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Monoidal Functors, Species and Hopf Algebras
 
Marcelo Aguiar Texas A&M University, College Station, TX
Swapneel Mahajan Indian Institute of Technology, Mumbai, India
A co-publication of the AMS and Centre de Recherches Mathématiques
Monoidal Functors, Species and Hopf Algebras
Hardcover ISBN:  978-0-8218-4776-3
Product Code:  CRMM/29
List Price: $175.00
MAA Member Price: $157.50
AMS Member Price: $140.00
eBook ISBN:  978-1-4704-1768-0
Product Code:  CRMM/29.E
List Price: $169.00
MAA Member Price: $152.10
AMS Member Price: $135.20
Hardcover ISBN:  978-0-8218-4776-3
eBook: ISBN:  978-1-4704-1768-0
Product Code:  CRMM/29.B
List Price: $344.00 $259.50
MAA Member Price: $309.60 $233.55
AMS Member Price: $275.20 $207.60
Monoidal Functors, Species and Hopf Algebras
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Monoidal Functors, Species and Hopf Algebras
Marcelo Aguiar Texas A&M University, College Station, TX
Swapneel Mahajan Indian Institute of Technology, Mumbai, India
A co-publication of the AMS and Centre de Recherches Mathématiques
Hardcover ISBN:  978-0-8218-4776-3
Product Code:  CRMM/29
List Price: $175.00
MAA Member Price: $157.50
AMS Member Price: $140.00
eBook ISBN:  978-1-4704-1768-0
Product Code:  CRMM/29.E
List Price: $169.00
MAA Member Price: $152.10
AMS Member Price: $135.20
Hardcover ISBN:  978-0-8218-4776-3
eBook ISBN:  978-1-4704-1768-0
Product Code:  CRMM/29.B
List Price: $344.00 $259.50
MAA Member Price: $309.60 $233.55
AMS Member Price: $275.20 $207.60
  • Book Details
     
     
    CRM Monograph Series
    Volume: 292010; 784 pp
    MSC: Primary 05; 16; 18; 20; 81;

    This research monograph integrates ideas from category theory, algebra and combinatorics. It is organized in three parts.

    Part I belongs to the realm of category theory. It reviews some of the foundational work of Bénabou, Eilenberg, Kelly and Mac Lane on monoidal categories and of Joyal and Street on braided monoidal categories, and proceeds to study higher monoidal categories and higher monoidal functors. Special attention is devoted to the notion of a bilax monoidal functor which plays a central role in this work.

    Combinatorics and geometry are the theme of Part II. Joyal's species constitute a good framework for the study of algebraic structures associated to combinatorial objects. This part discusses the category of species focusing particularly on the Hopf monoids therein. The notion of a Hopf monoid in species parallels that of a Hopf algebra and reflects the manner in which combinatorial structures compose and decompose. Numerous examples of Hopf monoids are given in the text. These are constructed from combinatorial and geometric data and inspired by ideas of Rota and Tits' theory of Coxeter complexes.

    Part III is of an algebraic nature and shows how ideas in Parts I and II lead to a unified approach to Hopf algebras. The main step is the construction of Fock functors from species to graded vector spaces. These functors are bilax monoidal and thus translate Hopf monoids in species to graded Hopf algebras. This functorial construction of Hopf algebras encompasses both quantum groups and the Hopf algebras of recent prominence in the combinatorics literature.

    The monograph opens a vast new area of research. It is written with clarity and sufficient detail to make it accessible to advanced graduate students.

    Titles in this series are co-published with the Centre de recherches mathématiques.

    Readership

    Graduate students and research mathematicians interested in category theory, algebraic combinatorics, Hopf algebras, and Coxeter groups.

