Book Description
Over the last few years, the study of complexes has become increasingly important. To date, however, most of the research is scattered throughout the literature or available only as lecture notes. Covers and Envelopes in the Category of Complexes of Modules collects these scattered notes and results into a single, concise volume that provides an account of recent developments in the theory and presents several new and important ideas. The author introduces the theory of complexes of modules using only elementary tools-making the field more accessible to non-specialists. He focuses the study on envelopes and covers in this category with respect to some well established and important classes of complexes. He places particular emphasis on DG-injective and DG-projective complexes and flat and DG-flat covers. Other topics covered include Zorn's Lemma for categories, preserving and reflecting covers by functors, orthogonality in the category of complexes, Gorenstein injective and projective complexes, and pure sequences of complexes. Along with its value as a collection of recent work in the field, Covers and Envelopes in the Category of Complexes of Modules presents powerful new ideas that will undoubtedly advance homological methods. Mathematicians-especially researchers in module theory and homological algebra-will welcome this volume as a reference guide and for its new and important results. ... Read more

Book 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

Book Description
This volume presents the proceedings of the workshop on highercategory theory and mathematical physics held at NorthwesternUniversity. Exciting new developments were presented with theaim of making them better known outside the community ofexperts. In particular, presentations in the style, "HigherCategories for the Working Mathematician", were encouraged. Thevolume is the first to bring together developments in highercategory theory with applications. This collection is a valuableintroduction to this topic---one that holds great promise forfuture developments in mathematics. ... Read more

Book Description
This digital document is an article from The Review of Metaphysics, published by Philosophy Education Society, Inc. on March 1, 1998. The length of the article is 743 words. The page length shown above is based on a typical 300-word page. The article is delivered in HTML format and is available in your Amazon.com Digital Locker immediately after purchase. You can view it with any web browser.

Citation Details
Title: A Realistic Theory of Categories: An Essay on Ontology.
Author: Robert Pasnau
Publication: The Review of Metaphysics (Refereed)
Date: March 1, 1998
Publisher: Philosophy Education Society, Inc.
Volume: v51Issue: n3Page: p666(2)

Article Type: Book Review

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Book Description

Based on several recent courses given to mathematical physics students, this volume is an introduction to bundle theory with the aim 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.

Book Description
This book addresses some methodological problems in the study of tense, aspect and action: How should linguists go about describing these categories and with what terminology? How does our work in this area relate to descriptions of language(s) in general? What research strategies should be explored? Bache discusses the interaction between language-specific grammars and universal grammar, including the problems of analytic directionality, semantic minimalism, and the general metalanguage of universal grammar. The book has several sources of inspiration: generative linguistics, structuralist phonology, glossematics, functional grammar, cognitive semantics and prototype theory. Bache argues strongly for the inclusion of a paradigmatic dimension in the study of the semantics of morphosyntactic categories. Rather than adhering to one particular linguistic school, Bache provides a general description of tense, aspect and action in the form of generalizations that should be accommodated in any theory. ... Read more

Book Description
Category theory has experienced a resurgence in popularity recently because of new links with topology and mathematical physics. This book provides a clearly written account of higher order category theory and presents operads and multicategories as a natural language for its study. Tom Leinster has included necessary background material and applications as well as appendices containing some of the more technical proofs that might have disrupted the flow of the text. ... Read more

Customer Reviews (1)

A very interesting overview
Structures such as braided monoidal categories, operads, and Hopf algebras are familiar to those who have studied topological quantum field theory, knot theory, string theory, and the renormalization procedure in quantum field theory. This book attempts, and succeeds, in presenting to the interested reader an overview of higher category theory, which subsumes the aforementioned topics. It is not however a book on applications, but instead details the purely mathematical aspects of higher category, clarifying for example the difference between `weak' n-categories and `strict' n-categories. The author though has not written a book in the typical "definition-theorem-proof" style, as he motivates the subject very well, and does not hesitate to use diagrams to get his point across. Indeed, he is careful to point out that the subject is inherently topological in its nature, and that diagrams used to illustrate higher-dimensional structures can be viewed as topological structures. The braided monoidal category that arises in knot theory is a perfect example of this.

