Item Details
Homological Algebra [electronic resource]: The Interplay of Homology With Distributive Lattices and Orthodox Semigroups
Marco Grandis
 Format
 EBook; Book; Online
 Published
 Singapore ; Hackensack, N.J. : World Scientific, [2012]
 Language
 English
 ISBN
 9789814407069, 9814407062
 Abstract
 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 'nondistributive' 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.
 Contents
 Introduction
 1. Coherence and models in homological algebra
 2. Puppeexact categories, orthodoxy. 2.8. Weak induction and the distributive expansion
 3. Involutive categories
 4. Categories of relations as REcategories
 5. Theories and models
 6. Homological theories and their universal models
 Appendix A. Some points of category theory
 Appendix B. A proof for the universal exact system.
 Description
 Mode of access: World wide Web.
 Notes
 Includes bibliographical references (pages 357361) and index.
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LEADER 03117cam a22003495i 4500001 u6420503003 SIRSI005 20170909041908.0006 m d007 cr n008 130620s2012 si a sb 001 0 eng da 2012552090a 9789814407069a 9814407062a (WaSeSS)ssj0000737763a BTCTA b eng c BTCTA d CDX d OCLCO d YDXCP d BWX d VGM d MUU d DLC d WaSeSSa lccopycata QA169 b .G73 2012a 512/.55 2 22a Grandis, Marco.a Homological algebra h [electronic resource] : b the interplay of homology with distributive lattices and orthodox semigroups / c Marco Grandis.a Singapore ; a Hackensack, N.J. : b World Scientific, c [2012]a Includes bibliographical references (pages 357361) and index.a Introduction  1. Coherence and models in homological algebra  2. Puppeexact categories, orthodoxy. 2.8. Weak induction and the distributive expansion  3. Involutive categories  4. Categories of relations as REcategories  5. Theories and models  6. Homological theories and their universal models  Appendix A. Some points of category theory  Appendix B. A proof for the universal exact system.a 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 'nondistributive' 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.a Mode of access: World wide Web.a Algebra, Homological.a Electronic books.a Ebook Central  Academic Completeu http://RE5QY4SB7X.search.serialssolutions.com/?V=1.0&L=RE5QY4SB7X&S=JCs&C=TC0000737763&T=marca 1a XX(6420503.1) w WEB i 64205031001 l INTERNET m UVALIB t INTERNET