  • Table of Contents
     
     
    • Chapters
    • Monoidal categories
    • Monoidal categories
    • Graded vector spaces
    • Monoidal functors
    • Operad Lax monoidal functors
    • Bilax monoidal functors in homological algebra
    • 2-monoidal categories
    • Higher monoidal categories
    • Hopf monoids in species
    • Monoidal structures on species
    • Deformations of Hopf monoids
    • The Coxeter complex of type $A$
    • Universal constructions of Hopf monoids
    • Hopf monoids from geometry
    • Hopf monoids from combinatorics
    • Hopf monoids in colored species
    • Fock functors
    • From species to graded vector spaces
    • Deformations of Fock functors
    • From Hopf monoids to Hopf algebras: Examples
    • Adjoints of the Fock functors
    • Decorated Fock functors and creation-annihilation
    • Colored Fock functors
    • Appendices
    • Categorical preliminaries
    • Operads
    • Pseudomonoids and the looping principle
    • Monoids and the simplicial category
  • Reviews
     
     
    • The book of Aguiar and Mahajan is a quantum leap toward the mathematics of the future. I strongly recommend it to all researchers in algebra, topology, and combinatorics.

      André Joyal
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Accessibility – to request an alternate format of an AMS title
Volume: 292010; 784 pp
MSC: Primary 05; 16; 18; 20; 81;

This research monograph integrates ideas from category theory, algebra and combinatorics. It is organized in three parts.

Part I belongs to the realm of category theory. It reviews some of the foundational work of Bénabou, Eilenberg, Kelly and Mac Lane on monoidal categories and of Joyal and Street on braided monoidal categories, and proceeds to study higher monoidal categories and higher monoidal functors. Special attention is devoted to the notion of a bilax monoidal functor which plays a central role in this work.

Combinatorics and geometry are the theme of Part II. Joyal's species constitute a good framework for the study of algebraic structures associated to combinatorial objects. This part discusses the category of species focusing particularly on the Hopf monoids therein. The notion of a Hopf monoid in species parallels that of a Hopf algebra and reflects the manner in which combinatorial structures compose and decompose. Numerous examples of Hopf monoids are given in the text. These are constructed from combinatorial and geometric data and inspired by ideas of Rota and Tits' theory of Coxeter complexes.

Part III is of an algebraic nature and shows how ideas in Parts I and II lead to a unified approach to Hopf algebras. The main step is the construction of Fock functors from species to graded vector spaces. These functors are bilax monoidal and thus translate Hopf monoids in species to graded Hopf algebras. This functorial construction of Hopf algebras encompasses both quantum groups and the Hopf algebras of recent prominence in the combinatorics literature.

The monograph opens a vast new area of research. It is written with clarity and sufficient detail to make it accessible to advanced graduate students.

Titles in this series are co-published with the Centre de recherches mathématiques.

Readership

Graduate students and research mathematicians interested in category theory, algebraic combinatorics, Hopf algebras, and Coxeter groups.

  • Chapters
  • Monoidal categories
  • Monoidal categories
  • Graded vector spaces
  • Monoidal functors
  • Operad Lax monoidal functors
  • Bilax monoidal functors in homological algebra
  • 2-monoidal categories
  • Higher monoidal categories
  • Hopf monoids in species
  • Monoidal structures on species
  • Deformations of Hopf monoids
  • The Coxeter complex of type $A$
  • Universal constructions of Hopf monoids
  • Hopf monoids from geometry
  • Hopf monoids from combinatorics
  • Hopf monoids in colored species
  • Fock functors
  • From species to graded vector spaces
  • Deformations of Fock functors
  • From Hopf monoids to Hopf algebras: Examples
  • Adjoints of the Fock functors
  • Decorated Fock functors and creation-annihilation
  • Colored Fock functors
  • Appendices
  • Categorical preliminaries
  • Operads
  • Pseudomonoids and the looping principle
  • Monoids and the simplicial category
  • The book of Aguiar and Mahajan is a quantum leap toward the mathematics of the future. I strongly recommend it to all researchers in algebra, topology, and combinatorics.

    André Joyal
Review Copy – for publishers of book reviews
Accessibility – to request an alternate format of an AMS title
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