From Groups to Categorial Algebra

Introduction to Protomodular and Mal’tsev Categories

Author: Dominique Bourn

Publisher: Birkhäuser

ISBN: 3319572199

Category: Mathematics

Page: 106

View: 6920

This book gives a thorough and entirely self-contained, in-depth introduction to a specific approach to group theory, in a large sense of that word. The focus lie on the relationships which a group may have with other groups, via “universal properties”, a view on that group “from the outside”. This method of categorical algebra, is actually not limited to the study of groups alone, but applies equally well to other similar categories of algebraic objects. By introducing protomodular categories and Mal’tsev categories, which form a larger class, the structural properties of the category Gp of groups, show how they emerge from four very basic observations about the algebraic litteral calculus and how, studied for themselves at the conceptual categorical level, they lead to the main striking features of the category Gp of groups. Hardly any previous knowledge of category theory is assumed, and just a little experience with standard algebraic structures such as groups and monoids. Examples and exercises help understanding the basic definitions and results throughout the text.
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: 3540459855

Category: Mathematics

Page: 382

View: 8019

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

Handbook of Categorical Algebra: Volume 2, Categories and Structures

Author: Francis Borceux

Publisher: Cambridge University Press

ISBN: 9780521441797

Category: Mathematics

Page: 443

View: 4278

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

Handbook of Categorical Algebra: Volume 1, Basic Category Theory

Author: Francis Borceux

Publisher: Cambridge University Press

ISBN: 9780521441780

Category: Mathematics

Page: 345

View: 7679

First of a 3-volume work giving a detailed account of what should be known by all working in, or using category theory. Volume 1 covers basic concepts.
Posted in Mathematics

Proceedings of the Conference on Categorical Algebra

La Jolla 1965

Author: S. Eilenberg,D. K. Harrison,H. Röhrl,S. MacLane

Publisher: Springer Science & Business Media

ISBN: 3642999026

Category: Mathematics

Page: 564

View: 8914

This volume contains the articles contributed to the Conference on Categorical Algebra, held June 7-12,1965, at the San Diego campus of the University of California under the sponsorship of the United States Air Force Office of Scientific Research. Of the thirty-seven mathemati cians, who were present seventeen presented their papers in the form of lectures. In addition, this volume contains papers contributed by other attending participants as well as by those who, after having planned to attend, were unable to do so. The editors hope to have achieved a representative, if incomplete, cover age of the present activities in Categorical Algebra within the United States by bringing together this group of mathematicians and by solici ting the articles contained in this volume. They also hope that these Proceedings indicate the trend of research in Categorical Algebra in this country. In conclusion, the editors wish to thank the participants and contrib. utors to these Proceedings for their continuous cooperation and encour agement. Our thanks are also due to the Springer-Verlag for publishing these Proceedings in a surprisingly short time after receiving the manu scripts.
Posted in Mathematics

Applications of Categorical Algebra


Author: Alex Heller

Publisher: American Mathematical Soc.

ISBN: 0821814176

Category: Algebra, Homological

Page: 231

View: 6370

Posted in Algebra, Homological

Handbook of Categorical Algebra 2

Categories and Structures

Author: N.A

Publisher: N.A


Category: Abelian categories

Page: 443

View: 8906

Posted in Abelian categories

The Logical Foundations of Mathematics

Foundations and Philosophy of Science and Technology Series

Author: William S. Hatcher

Publisher: Elsevier

ISBN: 1483189635

Category: Mathematics

Page: 330

View: 7292

The Logical Foundations of Mathematics offers a study of the foundations of mathematics, stressing comparisons between and critical analyses of the major non-constructive foundational systems. The position of constructivism within the spectrum of foundational philosophies is discussed, along with the exact relationship between topos theory and set theory. Comprised of eight chapters, this book begins with an introduction to first-order logic. In particular, two complete systems of axioms and rules for the first-order predicate calculus are given, one for efficiency in proving metatheorems, and the other, in a "natural deduction" style, for presenting detailed formal proofs. A somewhat novel feature of this framework is a full semantic and syntactic treatment of variable-binding term operators as primitive symbols of logic. Subsequent chapters focus on the origin of modern foundational studies; Gottlob Frege's formal system intended to serve as a foundation for mathematics and its paradoxes; the theory of types; and the Zermelo-Fraenkel set theory. David Hilbert's program and Kurt Gödel's incompleteness theorems are also examined, along with the foundational systems of W. V. Quine and the relevance of categorical algebra for foundations. This monograph will be of interest to students, teachers, practitioners, and researchers in mathematics.
Posted in Mathematics

