**CRM Monograph Series**

Volume: 29;
2010;
784 pp;
Hardcover

MSC: Primary 05; 16; 18; 20; 81;
Secondary 06; 51

**Print ISBN: 978-0-8218-4776-3
Product Code: CRMM/29**

List Price: $169.00

Individual Member Price: $135.20

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# Monoidal Functors, Species and Hopf Algebras

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*Marcelo Aguiar; Swapneel Mahajan*

A co-publication of the AMS and Centre de Recherches Mathématiques

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.

#### Table of Contents

# Table of Contents

## Monoidal Functors, Species and Hopf Algebras

#### Readership

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

#### 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