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As category theory approaches its first half-century, it continues to grow, finding new applications in areas that would have seemed inconceivable a generation ago, as well as in more traditional areas. The language, ideas, and techniques of category theory are well suited to discovering unifying structures in apparently different contexts. Representing this diversity of the field, this book contains the proceedings of an international conference on category theory. The subjects covered here range from topology and geometry to logic and theoretical computer science, from homotopy to braids and conformal field theory. Although generally aimed at experts in the various fields represented, the book will also provide an excellent opportunity for nonexperts to get a feel for the diversity of current applications of category theory. ... Read more

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This volume presents the proceedings of the workshop on higher category theory and mathematical physics held at Northwestern University. Exciting new developments were presented with the aim of making them better known outside the community of experts. In particular, presentations in the style, "Higher Categories for the Working Mathematician", were encouraged. The volume is the first to bring together developments in higher category theory with applications. This collection is a valuable introduction to this topic---one that holds great promise for future developments in mathematics. ... Read more

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Structured methodologies are a popular and powerful tool in information systems development. Many different ones exist, each employing a number of models and so a specification must be converted from one form to another during the development process. To solve this problem, Dr. Tse proposes a unifying framework behind popular structured models. He approaches the problem from the viewpoints of algebra and category theory. He not only develops the frameworks but also illustrates their practical and theoretical usefulness. Thus, this book will provide insight for software engineers into how methodologies can be formalized, and will open up a range of applications and problems for theoretical computer scientists. ... Read more

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With one exception, these papers are original and fullyrefereed research articles on various applications ofCategory Theory to Algebraic Topology,Logic and ComputerScience. The exception is an outstanding and lengthy surveypaper by Joyal/Street (80 pp) on a growing subject: it givesanaccount of classical Tannaka duality in such a way as tobe accessible to the general mathematical reader, and toprovide a key for entry to more recent developments andquantum groups. No expertise in eitherrepresentation theoryor category theory is assumed. Topics such as the Fouriercotransform, Tannaka duality for homogeneous spaces,braidedtensor categories, Yang-Baxter operators, Knot invariantsand quantum groups are introduced and studies.From the Contents: P.J. Freyd: Algebraically completecategories.- J.M.E. Hyland: First steps in synthetic domaintheory.- G. Janelidze, W. Tholen: How algebraic is thechange-of-base functor?.- A. Joyal, R. Street: Anintroduction 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 Leibnizrule in certain toposes.- S.H. Schanuel: Negative setshaveEuler characteristic and dimension.- ... Read more

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Mathematicians interested in understanding the directions of current research in set theory will not want to overlook this book, which contains the proceedings of the AMS Summer Research Conference on Axiomatic Set Theory, held in Boulder, Colorado, June 19--25, 1983. This was the first large meeting devoted exclusively to set theory since the legendary 1967 UCLA meeting, and a large majority of the most active research mathematicians in the field participated. All areas of set theory, including constructibility, forcing, combinatorics and descriptive set theory, were represented; many of the papers in the proceedings explore connections between areas. Readers should have a background of graduate-level set theory.

There is a paper by S.~Shelah applying proper forcing to obtain consistency results on combinatorial cardinal "invariants" below the continuum, and papers by R.~David and S.~Freidman on properties of $0^\#$. Papers by A.~Blass, H.-D.~Donder, T.~Jech and W.~Mitchell involve inner models with measurable cardinals and various combinatorial properties. T.~Carlson largely solves the pin-up problem, and D.~Velleman presents a novel construction of a Souslin tree from a morass. S.~Todorcevic obtains the strong failure of the \qedprinciple from the Proper Forcing Axiom and A.~Miller discusses properties of a new species of perfect-set forcing. H.~Becker and A.~Kechris attack the third Victoria Delfino problem while W.~Zwicker looks at combinatorics on $P_\kappa(\lambda)$ and J.~Henle studies infinite-exponent partition relations. A.~Blass shows that if every vector space has a basis then $AC$ holds. I.~Anellis treats the history of set theory, and W.~Fleissner presents set-theoretical axioms of use in general topology. ... Read more

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Topos theory provides an important setting and language for much of mathematical logic and set theory. It is well known that a typed language can be given for a topos to be regarded as a category of sets. This enables a fruitful interplay between category theory and set theory. However, one stumbling block to a logical approach to topos theory has been the treatment of geometric morphisms. This book presents a convenient and natural solution to this problem by developing the notion of a frame relative to an elementary topos. The authors show how this technique enables a logical approach to be taken to topics such as category theory relative to a topos and the relative Giraud theorem. The work is self-contained except that the authors presuppose a familiarity with basic category theory and topos theory. Logicians, set and category theorists, and computer scientist working in the field will find this work essential reading. ... Read more

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An exposition of building blocks for deformation theory of braided monoidal categories, giving rise to sequences of Vasseliv invariants of framed links, clarifying the interrelations between them. ... Read more

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This book offers a systematic account of syntactic categories - the building blocks of sentences and the units of grammatical analysis - and explains their place in different theories of language. It sets out and clarifies the conflicting definitions of competing frameworks which frequently make it hard or impossible to compare grammars.

