Category Structures
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Model Categories And Their Localizations
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Author : Philip S. Hirschhorn
language : en
Publisher: American Mathematical Soc.
Release Date : 2003
Model Categories And Their Localizations written by Philip S. Hirschhorn and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2003 with Mathematics categories.
The aim of this book is to explain modern homotopy theory in a manner accessible to graduate students yet structured so that experts can skip over numerous linear developments to quickly reach the topics of their interest. Homotopy theory arises from choosing a class of maps, called weak equivalences, and then passing to the homotopy category by localizing with respect to the weak equivalences, i.e., by creating a new category in which the weak equivalences are isomorphisms. Quillen defined a model category to be a category together with a class of weak equivalences and additional structure useful for describing the homotopy category in terms of the original category. This allows you to make constructions analogous to those used to study the homotopy theory of topological spaces. A model category has a class of maps called weak equivalences plus two other classes of maps, called cofibrations and fibrations. Quillen's axioms ensure that the homotopy category exists and that the cofibrations and fibrations have extension and lifting properties similar to those of cofibration and fibration maps of topological spaces. During the past several decades the language of model categories has become standard in many areas of algebraic topology, and it is increasingly being used in other fields where homotopy theoretic ideas are becoming important, including modern algebraic $K$-theory and algebraic geometry. All these subjects and more are discussed in the book, beginning with the basic definitions and giving complete arguments in order to make the motivations and proofs accessible to the novice. The book is intended for graduate students and research mathematicians working in homotopy theory and related areas.
Categories In Algebra Geometry And Mathematical Physics
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Author : Alexei Davydov
language : en
Publisher: American Mathematical Soc.
Release Date : 2007
Categories In Algebra Geometry And Mathematical Physics written by Alexei Davydov and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2007 with Mathematics categories.
Category theory has become the universal language of modern mathematics. This book is a collection of articles applying methods of category theory to the areas of algebra, geometry, and mathematical physics. Among others, this book contains articles on higher categories and their applications and on homotopy theoretic methods. The reader can learn about the exciting new interactions of category theory with very traditional mathematical disciplines.
Categories For The Working Mathematician
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Author : Saunders Mac Lane
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-04-17
Categories For The Working Mathematician written by Saunders Mac Lane and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-04-17 with Mathematics categories.
Categories for the Working Mathematician provides an array of general ideas useful in a wide variety of fields. Starting from the foundations, this book illuminates the concepts of category, functor, natural transformation, and duality. The book then turns to adjoint functors, which provide a description of universal constructions, an analysis of the representations of functors by sets of morphisms, and a means of manipulating direct and inverse limits. These categorical concepts are extensively illustrated in the remaining chapters, which include many applications of the basic existence theorem for adjoint functors. The categories of algebraic systems are constructed from certain adjoint-like data and characterized by Beck's theorem. After considering a variety of applications, the book continues with the construction and exploitation of Kan extensions. This second edition includes a number of revisions and additions, including two new chapters on topics of active interest. One is onsymmetric monoidal categories and braided monoidal categories and the coherence theorems for them. The second describes 2-categories and the higher dimensional categories which have recently come into prominence. The bibliography has also been expanded to cover some of the many other recent advances concerning categories.
Categories Types And Structures
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Author : Andrea Asperti
language : en
Publisher: MIT Press (MA)
Release Date : 1991
Categories Types And Structures written by Andrea Asperti and has been published by MIT Press (MA) this book supported file pdf, txt, epub, kindle and other format this book has been release on 1991 with Computers categories.
Category theory is a mathematical subject whose importance in several areas of computer science, most notably the semantics of programming languages and the design of programmes using abstract data types, is widely acknowledged. This book introduces category theory at a level appropriate for computer scientists and provides practical examples in the context of programming language design.
Knowledge Concepts And Categories
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Author : Koen Lamberts
language : en
Publisher: Psychology Press
Release Date : 2024-11-01
Knowledge Concepts And Categories written by Koen Lamberts and has been published by Psychology Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024-11-01 with Psychology categories.
This text brings together an overview of recent research on concepts and knowledge that abstracts across a variety of specific fields of cognitive psychology. Readers will find data from many different areas, including developmental psychology, formal modelling, neuropsychology and connectionism.
Early Category And Concept Development Making Sense Of The Blooming Buzzing Confusion
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Author : David H. Rakison Assistant Professor of Psychology Carnegie Mellon University
language : en
Publisher: Oxford University Press, USA
Release Date : 2003-01-09
Early Category And Concept Development Making Sense Of The Blooming Buzzing Confusion written by David H. Rakison Assistant Professor of Psychology Carnegie Mellon University and has been published by Oxford University Press, USA this book supported file pdf, txt, epub, kindle and other format this book has been release on 2003-01-09 with Psychology categories.