Quantum Groups

A Path to Current Algebra

Author: Ross Street

Publisher: Cambridge University Press

ISBN: 1139461443

Category: Mathematics

Page: N.A

View: 1753

Algebra has moved well beyond the topics discussed in standard undergraduate texts on 'modern algebra'. Those books typically dealt with algebraic structures such as groups, rings and fields: still very important concepts! However Quantum Groups: A Path to Current Algebra is written for the reader at ease with at least one such structure and keen to learn algebraic concepts and techniques. A key to understanding these new developments is categorical duality. A quantum group is a vector space with structure. Part of the structure is standard: a multiplication making it an 'algebra'. Another part is not in those standard books at all: a comultiplication, which is dual to multiplication in the precise sense of category theory, making it a 'coalgebra'. While coalgebras, bialgebras and Hopf algebras have been around for half a century, the term 'quantum group', along with revolutionary new examples, was launched by Drinfel'd in 1986.
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A Categorical Approach to Imprimitivity Theorems for $C^*$-Dynamical Systems

Author: Siegfried Echterhoff,S. Kaliszewski,John Quigg

Publisher: American Mathematical Soc.

ISBN: 0821838571

Category: Mathematics

Page: 169

View: 5759

Imprimitivity theorems provide a fundamental tool for studying the representation theory and structure of crossed-product $C^*$-algebras. In this work, we show that the Imprimitivity Theorem for induced algebras, Green's Imprimitivity Theorem for actions of groups, and Mansfield's Imprimitivity Theorem for coactions of groups can all be viewed as natural equivalences between various crossed-product functors among certain equivariant categories. The categories involved have $C^*$-algebras with actions or coactions (or both) of a fixed locally compact group $G$ as their objects, and equivariant equivalence classes of right-Hilbert bimodules as their morphisms. Composition is given by the balanced tensor product of bimodules.The functors involved arise from taking crossed products; restricting, inflating, and decomposing actions and coactions; inducing actions; and various combinations of these. Several applications of this categorical approach are also presented, including some intriguing relationships between the Green and Mansfield bimodules, and between restriction and induction of representations.
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Categorification and Higher Representation Theory

Author: Anna Beliakova,Aaron D. Lauda

Publisher: American Mathematical Soc.

ISBN: 1470424606

Category: Algebra

Page: 361

View: 7370

The emergent mathematical philosophy of categorification is reshaping our view of modern mathematics by uncovering a hidden layer of structure in mathematics, revealing richer and more robust structures capable of describing more complex phenomena. Categorified representation theory, or higher representation theory, aims to understand a new level of structure present in representation theory. Rather than studying actions of algebras on vector spaces where algebra elements act by linear endomorphisms of the vector space, higher representation theory describes the structure present when algebras act on categories, with algebra elements acting by functors. The new level of structure in higher representation theory arises by studying the natural transformations between functors. This enhanced perspective brings into play a powerful new set of tools that deepens our understanding of traditional representation theory. This volume exhibits some of the current trends in higher representation theory and the diverse techniques that are being employed in this field with the aim of showcasing the many applications of higher representation theory. The companion volume (Contemporary Mathematics, Volume 684) is devoted to categorification in geometry, topology, and physics.
Posted in Algebra

Algebraic Theories

A Categorical Introduction to General Algebra

Author: J. Adámek,J. Rosický,E. M. Vitale

Publisher: Cambridge University Press

ISBN: 1139491881

Category: Mathematics

Page: N.A

View: 2261

Algebraic theories, introduced as a concept in the 1960s, have been a fundamental step towards a categorical view of general algebra. Moreover, they have proved very useful in various areas of mathematics and computer science. This carefully developed book gives a systematic introduction to algebra based on algebraic theories that is accessible to both graduate students and researchers. It will facilitate interactions of general algebra, category theory and computer science. A central concept is that of sifted colimits - that is, those commuting with finite products in sets. The authors prove the duality between algebraic categories and algebraic theories and discuss Morita equivalence between algebraic theories. They also pay special attention to one-sorted algebraic theories and the corresponding concrete algebraic categories over sets, and to S-sorted algebraic theories, which are important in program semantics. The final chapter is devoted to finitary localizations of algebraic categories, a recent research area.
Posted in Mathematics

Representations of Algebraic Groups

Author: Jens Carsten Jantzen

Publisher: American Mathematical Soc.

