Hardcover ISBN:  9780821847763 
Product Code:  CRMM/29 
List Price:  $169.00 
MAA Member Price:  $152.10 
AMS Member Price:  $135.20 
Electronic ISBN:  9781470417680 
Product Code:  CRMM/29.E 
List Price:  $169.00 
MAA Member Price:  $152.10 
AMS Member Price:  $135.20 

Book DetailsCRM Monograph SeriesVolume: 29; 2010; 784 ppMSC: 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.ReadershipGraduate 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

2monoidal 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 creationannihilation

Colored Fock functors

Appendices

Categorical preliminaries

Operads

Pseudomonoids and the looping principle

Monoids and the simplicial category


Additional Material

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


RequestsReview Copy – for reviewers who would like to review an AMS bookAccessibility – to request an alternate format of an AMS title
 Book Details
 Table of Contents
 Additional Material
 Reviews
 Requests
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.
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

2monoidal 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 creationannihilation

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