Whether or not infants' earliest perception of the world is a "blooming, buzzing, confusion," it is not long before they come to perceive structure and order among the objects and events around them. At the core of this process, and cognitive development in general, is the ability to categorize--to group events, objects, or properties together--and to form mental representations, or concepts, that encapsulate the commonalities and structure of these categories. Categorization is the primary means of coding experience, underlying not only perceptual and reasoning processes, but also inductive inference and language. The aim of this book is to bring together the most recent findings and theories about the origins and early development of categorization and conceptual abilities. Despite recent advances in our understanding of this area, a number of hotly debated issues remain at the center of the controversy over categorization. Researchers continue to ask questions such as: Which mechanisms for categorization are available at birth and which emerge later? What are the relative roles of perceptual similarity and nonobservable properties in early classification? What is the role of contextual variation in categorization by infants and children? Do different experimental procedures reveal the same kind of knowledge? Can computational models simulate infant and child categorization? How do computational models inform behavioral research? What is the impact of language on category development? How does language partition the world? This book is the first to address these and other key cuestions within a single volume. The authors present a diverse set of views representing cutting-edge empirical and theoretical advances in the field. The result is a thorough review of empirical contributions to the literature, and a wealth of fresh theoretical perspectives on early categorization.
Formal Category Theory Adjointness For 2 Categories
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Author : J.W. Gray
language : en
Publisher: Springer
Release Date : 2006-11-15
Formal Category Theory Adjointness For 2 Categories written by J.W. Gray and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-11-15 with Mathematics categories.
Percepts Concepts And Categories
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Author : B. Burns
language : en
Publisher: Elsevier
Release Date : 1992-10-09
Percepts Concepts And Categories written by B. Burns and has been published by Elsevier this book supported file pdf, txt, epub, kindle and other format this book has been release on 1992-10-09 with Science categories.
The most important distinction derived from the computational view of thought is between structures and processes. So proclaimed Farah and Kosslyn in 1982, arguing that structures and processes cannot be examined in isolation and concluding that converging operations are required to isolate the structure-process pair that can explain a particular finding. The distinction between structure and process within the study of percepts, concepts and categories is considered in depth in this volume, with penetrating commentaries by fellow authors concluding each chapter. This interesting format achieves a broad coverage of the various aspects and implications of the structure-process distinction. It affords a salient indication of the diversity of positions as to the description and utility of distinguishing structures and processors. At the same time, it reveals that researchers specializing in areas of study ranging from simple structure and process involved in perceptual organization and texture to complex structure and process associated with reading graphs and chess expertise, do utilize such a distinction in similar ways. The analysis is organized into four major parts within the book: Early Visual Representation and Processing; Percepts, Concepts, Categories and Development; Categories, Concepts and Learning; and Higher-Order Representation and Processing.
Towards Higher Categories
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Author : John C. Baez
language : en
Publisher: Springer Science & Business Media
Release Date : 2009-09-23
Towards Higher Categories written by John C. Baez and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009-09-23 with Mathematics categories.
This IMA Volume in Mathematics and its Applications TOWARDS HIGHER CATEGORIES contains expository and research papers based on a highly successful IMA Summer Program on n-Categories: Foundations and Applications. We are grateful to all the participants for making this occasion a very productive and stimulating one. We would like to thank John C. Baez (Department of Mathematics, University of California Riverside) and J. Peter May (Department of Ma- ematics, University of Chicago) for their superb role as summer program organizers and editors of this volume. We take this opportunity to thank the National Science Foundation for its support of the IMA. Series Editors Fadil Santosa, Director of the IMA Markus Keel, Deputy Director of the IMA v PREFACE DEDICATED TO MAX KELLY, JUNE 5 1930 TO JANUARY 26 2007. This is not a proceedings of the 2004 conference “n-Categories: Fo- dations and Applications” that we organized and ran at the IMA during the two weeks June 7–18, 2004! We thank all the participants for helping make that a vibrant and inspiring occasion. We also thank the IMA sta? for a magni?cent job. There has been a great deal of work in higher c- egory theory since then, but we still feel that it is not yet time to o?er a volume devoted to the main topic of the conference.
A Handbook Of Model Categories
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Author : Scott Balchin
language : en
Publisher: Springer Nature
Release Date : 2021-10-29
A Handbook Of Model Categories written by Scott Balchin and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-10-29 with Mathematics categories.
This book outlines a vast array of techniques and methods regarding model categories, without focussing on the intricacies of the proofs. Quillen model categories are a fundamental tool for the understanding of homotopy theory. While many introductions to model categories fall back on the same handful of canonical examples, the present book highlights a large, self-contained collection of other examples which appear throughout the literature. In particular, it collects a highly scattered literature into a single volume. The book is aimed at anyone who uses, or is interested in using, model categories to study homotopy theory. It is written in such a way that it can be used as a reference guide for those who are already experts in the field. However, it can also be used as an introduction to the theory for novices.