ISBN: 082184377X


Page: 576

View: 402

The present book, which is a revised edition of the author's book published in 1987 by Academic Press, is intended to give the reader an introduction to the theory of algebraic representations of reductive algebraic groups. To develop appropriate techniques, the first part of the book is an introduction to the general theory of representations of algebraic group schemes. Here the author describes, among others, such important basic notions as induction functor, cohomology, quotients, Frobenius kernels, and reduction mod $p$. The second part of the book is devoted to the representation theory of reductive algebraic groups. It includes such topics as the description of simple modules, vanishing theorems, the Borel-Bott-Weil theorem and Weyl's character formula, Schubert schemes and line bundles on them. For this revised edition the author added several chapters describing some later developments, among them Schur algebras, Lusztig's conjecture, and Kazhdan-Lusztig polynomials, tilting modules, and representations of quantum groups.

Advanced Modern Algebra

Author: Joseph J. Rotman

Publisher: American Mathematical Soc.

ISBN: 0821847414

Category: Mathematics

Page: 1008

View: 6598

"This book is designed as a text for the first year of graduate algebra, but it can also serve as a reference since it contains more advanced topics as well. This second edition has a different organization than the first. It begins with a discussion of the cubic and quartic equations, which leads into permutations, group theory, and Galois theory (for finite extensions; infinite Galois theory is discussed later in the book). The study of groups continues with finite abelian groups (finitely generated groups are discussed later, in the context of module theory), Sylow theorems, simplicity of projective unimodular groups, free groups and presentations, and the Nielsen-Schreier theorem (subgroups of free groups are free). The study of commutative rings continues with prime and maximal ideals, unique factorization, noetherian rings, Zorn's lemma and applications, varieties, and Gr'obner bases. Next, noncommutative rings and modules are discussed, treating tensor product, projective, injective, and flat modules, categories, functors, and natural transformations, categorical constructions (including direct and inverse limits), and adjoint functors. Then follow group representations: Wedderburn-Artin theorems, character theory, theorems of Burnside and Frobenius, division rings, Brauer groups, and abelian categories. Advanced linear algebra treats canonical forms for matrices and the structure of modules over PIDs, followed by multilinear algebra. Homology is introduced, first for simplicial complexes, then as derived functors, with applications to Ext, Tor, and cohomology of groups, crossed products, and an introduction to algebraic K-theory. Finally, the author treats localization, Dedekind rings and algebraic number theory, and homological dimensions. The book ends with the proof that regular local rings have unique factorization."--Publisher's description.
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From a Geometrical Point of View

A Study of the History and Philosophy of Category Theory

Author: Jean-Pierre Marquis

Publisher: Springer Science & Business Media

ISBN: 1402093845

Category: Science

Page: 310

View: 4004

From a Geometrical Point of View explores historical and philosophical aspects of category theory, trying therewith to expose its significance in the mathematical landscape. The main thesis is that Klein’s Erlangen program in geometry is in fact a particular instance of a general and broad phenomenon revealed by category theory. The volume starts with Eilenberg and Mac Lane’s work in the early 1940’s and follows the major developments of the theory from this perspective. Particular attention is paid to the philosophical elements involved in this development. The book ends with a presentation of categorical logic, some of its results and its significance in the foundations of mathematics. From a Geometrical Point of View aims to provide its readers with a conceptual perspective on category theory and categorical logic, in order to gain insight into their role and nature in contemporary mathematics. It should be of interest to mathematicians, logicians, philosophers of mathematics and science in general, historians of contemporary mathematics, physicists and computer scientists.
Posted in Science

Representation Theory

A Homological Algebra Point of View

Author: Alexander Zimmermann

Publisher: Springer

ISBN: 3319079689

Category: Mathematics

Page: 707

View: 9306

Introducing the representation theory of groups and finite dimensional algebras, first studying basic non-commutative ring theory, this book covers the necessary background on elementary homological algebra and representations of groups up to block theory. It further discusses vertices, defect groups, Green and Brauer correspondences and Clifford theory. Whenever possible the statements are presented in a general setting for more general algebras, such as symmetric finite dimensional algebras over a field. Then, abelian and derived categories are introduced in detail and are used to explain stable module categories, as well as derived categories and their main invariants and links between them. Group theoretical applications of these theories are given – such as the structure of blocks of cyclic defect groups – whenever appropriate. Overall, many methods from the representation theory of algebras are introduced. Representation Theory assumes only the most basic knowledge of linear algebra, groups, rings and fields and guides the reader in the use of categorical equivalences in the representation theory of groups and algebras. As the book is based on lectures, it will be accessible to any graduate student in algebra and can be used for self-study as well as for classroom use.
Posted in Mathematics