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Softcover ISBN:  9781470478117 
Product Code:  SURV/285 
List Price:  $135.00 
MAA Member Price:  $121.50 
AMS Member Price:  $108.00 
eBook ISBN:  9781470478483 
Product Code:  SURV/285.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 
Softcover ISBN:  9781470478117 
eBook ISBN:  9781470478483 
Product Code:  SURV/285.B 
List Price:  $260.00 $197.50 
MAA Member Price:  $234.00 $177.75 
AMS Member Price:  $208.00 $158.00 

Book DetailsMathematical Surveys and MonographsVolume: 285; 2024; 598 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, and Volume III: From Categories to Structured Ring Spectra—this book) 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 is a detailed study of enriched monoidal categories, pointed diagram categories, and enriched multicategories. Using this machinery, Part 2 discusses the rich interconnection between the higher ringlike categories, homotopy theory, and algebraic \(K\)theory. Starting with a chapter on homotopy theory background, the first half of Part 2 constructs the Segal \(K\)theory functor and the ElmendorfMandell \(K\)theory multifunctor from permutative categories to symmetric spectra. For the latter, the detailed treatment here includes identification and correction of some subtle errors concerning its extended domain. The second half applies the \(K\)theory multifunctor to small ring, bipermutative, braided ring, and \(E_n\)monoidal categories to obtain, respectively, strict ring, \(E_{\infty}\), \(E_2\), and \(E_n\)symmetric spectra.
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

Enriched monoidal categories and multicategories

Enriched monoidal categories

Change of enrichment

Selfenrichment and enriched Yoneda

Pointed objects, smash products, and pointed homs

Multicategories

Enriched multicategories

Algebraic $K$theory

Homotopy theory background

Segal $K$theory of permutative categories

Categories of $\mathcal{G}_*$objects

ElmendorfMandell $K$theory of permutative categories

$K$theory of ring and bipermutative categories

$K$theory of braided ring categories

$K$theory of $E_n$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, and Volume III: From Categories to Structured Ring Spectra—this book) 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 is a detailed study of enriched monoidal categories, pointed diagram categories, and enriched multicategories. Using this machinery, Part 2 discusses the rich interconnection between the higher ringlike categories, homotopy theory, and algebraic \(K\)theory. Starting with a chapter on homotopy theory background, the first half of Part 2 constructs the Segal \(K\)theory functor and the ElmendorfMandell \(K\)theory multifunctor from permutative categories to symmetric spectra. For the latter, the detailed treatment here includes identification and correction of some subtle errors concerning its extended domain. The second half applies the \(K\)theory multifunctor to small ring, bipermutative, braided ring, and \(E_n\)monoidal categories to obtain, respectively, strict ring, \(E_{\infty}\), \(E_2\), and \(E_n\)symmetric spectra.
Graduate students and researchers interested in category theory and algebraic \(K\)theory.

Enriched monoidal categories and multicategories

Enriched monoidal categories

Change of enrichment

Selfenrichment and enriched Yoneda

Pointed objects, smash products, and pointed homs

Multicategories

Enriched multicategories

Algebraic $K$theory

Homotopy theory background

Segal $K$theory of permutative categories

Categories of $\mathcal{G}_*$objects

ElmendorfMandell $K$theory of permutative categories

$K$theory of ring and bipermutative categories

$K$theory of braided ring categories

$K$theory of $E_n$monoidal categories

Bibliography and indices

Open questions

Bibliography

List of main facts

List of notations

Index