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Softcover ISBN:  9781470478100 
Product Code:  SURV/284 
List Price:  $135.00 
MAA Member Price:  $121.50 
AMS Member Price:  $108.00 
eBook ISBN:  9781470478476 
Product Code:  SURV/284.E 
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MAA Member Price:  $112.50 
AMS Member Price:  $100.00 
Softcover ISBN:  9781470478100 
eBook ISBN:  9781470478476 
Product Code:  SURV/284.B 
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Book DetailsMathematical Surveys and MonographsVolume: 284; 2024; 404 ppMSC: Primary 18; 19; 55
Bimonoidal categories are categorical analogues of rings without additive inverses. They have been actively studied in category theory, homotopy theory, and algebraic \(K\)theory since around 1970. There is an abundance of new applications and questions of bimonoidal categories in mathematics and other sciences. The three books published by the AMS in the Mathematical Surveys and Monographs series under the title Bimonoidal Categories, \(E_n\)Monoidal Categories, and Algebraic \(K\)Theory (Volume I: Symmetric Bimonoidal Categories and Monoidal Bicategories, Volume II: Braided Bimonoidal Categories with Applications—this book, and Volume III: From Categories to Structured Ring Spectra) provide a unified treatment of bimonoidal and higher ringlike categories, their connection with algebraic \(K\)theory and homotopy theory, and applications to quantum groups and topological quantum computation. With ample background material, extensive coverage, detailed presentation of both wellknown and new theorems, and a list of open questions, this work is a userfriendly resource for beginners and experts alike.
Part 1 of this book studies braided bimonoidal categories, with applications to quantum groups and topological quantum computation. It is proved that the categories of modules over a braided bialgebra, of Fibonacci anyons, and of Ising anyons form braided bimonoidal categories. Two coherence theorems for braided bimonoidal categories are proved, confirming the BlassGurevich Conjecture. The rest of this part discusses braided analogues of Baez's Conjecture and the monoidal bicategorical matrix construction in Volume I: Symmetric Bimonoidal Categories and Monoidal Bicategories. Part 2 studies ring and bipermutative categories in the sense of ElmendorfMandell, braided ring categories, and \(E_n\)monoidal categories, which combine \(n\)fold monoidal categories with ring categories.
ReadershipGraduate students and researchers interested in category theory and algebraic \(K\)theory.
This item is also available as part of a set: 
Table of Contents

Braided bimonoidal categories

Preliminaries on braided structures

Braided bimonoidal categories

Applications to quantum groups and topological quantum computation

Bimonoidal centers

Coherence of braided bimonoidal categories

Strictification of tight braided bimonoidal categories

The braided Baez conjecture

Monoidal bicategorification

$E_n$monoidal categories

Ring, bipermutative, and braided ring categories

Iterated and $E_n$monoidal categories

Bibliography and indices

Open questions

Bibliography

List of main facts

List of notations

Index


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Bimonoidal categories are categorical analogues of rings without additive inverses. They have been actively studied in category theory, homotopy theory, and algebraic \(K\)theory since around 1970. There is an abundance of new applications and questions of bimonoidal categories in mathematics and other sciences. The three books published by the AMS in the Mathematical Surveys and Monographs series under the title Bimonoidal Categories, \(E_n\)Monoidal Categories, and Algebraic \(K\)Theory (Volume I: Symmetric Bimonoidal Categories and Monoidal Bicategories, Volume II: Braided Bimonoidal Categories with Applications—this book, and Volume III: From Categories to Structured Ring Spectra) provide a unified treatment of bimonoidal and higher ringlike categories, their connection with algebraic \(K\)theory and homotopy theory, and applications to quantum groups and topological quantum computation. With ample background material, extensive coverage, detailed presentation of both wellknown and new theorems, and a list of open questions, this work is a userfriendly resource for beginners and experts alike.
Part 1 of this book studies braided bimonoidal categories, with applications to quantum groups and topological quantum computation. It is proved that the categories of modules over a braided bialgebra, of Fibonacci anyons, and of Ising anyons form braided bimonoidal categories. Two coherence theorems for braided bimonoidal categories are proved, confirming the BlassGurevich Conjecture. The rest of this part discusses braided analogues of Baez's Conjecture and the monoidal bicategorical matrix construction in Volume I: Symmetric Bimonoidal Categories and Monoidal Bicategories. Part 2 studies ring and bipermutative categories in the sense of ElmendorfMandell, braided ring categories, and \(E_n\)monoidal categories, which combine \(n\)fold monoidal categories with ring categories.
Graduate students and researchers interested in category theory and algebraic \(K\)theory.

Braided bimonoidal categories

Preliminaries on braided structures

Braided bimonoidal categories

Applications to quantum groups and topological quantum computation

Bimonoidal centers

Coherence of braided bimonoidal categories

Strictification of tight braided bimonoidal categories

The braided Baez conjecture

Monoidal bicategorification

$E_n$monoidal categories

Ring, bipermutative, and braided ring categories

Iterated and $E_n$monoidal categories

Bibliography and indices

Open questions

Bibliography

List of main facts

List of notations

Index