Homological Algebra The Interplay Of Homology With Distributive Lattices And Orthodox Semigroups
DOWNLOAD
AUDIOBOOK
READ ONLINE
Download Homological Algebra The Interplay Of Homology With Distributive Lattices And Orthodox Semigroups PDF/ePub or read online books in Mobi eBooks. Click Download or Read Online button to get Homological Algebra The Interplay Of Homology With Distributive Lattices And Orthodox Semigroups book now. This website allows unlimited access to, at the time of writing, more than 1.5 million titles, including hundreds of thousands of titles in various foreign languages. If the content not found or just blank you must refresh this page
Homological Algebra
DOWNLOAD
AUDIOBOOK
READ ONLINE
Author : Marco Grandis
language : en
Publisher: World Scientific
Release Date : 2012
Homological Algebra written by Marco Grandis and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012 with Mathematics categories.
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.
Homological Algebra In Strongly Non Abelian Settings
DOWNLOAD
AUDIOBOOK
READ ONLINE
Author : Marco Grandis
language : en
Publisher: World Scientific
Release Date : 2013-01-11
Homological Algebra In Strongly Non Abelian Settings written by Marco Grandis and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-01-11 with Mathematics categories.
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
DOWNLOAD
AUDIOBOOK
READ ONLINE
Author : Marco Grandis
language : en
Publisher: World Scientific
Release Date : 2012-06-08
Homological Algebra The Interplay Of Homology With Distributive Lattices And Orthodox Semigroups written by Marco Grandis and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-06-08 with Mathematics categories.
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.
Homological Algebra
DOWNLOAD
AUDIOBOOK
READ ONLINE
Author : Marco Grandis
language : en
Publisher: World Scientific
Release Date : 2013
Homological Algebra written by Marco Grandis and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013 with Mathematics categories.
This book proposes a study of semi-exact homological categories as a basis for a generalized homological algebra. The aim is to extend homological notions to deeply non-abelian situations, where satellites and spectral sequences produced by unstable homotopy can still be studied.
Elementary Overview Of Mathematical Structures An Algebra Topology And Categories
DOWNLOAD
AUDIOBOOK
READ ONLINE
Author : Marco Grandis
language : en
Publisher: World Scientific
Release Date : 2020-08-12
Elementary Overview Of Mathematical Structures An Algebra Topology And Categories written by Marco Grandis and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-08-12 with Mathematics categories.
'The presentation is modeled on the discursive style of the Bourbaki collective, and the coverage of topics is rich and varied. Grandis has provided a large selection of exercises and has sprinkled orienting comments throughout. For an undergraduate library where strong students seek an overview of a significant portion of mathematics, this would be an excellent acquisition. Summing up: Recommended.'CHOICESince the last century, a large part of Mathematics is concerned with the study of mathematical structures, from groups to fields and vector spaces, from lattices to Boolean algebras, from metric spaces to topological spaces, from topological groups to Banach spaces.More recently, these structured sets and their transformations have been assembled in higher structures, called categories.We want to give a structural overview of these topics, where the basic facts of the different theories are unified through the 'universal properties' that they satisfy, and their particularities stand out, perhaps even more.This book can be used as a textbook for undergraduate studies and for self-study. It can provide students of Mathematics with a unified perspective of subjects which are often kept apart. It is also addressed to students and researchers of disciplines having strong interactions with Mathematics, like Physics and Chemistry, Statistics, Computer Science, Engineering.
Manifolds And Local Structures A General Theory
DOWNLOAD
AUDIOBOOK
READ ONLINE
Author : Marco Grandis
language : en
Publisher: World Scientific
Release Date : 2021-02-10
Manifolds And Local Structures A General Theory written by Marco Grandis and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-02-10 with Mathematics categories.
Local structures, like differentiable manifolds, fibre bundles, vector bundles and foliations, can be obtained by gluing together a family of suitable 'elementary spaces', by means of partial homeomorphisms that fix the gluing conditions and form a sort of 'intrinsic atlas', instead of the more usual system of charts living in an external framework.An 'intrinsic manifold' is defined here as such an atlas, in a suitable category of elementary spaces: open euclidean spaces, or trivial bundles, or trivial vector bundles, and so on.This uniform approach allows us to move from one basis to another: for instance, the elementary tangent bundle of an open Euclidean space is automatically extended to the tangent bundle of any differentiable manifold. The same holds for tensor calculus.Technically, the goal of this book is to treat these structures as 'symmetric enriched categories' over a suitable basis, generally an ordered category of partial mappings.This approach to gluing structures is related to Ehresmann's one, based on inductive pseudogroups and inductive categories. A second source was the theory of enriched categories and Lawvere's unusual view of interesting mathematical structures as categories enriched over a suitable basis.
