with a practical flavor while stimulating original ideas and solutions.Reflecting the latest innovations at the interface between mathematics and computer science, the work will interest researchers and advanced students in both fields. ... Read more