Categories and Sheaves

Author: Masaki Kashiwara,Pierre Schapira

Publisher: Springer Science & Business Media

ISBN: 3540279490

Category: Mathematics

Page: 498

View: 3064

Categories and sheaves appear almost frequently in contemporary advanced mathematics. This book covers categories, homological algebra and sheaves in a systematic manner starting from scratch and continuing with full proofs to the most recent results in the literature, and sometimes beyond. The authors present the general theory of categories and functors, emphasizing inductive and projective limits, tensor categories, representable functors, ind-objects and localization.
Posted in Mathematics

Motives

Author: Uwe Jannsen,Steven Kleiman,Jean Pierre Serre

Publisher: American Mathematical Soc.

ISBN: 0821827987

Category: Mathematics

Page: 676

View: 8236

'Motives' were introduced in the mid-1960s by Grothendieck to explain the analogies among the various cohomology theories for algebraic varieties, to play the role of the missing rational cohomology, and to provide a blueprint for proving Weil's conjectures about the zeta function of a variety over a finite field. Over the last ten years or so, researchers in various areas - Hodge theory, algebraic $K$-theory, polylogarithms, automorphic forms, $L$-functions, $\ell$-adic representations, trigonometric sums, and algebraic cycles - have discovered that an enlarged (and in part conjectural) theory of 'mixed' motives indicates and explains phenomena appearing in each area.Thus the theory holds the potential of enriching and unifying these areas. This is one of two volumes containing the revised texts of nearly all the lectures presented at the AMS-IMS-SIAM Joint Summer Research Conference on Motives, held in Seattle, in 1991. A number of related works are also included, making for a total of forty-seven papers, from general introductions to specialized surveys to research papers.
Posted in Mathematics

Equivariant Sheaves and Functors

Author: Joseph Bernstein,Valery Lunts

Publisher: Springer

ISBN: 3540484302

Category: Mathematics

Page: 146

View: 4135

The equivariant derived category of sheaves is introduced. All usual functors on sheaves are extended to the equivariant situation. Some applications to the equivariant intersection cohomology are given. The theory may be useful to specialists in representation theory, algebraic geometry or topology.
Posted in Mathematics

Methods of Homological Algebra

Author: Sergei I. Gelfand,Yuri I. Manin

Publisher: Springer Science & Business Media

ISBN: 3662124920

Category: Mathematics

Page: 372

View: 5156

This modern approach to homological algebra by two leading writers in the field is based on the systematic use of the language and ideas of derived categories and derived functors. It describes relations with standard cohomology theory and provides complete proofs. Coverage also presents basic concepts and results of homotopical algebra. This second edition contains numerous corrections.
Posted in Mathematics

Proceedings

Author: N.A

Publisher: N.A

ISBN: 9780780350649

Category: Computers

Page: 538

View: 3851

Posted in Computers

Handbook of Categorical Algebra: Volume 3, Sheaf Theory

Author: Francis Borceux

Publisher: Cambridge University Press

ISBN: 9780521441803

Category: Mathematics

Page: 522

View: 2085

The Handbook of Categorical Algebra is intended to give, in three volumes, a rather detailed account of what, ideally, everybody working in category theory should know, whatever the specific topic of research they have chosen. The book is planned also to serve as a reference book for both specialists in the field and all those using category theory as a tool. Volume 3 begins with the essential aspects of the theory of locales, proceeding to a study in chapter 2 of the sheaves on a locale and on a topological space, in their various equivalent presentations: functors, etale maps or W-sets. Next, this situation is generalized to the case of sheaves on a site and the corresponding notion of Grothendieck topos is introduced. Chapter 4 relates the theory of Grothendieck toposes with that of accessible categories and sketches, by proving the existence of a classifying topos for all coherent theories.
Posted in Mathematics

Mathematical Reviews

Author: N.A

Publisher: N.A

ISBN: N.A

Category: Mathematics

Page: N.A

View: 495

Posted in Mathematics

Definable Additive Categories

Purity and Model Theory

Author: Mike Prest

Publisher: American Mathematical Soc.

ISBN: 0821847678

Category: Mathematics

Page: 109

View: 9480

Most of the model theory of modules works, with only minor modifications, in much more general additive contexts (such as functor categories, categories of comodules, categories of sheaves). Furthermore, even within a given category of modules, many subcategories form a ``self-sufficient'' context in which the model theory may be developed without reference to the larger category of modules. The notion of a definable additive category covers all these contexts. The (imaginaries) language which one uses for model theory in a definable additive category can be obtained from the category (of structures and homomorphisms) itself, namely, as the category of those functors to the category of abelian groups which commute with products and direct limits. Dually, the objects of the definable category--the modules (or functors, or comodules, or sheaves)--to which that model theory applies may be recovered as the exact functors from the, small abelian, category (the category of pp-imaginaries) which underlies that language.
Posted in Mathematics

Ind-Sheaves

Author: Masaki Kashiwara,Pierre Schapira

Publisher: Societe Mathematique De France

ISBN: N.A

Category: Mathematics

Page: 136

View: 4355

Posted in Mathematics

Categorical Algebra and its Applications

Proceedings of a Conference, Held in Louvain-la-Neuve, Belgium, July 26 - August 1, 1987