Algebraic Topology A Structural Introduction
DOWNLOAD
AUDIOBOOK
READ ONLINE
Author : Marco Grandis
language : en
Publisher: World Scientific
Release Date : 2021-12-24
Algebraic Topology A Structural Introduction written by Marco Grandis and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-12-24 with Mathematics categories.
Algebraic Topology is a system and strategy of partial translations, aiming to reduce difficult topological problems to algebraic facts that can be more easily solved. The main subject of this book is singular homology, the simplest of these translations. Studying this theory and its applications, we also investigate its underlying structural layout - the topics of Homological Algebra, Homotopy Theory and Category Theory which occur in its foundation.This book is an introduction to a complex domain, with references to its advanced parts and ramifications. It is written with a moderate amount of prerequisites — basic general topology and little else — and a moderate progression starting from a very elementary beginning. A consistent part of the exposition is organised in the form of exercises, with suitable hints and solutions.It can be used as a textbook for a semester course or self-study, and a guidebook for further study.
Category Theory And Applications A Textbook For Beginners Second Edition
DOWNLOAD
AUDIOBOOK
READ ONLINE
Author : Marco Grandis
language : en
Publisher: World Scientific
Release Date : 2021-03-05
Category Theory And Applications A Textbook For Beginners Second Edition written by Marco Grandis and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-03-05 with Mathematics categories.
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 better 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 the 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.A reader should have some elementary knowledge of these three subjects, or at least two of them, in order to be able to follow the main examples, appreciate the unifying power of the categorical approach, and discover the subterranean links brought to light and formalised by this perspective.Applications of Category Theory form a vast and differentiated domain. This book wants to present the basic applications in Algebra and Topology, with a choice of more advanced ones, based on the interests of the author. References are given for applications in many other fields.In this second edition, the book has been entirely reviewed, adding many applications and exercises. All non-obvious exercises have now a solution (or a reference, in the case of an advanced topic); solutions are now collected in the last chapter.
Higher Dimensional Categories From Double To Multiple Categories
DOWNLOAD
AUDIOBOOK
READ ONLINE
Author : Grandis Marco
language : en
Publisher: World Scientific
Release Date : 2019-09-09
Higher Dimensional Categories From Double To Multiple Categories written by Grandis Marco and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-09-09 with Mathematics categories.
The study of higher dimensional categories has mostly been developed in the globular form of 2-categories, n-categories, omega-categories and their weak versions. Here we study a different form: double categories, n-tuple categories and multiple categories, with their weak and lax versions.We want to show the advantages of this form for the theory of adjunctions and limits. Furthermore, this form is much simpler in higher dimension, starting with dimension three where weak 3-categories (also called tricategories) are already quite complicated, much more than weak or lax triple categories.This book can be used as a textbook for graduate and postgraduate studies, and as a basis for research. Notions are presented in a 'concrete' way, with examples and exercises; the latter are endowed with a solution or hints. Part I, devoted to double categories, starts at basic category theory and is kept at a relatively simple level. Part II, on multiple categories, can be used independently by a reader acquainted with 2-dimensional categories.
Topological Data Analysis
DOWNLOAD
AUDIOBOOK
READ ONLINE
Author : Nils A. Baas
language : en
Publisher: Springer Nature
Release Date : 2020-06-25
Topological Data Analysis written by Nils A. Baas and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-06-25 with Mathematics categories.
This book gathers the proceedings of the 2018 Abel Symposium, which was held in Geiranger, Norway, on June 4-8, 2018. The symposium offered an overview of the emerging field of "Topological Data Analysis". This volume presents papers on various research directions, notably including applications in neuroscience, materials science, cancer biology, and immune response. Providing an essential snapshot of the status quo, it represents a valuable asset for practitioners and those considering entering the field.