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Mathematical Surveys and Monographs
Categories and Representation Theory: With A Focus on 2-Categorical Covering Theory
Hideto Asashiba: Shizuoka University, Suruga-ku, Shizuoka, Japan and Kyoto University, Sakyo-ku, Kyoto, Japan and Osaka Central Advanced Mathematical Institute, Sumiyoshi-ku, Osaka, Japan
Digital content for Categories and Representation Theory: With A Focus on 2-Categorical Covering Theory
This book gives a self-contained account of applications of category theory to the theory of representations of algebras. Its main focus is on 2-categorical techniques, including 2-categorical covering theory. The book has few prerequisites beyond linear algebra and elementary ring theory, but familiarity with the basics of representations of quivers and of category theory will be helpful. In addition to providing an introduction to category theory, the book develops useful tools such as quivers, adjoints, string diagrams, and tensor products over a small category; gives an exposition of new advances such as a 2-categorical generalization of Cohen-Montgomery duality in pseudo-actions of a group; and develops the moderation level of categories, first proposed by Levy, to avoid the set theoretic paradox in category theory.
The book is accessible to advanced undergraduate and graduate students who would like to study the representation theory of algebras, and it contains many exercises. It can be used as the textbook for an introductory course on the category theoretic approach with an emphasis on 2-categories, and as a reference for researchers in algebra interested in derived equivalences and covering theory.
Readership
Undergraduate and graduate students interested in the representation theory of algebras and 2-categorical covering theory.
Table Of Contents
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Front/Back Matter
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View this volume's front and back matter
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Chapters
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Categoriespp. 1 - 23
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Representationspp. 25 - 59
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Classical covering theorypp. 61 - 64
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Basics of 2-categoriespp. 65 - 94
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2-categorical covering theory under pseudo-actions of a grouppp. 95 - 145
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Computations of orbit categories and smash productspp. 147 - 162
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Relationships between module categoriespp. 163 - 185
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2-categorical covering theory under colax actions of a categorypp. 187 - 192
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Set theory for the foundation of category theorempp. 193 - 210
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Supplement to the original versionpp. 211 - 232
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DOI: https://doi.org/10.1090/surv/271
MSC: Primary 18-01; 18-02; 18A50; 18M30; 18N10; 16-02; 16D90; 16G20; 16W22; 16W50
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