Gisa Rauh describes the history and nature of traditional and contemporary accounts and definitions of grammatical categories. She explains their properties and use in generative, cognitive, and functional theories, and considers their function in language typology. She distinguishes between the cognitive functions of categories that relate to traditional parts of speech and serve to structure a language's lexicon; and those which determine the syntactic behaviour of the linguistic items they specify.

Professor Rauh illustrates her account with a wide range of examples. Her clear and balanced exposition will be welcomed by students and scholars in all branches of linguistics as well as by those in related subjects such as computational science and the philosophy of language. ... Read more

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Drawing on social science perspectives, "Contested Categories" presents a series of empirical studies that engage with the often shifting and day-to-day realities of life sciences categories. In doing so, it shows how such categories remain contested and dynamic, and that the boundaries they create are subject to negotiation as well as re-configuration and re-stabilization processes. Organized around the themes of biological substances and objects, personhood and the genomic body and the creation and dispersion of knowledge, each of the volume's chapters reveals the elusive nature of fixity with regard to life science categories. With contributions from an international team of scholars, this book will be essential reading for anyone interested in the social, legal, policy and ethical implications of science and technology and the life sciences. ... Read more

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This edited volume gives an overview of the concepts andmethods used in current research in quantum logic, viewed both as anoperational physical theory and in terms of purely mathematicalstructure. Far from being simply a collation of research papers, itconsists of 11 specially commissioned essays that, taken together,provide both an introduction to quantum logic and an overview ofcurrent research in this subject. The contributors, who include someof the most distinguished names in the field, discuss topics rangingfrom the algebraic theory of orthomodular lattices to recentapplications of category-theoretic methods and connections withtheoretical computer science. Moreover, an historical overview of thefield and an extensive citation and concept index are included.
Audience: This volume, dedicated to D.J. Foulis in honour of hisseminal contributions to quantum logic, should interest puremathematicians, theoretical physicists, computer scientists andphilosophers of science. ... Read more

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Algebraic K-Theory has become an increasingly active area of research. With its connections to algebra, algebraic geometry, topology, and number theory, it has implications for a wide variety of researchers and graduate students in mathematics. The book is based on lectures given at the author's home institution, the Tata Institute in Bombay, and elsewhere. A detailed appendix on topology was provided in the first edition to make the treatment accessible to readers with a limited background in topology. This new edition also includes an appendix on algebraic geometry that contains the required definitions and results needed to understand the core of the book; this makes the book accessible to a wider audience.

A central part of the book is a detailed exposition of the ideas of Quillen as contained in his classic papers “Higher Algebraic K-Theory, I, II.” A more elementary proof of the theorem of Merkujev--Suslin is given in this edition; this makes the treatment of this topic self-contained. An applications is also given to modules of finite length and finite projective dimension over the local ring of a normal surface singularity. These results lead the reader to some interesting conclusions regarding the Chow group of varieties.

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Category Theory has, in recent years, become increasingly important and popular in computer science, and many universities now introduce Category Theory as part of the curriculum for undergraduate computer science students. Here, the theory is developed in a straightforward way, and is enriched with many examples from computer science. ... Read more

Customer Reviews (2)

a recommendation of Category Theory texts for CS/IT
In September 1997 we needed a book on Category Theory for our first yearundergraduate class in the B.A. (Mod) honors degree in Information andCommunications Technology (ICT) at the University of Dublin, TrinityCollege, Dublin 2, Ireland. This book was at that time the only one thatsatisfied our requirements. Now we have chosen (Lawvere and Schanuel 1997)in addition. It is our opinion that one ought to start with the latter, amost excellent introduction of great profundity, and, for application tocomputing, use the Walters text. It is hard to beat this combination for afirst year undergraduate course, as far as we know at this time (Sept.98)

A Very comprehensive textbook for beginners computer sci.
The Book begins with the plain definition of a category, as does any other book. However, it points out a category as a kind of (abstract) Data Type. Distributive Categories are discussed as a milestone for developing the basic concepts in computation, as those of imperative programs and Data Types. The Book has a lot of examples (from computation) and the author took care of drawning conclusions from them before develop an abstract framework. The concepts of automata and automata with inputs are shown (the later in a functorial category). Grammars and Graphs are discussed as well. The book has a very good introduction to the concept of freeness and adjunctions. Its latest chapter treats the computational category theory in the context of Knuth-Bendix procedure. The exercises present in the book are great !! They guide the student gradualy into deeper questions without any frustation. There are very easy exercises which have the only goal of finding out ones undersating of a new definition. ... Read more