The author introduces higher-dimensional category theory as one that uses "higher-dimensional arrows", in analogy to ordinary category as one that uses 1-dimensional arrows. Higher-dimensional category theory or `n-category theory,' is viewed as a generalization of the notion of category. To motivate the concept of a weak n-category, the author reminds the reader of the attempt to prove to what extent the loop group in differential topology is in fact a topological group. The composition of paths in the loop groups is not associative, but rather associative up to homotopy. Associativity does hold in strict n-categories but not in weak n-categories. As another example of non-associativity, the author discusses the fundamental omega-groupoid, which is the higher-dimensional category arising from a topological space. Several examples of weak n-categories are given in the motivating chapter of the book.

The first chapter is an overview of classical category theory, most of which may be review for readers familiar with it. Of particular importance is the notion of a monoidal category, which for the case of a strict monoidal category, generalizes the familiar tensor product operation. The tensor operation is generalized to that of a functor on a category that obeys strict associativity and unit laws. Weak monoidal categories are also defined, where the functor now obeys associativity and unit laws only up to isomorphism. These are the `coherence isomorphisms', and these satisfy the `pentagon' and `triangle axioms.' Modules over commuative rings with the usual tensor product are monoidal categories.

The author introduces operads in chapter two, concentrating first on multicategories, which are collections of objects on which are defined maps or "arrows", and compositions that satisfy associativity and unit properties. Operads are multicategories with only one object, and can be viewed as an abstraction of a set of composable functions of several variables where the variables can be permuted. Several examples are given of multicategories with many objects, including how a monoidal category can give rise to a multicategory. Operads appear in physical applications, such as string field theory and conformal field theory, which are not discussed in the book, but the author gives many examples of operads that make their properties readily apparent. One of these involves iterated loop spaces, where operads arose historically.

After a further discussion of monoidal categories in chapter 3, the author spends part two of the book solely on operads. One of the first goals of the author is formalize the notion of an input type, so as to allow more than just finite sequences of objects. For each input type he defines a theory of operads and multicategories, which yields the "plain" operad when the inputs are finite sequences. The author also discusses how to start with a monad T on any category and construct `T-multicategories'. T-operads are then T-multicategories with one object, and algebras can be associated to T-multicategories. These algebras are an analog of the "representation" or "model" for the T-multicategory. The author's work on "free category" or `fc-multicategories', which are 2-dimensional examples of these generalized multicategories. Fc-multicategories are T-multicategories on the free category (fc) monad on the category of directed graphs. As a very interesting example of an fc-multicategory, the author discusses one which encapsulates (in a single structure) rings, homomorphisms of rings, modules over rings, homomorphisms of modules, and tensor products of modules.

Also discussed in this part is the notion of an `opetope' (for "operation polytope"), which are a kind of generalization of the simplices of simplicial geometry. The opetopes are thus the "polytopes" of higher-dimensional category theory, and are defined by first taking for every natural number and defining a category and monad inductively. The zeroth category is Set and the zeroth monad is the identity. This gives rise to an infinite sequence of opetopes, with the zeroth opetope being 1. The nth category is then canonically isomorphic to Set modulo the nth opetope. A 2-opetope is the natural numbers, while a 3-opetope is the collection of trees. The author shows how to construct a category of n-dimensional pasting diagrams for each natural number n, where for n = 1 is the category of finite totally ordered sets, and for n = 2, the category of trees. The geometric connotations of the pasting diagrams are obvious, as well as their analogy to simplicial objects. An opetopic n-pasting diagram is defined as an (n+1)-opetope for each natural number n. 2-pasting diagrams correspond to trees, and the author shows how to construct `stable trees', i.e. those trees whose vertices have at least two branches coming out of them. The relation of stable trees to the constructions of Stasheff are discussed, along with the connection of opetopes to the construction of weak n-categories. ... Read more

Book 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.
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Book 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