Grothendieck Construction Of Bipermutative Indexed Categories
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Grothendieck Construction Of Bipermutative Indexed Categories
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Author : Donald Ying Yau
language : en
Publisher:
Release Date : 2024
Grothendieck Construction Of Bipermutative Indexed Categories written by Donald Ying Yau and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024 with Grothendieck categories categories.
"The Grothendieck construction provides an explicit link between indexed categories and opfibrations. It is a fundamental concept in category theory and related fields with far reaching applications. Bipermutative categories are categorifications of rings. They play a central role in algebraic K-theory and infinite loop space theory"--
Grothendieck Construction Of Bipermutative Indexed Categories
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Author : Donald Yau
language : en
Publisher: CRC Press
Release Date : 2023-12-06
Grothendieck Construction Of Bipermutative Indexed Categories written by Donald Yau and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-12-06 with Mathematics categories.
The Grothendieck construction provides an explicit link between indexed categories and opfibrations. It is a fundamental concept in category theory and related fields with far-reaching applications. Bipermutative categories are categorifications of rings. They play a central role in algebraic K-theory and infinite loop space theory. This monograph is a detailed study of the Grothendieck construction over a bipermutative category in the context of categorically enriched multicategories, with new and important applications to inverse K-theory and pseudo symmetric E∞-algebras. After carefully recalling preliminaries in enriched categories, bipermutative categories, and enriched multicategories, we show that the Grothendieck construction over a small tight bipermutative category is a pseudo symmetric Cat-multifunctor and generally not a Cat-multifunctor in the symmetric sense. Pseudo symmetry of Cat-multifunctors is a new concept we introduce in this work. The following features make it accessible as a graduate text or reference for experts: Complete definitions and proofs Self-contained background. Parts of Chapters 1–3, 7, 9, and 10 contain background material from the research literature Extensive cross-references Connections between chapters. Each chapter has its own introduction discussing not only the topics of that chapter but also its connection with other chapters Open questions. Appendix A contains open questions that arise from the material in the text and are suitable for graduate students This book is suitable for graduate students and researchers with an interest in category theory, algebraic K-theory, homotopy theory, and related fields. The presentation is thorough and self-contained, with complete details and background material for non-expert readers.
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.
Galois Extensions Of Structured Ring Spectra Stably Dualizable Groups
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Author : John Rognes
language : en
Publisher: American Mathematical Soc.
Release Date : 2008
Galois Extensions Of Structured Ring Spectra Stably Dualizable Groups written by John Rognes 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 2008 with Mathematics categories.
The author introduces the notion of a Galois extension of commutative $S$-algebras ($E_\infty$ ring spectra), often localized with respect to a fixed homology theory. There are numerous examples, including some involving Eilenberg-Mac Lane spectra of commutative rings, real and complex topological $K$-theory, Lubin-Tate spectra and cochain $S$-algebras. He establishes the main theorem of Galois theory in this generality. Its proof involves the notions of separable and etale extensions of commutative $S$-algebras, and the Goerss-Hopkins-Miller theory for $E_\infty$ mapping spaces. He shows that the global sphere spectrum $S$ is separably closed, using Minkowski's discriminant theorem, and he estimates the separable closure of its localization with respect to each of the Morava $K$-theories. He also defines Hopf-Galois extensions of commutative $S$-algebras and studies the complex cobordism spectrum $MU$ as a common integral model for all of the local Lubin-Tate Galois extensions. The author extends the duality theory for topological groups from the classical theory for compact Lie groups, via the topological study by J. R. Klein and the $p$-complete study for $p$-compact groups by T. Bauer, to a general duality theory for stably dualizable groups in the $E$-local stable homotopy category, for any spectrum $E$.
Infinity Operads And Monoidal Categories With Group Equivariance
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Author : Donald Yau
language : en
Publisher: World Scientific
Release Date : 2021-12-02
Infinity Operads And Monoidal Categories With Group Equivariance written by Donald Yau and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-12-02 with Mathematics categories.
This monograph provides a coherent development of operads, infinity operads, and monoidal categories, equipped with equivariant structures encoded by an action operad. A group operad is a planar operad with an action operad equivariant structure. In the first three parts of this monograph, we establish a foundation for group operads and for their higher coherent analogues called infinity group operads. Examples include planar, symmetric, braided, ribbon, and cactus operads, and their infinity analogues. For example, with the tools developed here, we observe that the coherent ribbon nerve of the universal cover of the framed little 2-disc operad is an infinity ribbon operad.In Part 4 we define general monoidal categories equipped with an action operad equivariant structure and provide a unifying treatment of coherence and strictification for them. Examples of such monoidal categories include symmetric, braided, ribbon, and coboundary monoidal categories, which naturally arise in the representation theory of quantum groups and of coboundary Hopf algebras and in the theory of crystals of finite dimensional complex reductive Lie algebras.