Author: Francis Borceux

Publisher: Springer

ISBN: 9783540503620

Category: Mathematics

Page: 382

View: 2570

Categorical algebra and its applications contain several fundamental papers on general category theory, by the top specialists in the field, and many interesting papers on the applications of category theory in functional analysis, algebraic topology, algebraic geometry, general topology, ring theory, cohomology, differential geometry, group theory, mathematical logic and computer sciences. The volume contains 28 carefully selected and refereed papers, out of 96 talks delivered, and illustrates the usefulness of category theory today as a powerful tool of investigation in many other areas.
Posted in Mathematics

Algebras and Representation Theory

Author: Karin Erdmann,Thorsten Holm

Publisher: Springer

ISBN: 3319919989

Category: Mathematics

Page: 298

View: 5794

This carefully written textbook provides an accessible introduction to the representation theory of algebras, including representations of quivers. The book starts with basic topics on algebras and modules, covering fundamental results such as the Jordan-Hölder theorem on composition series, the Artin-Wedderburn theorem on the structure of semisimple algebras and the Krull-Schmidt theorem on indecomposable modules. The authors then go on to study representations of quivers in detail, leading to a complete proof of Gabriel's celebrated theorem characterizing the representation type of quivers in terms of Dynkin diagrams. Requiring only introductory courses on linear algebra and groups, rings and fields, this textbook is aimed at undergraduate students. With numerous examples illustrating abstract concepts, and including more than 200 exercises (with solutions to about a third of them), the book provides an example-driven introduction suitable for self-study and use alongside lecture courses.
Posted in Mathematics

Homological Algebra

Author: S.I. Gelfand,Yu.I. Manin

Publisher: Springer Science & Business Media

ISBN: 9783540533733

Category: Mathematics

Page: 222

View: 4859

This book, the first printing of which was published as volume 38 of the Encyclopaedia of Mathematical Sciences, presents a modern approach to homological algebra, based on the systematic use of the terminology and ideas of derived categories and derived functors. The book contains applications of homological algebra to the theory of sheaves on topological spaces, to Hodge theory, and to the theory of modules over rings of algebraic differential operators (algebraic D-modules). The authors Gelfand and Manin explain all the main ideas of the theory of derived categories. Both authors are well-known researchers and the second, Manin, is famous for his work in algebraic geometry and mathematical physics. The book is an excellent reference for graduate students and researchers in mathematics and also for physicists who use methods from algebraic geometry and algebraic topology.
Posted in Mathematics

Categorical Foundations

Special Topics in Order, Topology, Algebra, and Sheaf Theory

Author: Maria Cristina Pedicchio,Walter Tholen,G. C. Rota

Publisher: Cambridge University Press

ISBN: 9780521834148

Category: Mathematics

Page: 417

View: 6530

The book offers categorical introductions to order, topology, algebra and sheaf theory, suitable for graduate students, teachers and researchers of pure mathematics.
Posted in Mathematics

Representations of Groups

Canadian Mathematical Society Annual Seminar, June 15-24, 1994, Banff, Alberta, Canada

Author: Bruce Normansell Allison,Gerald Howard Cliff,Canadian Mathematical Society

Publisher: American Mathematical Soc.

ISBN: 9780821803110

Category: Mathematics

Page: 385

View: 8956

Representations of Groups contains papers presented at the Canadian Mathematical Society Annual Seminar held in June 1994, in Banff, Alberta. The material addresses representations of Lie groups, algebraic groups, finite groups, and quantum groups and the relationships among these areas. With both survey and research articles, this book offers the latest results on various aspects of representation theory of groups.
Posted in Mathematics

Topos Theory

Author: P.T. Johnstone

Publisher: Courier Corporation

ISBN: 0486493369

Category: Mathematics

Page: 400

View: 3821

Focusing on topos theory's integration of geometric and logical ideas into the foundations of mathematics and theoretical computer science, this volume explores internal category theory, topologies and sheaves, geometric morphisms, and other subjects. 1977 edition.
Posted in Mathematics

Sheaves in Topology

Author: Alexandru Dimca

Publisher: Springer Science & Business Media

ISBN: 3642188680

Category: Mathematics

Page: 240

View: 3920

Constructible and perverse sheaves are the algebraic counterpart of the decomposition of a singular space into smooth manifolds. This introduction to the subject can be regarded as a textbook on modern algebraic topology, treating the cohomology of spaces with sheaf (as opposed to constant) coefficients. The author helps readers progress quickly from the basic theory to current research questions, thoroughly supported along the way by examples and exercises.
Posted in Mathematics

Handbook of Categorical Algebra: Volume 2, Categories and Structures

Author: Francis Borceux

Publisher: Cambridge University Press

ISBN: 9780521441797

Category: Mathematics

Page: 443

View: 2804

The Handbook of Categorical Algebra is designed to give, in three volumes, a detailed account of what should be known by everybody working in, or using, category theory. As such it will be a unique reference. The volumes are written in sequence. The second, which assumes familiarity with the material in the first, introduces important classes of categories that have played a fundamental role in the subject's development and applications. In addition, after several chapters discussing specific categories, the book develops all the major concepts concerning Benabou's ideas of fibred categories. There is ample material here for a graduate course in category theory, and the book should also serve as a reference for users.
Posted in Mathematics