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This is the first book on Abelian Group Theory (or Group Theory) to cover elementary results in Abelian Groups. It contains comprehensive coverage of almost all the topics related to the theory and is designed to be used as a course book for students at both undergraduate and graduate level. The text caters to students of differing capabilities by categorising the exercises in each chapter according to their level of difficulty starting with simple exercises (marked S1, S2 etc), of medium difficulty (M1, M2 etc) and ending with difficult exercises (D1, D2 etc). Solutions for all of the exercises are included. This book should also appeal to experts in the field as an excellent reference to a large number of examples in Group Theory. ... Read more

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A Handbook of Categorical Algebra, in three volumes, is a detailed account of everything a mathematician needs to know aboutcategory theory. Each volume is self-contained and is accessible to graduate students with a good background in mathematics.Volume 1 is devoted to general concepts.After introducing the terminology and proving the fundamental results concerning limits, adjoint functors and Kan extensions, the categories of fractions are studied in detail; special consideration is paid to the case of localizations. The remainder of the first volume studies various "refinements" of the fundamental concepts of category and functor. ... Read more

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This book contains papers presented at the NSF/CBMS Regional Conference on Coordinates in Operator Algebras, held at Texas Christian University in Fort Worth in May 1990. During the conference, in addition to a series of ten lectures by Paul S. Muhly (which will be published in a CBMS Regional Conference Series volume), twenty-eight lectures were delivered by conference participants on a broad range of topics of current interest in operator algebras and operator theory. This volume contains slightly expanded versions of most of those lectures. Participants were encouraged to bring open problems to the conference, and, as a result, over one hundred problems and questions are scattered throughout this volume. Readers will appreciate this book for the overview it provides of current topics and methods of operator algebras and operator theory. ... Read more

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This book deals with algebraic topology, homotopy theory andsimple homotopy theory of infinite CW-complexes with ends. Contrary to most other works on these subjects, the current volumedoes not use inverse systems to treat these topics. Here, the homotopytheory is approached without the rather sophisticated notion ofpro-category. Spaces with ends are studied only by using appropriateconstructions such as spherical objects of CW-complexes in thecategory of spaces with ends, and all arguments refer directly to thiscategory. In this way, infinite homotopy theory is presented as anatural extension of classical homotopy theory. In particular, thisbook introduces the construction of the proper groupoid of a spacewith ends and then the cohomology with local coefficients is definedby the enveloping ringoid of the proper fundamental groupoid. This volume will be of interest to researchers whose work involvesalgebraic topology, category theory, homological algebra, generaltopology, manifolds, and cell complexes. ... Read more

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The purpose of the book is to take stock of the situation concerning Algebra via Category Theory in the last fifteen years, where the new and synthetic notions of Mal'cev, protomodular, homological and semi-abelian categories emerged. These notions force attention on the fibration of points and allow a unified treatment of the main algebraic: homological lemmas, Noether isomorphisms, commutator theory.
The book gives full importance to examples and makes strong connections with Universal Algebra. One of its aims is to allow appreciating how productive the essential categorical constraint is: knowing an object, not from inside via its elements, but from outside via its relations with its environment.
The book is intended to be a powerful tool in the hands of researchers in category theory, homology theory and universal algebra, as well as a textbook for graduate courses on these topics. ... Read more

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This book develops thorough and complete foundations for the method of almost etale extensions, which is at the basis of Faltings' approach to p-adic Hodge theory. The central notion is that of an 'almost ring'. Almost rings are the commutative unitary monoids in a tensor category obtained as a quotient V-Mod/S of the category V-Mod of modules over a fixed ring V; the subcategory S consists of all modules annihilated by a fixed ideal m of V, satisfying certain natural conditions. The reader is assumed to be familiar with general categorical notions, some basic commutative algebra and some advanced homological algebra (derived categories, simplicial methods). Apart from these general prerequisites, the text is as self-contained as possible. One novel feature of the book - compared with Faltings' earlier treatment - is the systematic exploitation of the cotangent complex, especially for the study of deformations of almost algebras. ... Read more

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Proof theory and category theory were first drawn together byLambek some 30 years ago but, until now, the most fundamental notionsof category theory (as opposed to their embodiments in logic) have notbeen explained systematically in terms of proof theory. Here it isshown that these notions, in particular the notion of adjunction, canbe formulated in such as way as to be characterised by compositionelimination. Among the benefits of these composition-free formulationsare syntactical and simple model-theoretical, geometrical decisionprocedures for the commuting of diagrams of arrows. Compositionelimination, in the form of Gentzen's cut elimination, takes incategories, and techniques inspired by Gentzen are shown to work evenbetter in a purely categorical context than in logic. An acquaintancewith the basic ideas of general proof theory is relied on only for thesake of motivation, however, and the treatment of matters related tocategories is also in general self contained. Besides familiar topics,presented in a novel, simple way, the monograph also contains newresults. It can be used as an introductory text in categorical prooftheory. ... Read more

Customer Reviews (1)