2 Dimensional Categories
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Author : Niles Johnson
language : en
Publisher: Oxford University Press (UK)
Release Date : 2021
2 Dimensional Categories written by Niles Johnson and has been published by Oxford University Press (UK) this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021 with Computers categories.
Category theory emerged in the 1940s in the work of Samuel Eilenberg and Saunders Mac Lane. It describes relationships between mathematical structures. Outside of pure mathematics, category theory is an important tool in physics, computer science, linguistics, and a quickly-growing list of other sciences. This book is about 2-dimensional categories, which add an extra dimension of richness and complexity to category theory. 2-Dimensional Categories is an introduction to 2-categories and bicategories, assuming only the most elementary aspects of category theory. A review of basic category theory is followed by a systematic discussion of 2-/bicategories, pasting diagrams, lax functors, 2-/bilimits, the Duskin nerve, 2-nerve, internal adjunctions, monads in bicategories, 2-monads, biequivalences, the Bicategorical Yoneda Lemma, and the Coherence Theorem for bicategories. Grothendieck fibrations and the Grothendieck construction are discussed next, followed by tricategories, monoidal bicategories, the Gray tensor product, and double categories. Completely detailed proofs of several fundamental but hard-to-find results are presented for the first time. With exercises and plenty of motivation and explanation, this book is useful for both beginners and experts.
Simplicial Homotopy Theory
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Author : Paul Gregory Goerss
language : en
Publisher: Springer Science & Business Media
Release Date : 1999
Simplicial Homotopy Theory written by Paul Gregory Goerss 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 1999 with Mathematics categories.
Since the beginning of the modern era of algebraic topology, simplicial methods have been used systematically and effectively for both computation and basic theory. With the development of Quillen's concept of a closed model category and, in particular, a simplicial model category, this collection of methods has become the primary way to describe non-abelian homological algebra and to address homotopy-theoretical issues in a variety of fields, including algebraic K-theory. This book supplies a modern exposition of these ideas, emphasizing model category theoretical techniques. Discussed here are the homotopy theory of simplicial sets, and other basic topics such as simplicial groups, Postnikov towers, and bisimplicial sets. The more advanced material includes homotopy limits and colimits, localization with respect to a map and with respect to a homology theory, cosimplicial spaces, and homotopy coherence. Interspersed throughout are many results and ideas well-known to experts, but uncollected in the literature. Intended for second-year graduate students and beyond, this book introduces many of the basic tools of modern homotopy theory. An extensive background in topology is not assumed.
Infinity Properads And Infinity Wheeled Properads
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Author : Philip Hackney
language : en
Publisher: Springer
Release Date : 2015-09-07
Infinity Properads And Infinity Wheeled Properads written by Philip Hackney and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-09-07 with Mathematics categories.
The topic of this book sits at the interface of the theory of higher categories (in the guise of (∞,1)-categories) and the theory of properads. Properads are devices more general than operads and enable one to encode bialgebraic, rather than just (co)algebraic, structures. The text extends both the Joyal-Lurie approach to higher categories and the Cisinski-Moerdijk-Weiss approach to higher operads, and provides a foundation for a broad study of the homotopy theory of properads. This work also serves as a complete guide to the generalised graphs which are pervasive in the study of operads and properads. A preliminary list of potential applications and extensions comprises the final chapter. Infinity Properads and Infinity Wheeled Properads is written for mathematicians in the fields of topology, algebra, category theory, and related areas. It is written roughly at the second year graduate level, and assumes a basic knowledge of category theory.
A First Course In Analysis
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Author : Donald Yau
language : en
Publisher: World Scientific
Release Date : 2013
A First Course In Analysis written by Donald Yau 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 is an introductory text on real analysis for undergraduate students. The prerequisite for this book is a solid background in freshman calculus in one variable. The intended audience of this book includes undergraduate mathematics majors and students from other disciplines who use real analysis. Since this book is aimed at students who do not have much prior experience with proofs, the pace is slower in earlier chapters than in later chapters. There are hundreds of exercises, and hints for some of them are included.
Colored Operads
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Author : Donald Yau
language : en
Publisher: American Mathematical Soc.
Release Date : 2016-02-29
Colored Operads written by Donald Yau 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 2016-02-29 with Mathematics categories.
The subject of this book is the theory of operads and colored operads, sometimes called symmetric multicategories. A (colored) operad is an abstract object which encodes operations with multiple inputs and one output and relations between such operations. The theory originated in the early 1970s in homotopy theory and quickly became very important in algebraic topology, algebra, algebraic geometry, and even theoretical physics (string theory). Topics covered include basic graph theory, basic category theory, colored operads, and algebras over colored operads. Free colored operads are discussed in complete detail and in full generality. The intended audience of this book includes students and researchers in mathematics and other sciences where operads and colored operads are used. The prerequisite for this book is minimal. Every major concept is thoroughly motivated. There are many graphical illustrations and about 150 exercises. This book can be used in a graduate course and for independent study.