homological algebra in strongly non abelian settings

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Homological Algebra

Author : Marco Grandis
ISBN : 9789814425933
Genre : Mathematics
File Size : 31. 72 MB
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We propose here a study of ‘semiexact’ and ‘homological' categories as a basis for a generalised homological algebra. Our aim is to extend the homological notions to deeply non-abelian situations, where satellites and spectral sequences can still be studied. This is a sequel of a book on ‘Homological Algebra, The interplay of homology with distributive lattices and orthodox semigroups’, published by the same Editor, but can be read independently of the latter. The previous book develops homological algebra in p-exact categories, i.e. exact categories in the sense of Puppe and Mitchell — a moderate generalisation of abelian categories that is nevertheless crucial for a theory of ‘coherence’ and ‘universal models’ of (even abelian) homological algebra. The main motivation of the present, much wider extension is that the exact sequences or spectral sequences produced by unstable homotopy theory cannot be dealt with in the previous framework. According to the present definitions, a semiexact category is a category equipped with an ideal of ‘null’ morphisms and provided with kernels and cokernels with respect to this ideal. A homological category satisfies some further conditions that allow the construction of subquotients and induced morphisms, in particular the homology of a chain complex or the spectral sequence of an exact couple. Extending abelian categories, and also the p-exact ones, these notions include the usual domains of homology and homotopy theories, e.g. the category of ‘pairs’ of topological spaces or groups; they also include their codomains, since the sequences of homotopy ‘objects’ for a pair of pointed spaces or a fibration can be viewed as exact sequences in a homological category, whose objects are actions of groups on pointed sets. Homological Algebra: The Interplay of Homology with Distributive Lattices and Orthodox Semigroups Contents:IntroductionSemiexact categoriesHomological CategoriesSubquotients, Homology and Exact CouplesSatellitesUniversal ConstructionsApplications to Algebraic TopologyHomological Theories and Biuniversal ModelsAppendix A. Some Points of Category Theory Readership: Graduate students, professors and researchers in pure mathematics, in particular category theory and algebraic topology. Keywords:Non Abelian Homological Algebra;Spectral Sequences;Distributive Lattices;Orthodox Semigroups;Categories of RelationsReviews: “The range of applications and examples is considerable and many are outside the reach of more standard forms of homological algebra, but the methods used here also give insight as to 'why' the classical theory works and how its results can be interpreted.” Zentralblatt MATH

Homological Algebra

Author : Marco Grandis
ISBN : 9789814407069
Genre : Mathematics
File Size : 40. 2 MB
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In this book we want to explore aspects of coherence in homological algebra, that already appear in the classical situation of abelian groups or abelian categories. Lattices of subobjects are shown to play an important role in the study of homological systems, from simple chain complexes to all the structures that give rise to spectral sequences. A parallel role is played by semigroups of endorelations. These links rest on the fact that many such systems, but not all of them, live in distributive sublattices of the modular lattices of subobjects of the system. The property of distributivity allows one to work with induced morphisms in an automatically consistent way, as we prove in a 'Coherence Theorem for homological algebra'. (On the contrary, a 'non-distributive' homological structure like the bifiltered chain complex can easily lead to inconsistency, if one explores the interaction of its two spectral sequences farther than it is normally done.) The same property of distributivity also permits representations of homological structures by means of sets and lattices of subsets, yielding a precise foundation for the heuristic tool of Zeeman diagrams as universal models of spectral sequences. We thus establish an effective method of working with spectral sequences, called 'crossword chasing', that can often replace the usual complicated algebraic tools and be of much help to readers that want to apply spectral sequences in any field.

Category Theory And Applications

Author : Marco Grandis
ISBN : 9789813231085
Genre :
File Size : 56. 27 MB
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Category Theory now permeates most of Mathematics, large parts of theoretical Computer Science and parts of theoretical Physics. Its unifying power brings together different branches, and leads to a deeper understanding of their roots. This book is addressed to students and researchers of these fields and can be used as a text for a first course in Category Theory. It covers its basic tools, like universal properties, limits, adjoint functors and monads. These are presented in a concrete way, starting from examples and exercises taken from elementary Algebra, Lattice Theory and Topology, then developing the theory together with new exercises and applications. Applications of Category Theory form a vast and differentiated domain. This book wants to present the basic applications and a choice of more advanced ones, based on the interests of the author. References are given for applications in many other fields. Contents: Introduction Categories, Functors and Natural Transformations Limits and Colimits Adjunctions and Monads Applications in Algebra Applications in Topology and Algebraic Topology Applications in Homological Algebra Hints at Higher Dimensional Category Theory References Indices Readership: Graduate students and researchers of mathematics, computer science, physics. Keywords: Category TheoryReview: Key Features: The main notions of Category Theory are presented in a concrete way, starting from examples taken from the elementary part of well-known disciplines: Algebra, Lattice Theory and Topology The theory is developed presenting other examples and some 300 exercises; the latter are endowed with a solution, or a partial solution, or adequate hints Three chapters and some extra sections are devoted to applications

Foundations Of Relative Homological Algebra

Author : Samuel Eilenberg
ISBN : 9780821812556
Genre : Algebra, Homological
File Size : 72. 89 MB
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Nonabelian Algebraic Topology

Author : Ronald Brown
ISBN : 3037190833
Genre : Mathematics
File Size : 31. 14 MB
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A Singular Introduction To Commutative Algebra

Author : Gert-Martin Greuel
ISBN : 9783662049631
Genre : Mathematics
File Size : 47. 42 MB
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This book can be understood as a model for teaching commutative algebra, and takes into account modern developments such as algorithmic and computational aspects. As soon as a new concept is introduced, the authors show how the concept can be worked on using a computer. The computations are exemplified with the computer algebra system Singular, developed by the authors. Singular is a special system for polynomial computation with many features for global as well as for local commutative algebra and algebraic geometry. The book includes a CD containing Singular as well as the examples and procedures explained in the book.

Operator Algebras Operator Theory And Applications

Author : J. J. Grobler
ISBN : 3034601743
Genre : Mathematics
File Size : 87. 27 MB
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Introduction To Commutative Algebra And Algebraic Geometry

Author : Ernst Kunz
ISBN : 0817630651
Genre : Mathematics
File Size : 61. 98 MB
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Mal Cev Protomodular Homological And Semi Abelian Categories

Author : Francis Borceux
ISBN : 1402019610
Genre : Mathematics
File Size : 23. 75 MB
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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.

A Non Hausdorff Completion

Author : Saul Lubkin
ISBN : 9789814667401
Genre : Mathematics
File Size : 70. 19 MB
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This book introduces entirely new invariants never considered before, in homological algebra and commutative (and even non-commutative) algebra. The C-completion C(M), and higher C-completions, Cn(M), are defined for an arbitrary left module M over a topological ring A. Spectral sequences are defined that use these invariants. Given a left module over a topological ring A, under mild conditions the usual Hausdorff completion: M^ can be recovered from the C-completion C(M), by taking the quotient module by the closure of {0}. The new invariants and tools in this book are expected to be used in the study of p-adic cohomology in algebraic geometry; and also in the study of p-adic Banach spaces — by replacing the cumbersome "complete tensor product" of p-adic Banach spaces, with the more sophisticated "C-complete tensor product", discussed in this book. It is also not unlikely that the further study of these new invariants may well develop into a new branch of abstract mathematics - connected with commutative algebra, homological algebra, and algebraic topology.

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