Be prepared
This monograph gives an excellent introduction to a very abtruse and difficult subject. Readers familiar with the K-theory of operators due to Brown-Douglas-Fillmore and the Kadison-Ringrose theory of operator algebras should have ample background to read this book. These background works are very advanced mathematical subjects and so this book should be viewed as one written for specialists only.KK-theory can be thought of in some sense as a unification of K-homology and K-cohomology. This takes place via the KK bivariant groups of the mathematician G.G. Kasparov. Ordinary K-theory has its origins of course in the theory of elliptic operators via the work of M.Atiyah and I.Singer on index theory. The authors of this book however do not motivate their book at all with any history or with any discussion of elliptic operators but instead proceed from a purely algebraic/axiomatic point of view. This approach is well suited to mathematicians coming to the subject for the first time since no doubt they are familiar with the BDF theory if they are choosing to learn KK-theory. However, KK-theory is now finding applications in physics, such as superstring and dual theories, and physicists would not in general be familiar with the BDF/Kadison-Ringrose works. For physicists then this book might be very difficult to follow. That being said this is an interesting subject, and the authors do a good job for those who are prepared. The product on KK groups that Kasparov constructed can be used to obtain all the usual products in topological K-theory, such as the cap, cup, and slant products. Reading the book will also prepare one for the fascinating area of non-commutative geometry ala Alain Connes. ... Read more

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Based on several recent courses given to mathematical physics students, this volume is an introduction to bundle theory. It aims to provide newcomers to the field with solid foundations in topological K-theory. A fundamental theme, emphasized in the book, centers around the gluing of local bundle data related to bundles into a global object. One renewed motivation for studying this subject, comes from quantum field theory, where topological invariants play an important role.

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The already broad range of applications of ring theory has been enhanced in the eighties by the increasing interest in algebraic structures of considerable complexity, the so-called class of quantum groups. One of the fundamental properties of quantum groups is that they are modelled by associative coordinate rings possessing a canonical basis, which allows for the use of algorithmic structures based on Groebner bases to study them. This book develops these methods in a self-contained way, concentrating on an in-depth study of the notion of a vast class of non-commutative rings (encompassing most quantum groups), the so-called Poincaré-Birkhoff-Witt rings. We include algorithms which treat essential aspects like ideals and (bi)modules, the calculation of homological dimension and of the Gelfand-Kirillov dimension, the Hilbert-Samuel polynomial, primality tests for prime ideals, etc. ... Read more

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A basic problem for the interconnection of communications mediais to design interconnection networks for specific needs. For example,to minimize delay and to maximize reliability, networks are requiredthat have minimum diameter and maximum connectivity under certainconditions. The book provides a recent solution to this problem. Thesubject of all five chapters is the interconnection problem. The firsttwo chapters deal with Cayley digraphs which are candidates fornetworks of maximum connectivity with given degree and number ofnodes. Chapter 3 addresses Bruijn digraphs, Kautz digraphs, and theirgeneralizations, which are candidates for networks of minimum diameterand maximum connectivity with given degree and number of nodes.Chapter 4 studies double loop networks, and Chapter 5 considersbroadcasting and the Gossiping problem. All the chapters emphasize thecombinatorial aspects of network theory.
Audience: A vital reference for graduate students andresearchers in applied mathematics and theoretical computer science. ... Read more

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Ontology was once understood to be the philosophical inquiry into the structure of reality: the analysis and categorization of ‘what there is’. Recently, however, a field called ‘ontology’ has become part of the rapidly growing research industry in information technology. The two fields have more in common than just their name.

Theory and Applications of Ontology is a two-volume anthology that aims to further an informed discussion about the relationship between ontology in philosophy and ontology in information technology. It fills an important lacuna in cutting-edge research on ontology in both fields, supplying stage-setting overview articles on history and method, presenting directions of current research in either field, and highlighting areas of productive interdisciplinary contact.

Theory and Applications of Ontology: Philosophical Perspectives presents ontology in philosophy in ways that computer scientists are not likely to find elsewhere. The volume offers an overview of current research traditions in ontology, contrasting analytical, phenomenological, and hermeneutic approaches. It introduces the reader to current philosophical research on those categories of everyday and scientific reasoning that are most relevant to present and future research in information technology.

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Presents the most general approach to combinatorial construction of topological quantum field theories in three dimensions. Softcover. ... Read more

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This book presents an innovative theory of syntactic categories and the lexical classes they define. It revives the traditional idea that these are to be distinguished notionally (semantically). The author proposes a notation based on semantic features that accounts for the syntactic behavior of classes. The book also presents a case for considering this classification--again in a rather traditional vein--to be basic to determining the syntactic structure of sentences. ... Read more

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The first two chapters of this book offer a modern, self-contained exposition of the elementary theory of triangulated categories and their quotients. The simple, elegant presentation of these known results makes these chapters eminently suitable as a text for graduate students. The remainder of the book is devoted to new research, providing, among other material, some remarkable improvements on Brown's classical representability theorem. In addition, the author introduces a class of triangulated categories"--the "well generated triangulated categories"--and studies their properties. This exercise is particularly worthwhile in that many examples of triangulated categories are well generated, and the book proves several powerful theorems for this broad class. These chapters will interest researchers in the fields of algebra, algebraic geometry, homotopy theory, and mathematical physics. ... Read more

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This textbook explains the basic principles of categorical type theory and the techniques used to derive categorical semantics for specific type theories. It introduces the reader to ordered set theory, lattices and domains, and this material provides plenty of examples for an introduction to category theory, which covers categories, functors, natural transformations, the Yoneda lemma, cartesian closed categories, limits, adjunctions and indexed categories. Four kinds of formal system are considered in detail, namely algebraic, functional, polymorphic functional, and higher order polymorphic functional type theory. For each of these the categorical semantics are derived and results about the type systems are proved categorically. Issues of soundness and completeness are also considered. Aimed at advanced undergraduates and beginning graduates, this book will be of interest to theoretical computer scientists, logiciansand mathematicians specializing in category theory. ... Read more

Customer Reviews (1)

Excellent introduction to categories for computer scientists
The book gives you all of the cateogry theory you need to study type theory. The examples are from domains that are comfortable for computer scientists. The difficult proofs are given in great detail, while other books often gloss over the details. ... Read more