**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

AMS Member Price: $135.20

MAA Member Price: $152.10

**Electronic ISBN: 978-1-4704-1768-0
Product Code: CRMM/29.E**

List Price: $169.00

AMS Member Price: $135.20

MAA Member Price: $152.10

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#### Supplemental Materials

# 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.

#### Readership

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

#### Reviews & Endorsements

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

#### Table of Contents

# Table of Contents

## Monoidal Functors, Species and Hopf Algebras

- Cover Cover11
- Title page i2
- Contents iii4
- List of tables ix10
- List of figures xi12
- Foreword by Kenneth Brown and Stephen Chase xiii14
- Foreword by André Joyal xv16
- Introduction xxi22
- Acknowledgments li52
- Monoidal categories 154
- Monoidal categories 356
- Graded vector spaces 2174
- Monoidal functors 61114
- Operad Lax monoidal functors 119172
- Bilax monoidal functors in homological algebra 137190
- 2-monoidal categories 161214
- Higher monoidal categories 207260
- Hopf monoids in species 233286
- Monoidal structures on species 235288
- Deformations of Hopf monoids 283336
- The Coxeter complex of type 𝐴 305358
- Universal constructions of Hopf monoids 363416
- Hopf monoids from geometry 401454
- Hopf monoids from combinatorics 443496
- Hopf monoids in colored species 487540
- Fock functors 517570
- From species to graded vector spaces 519572
- Deformations of Fock functors 547600
- From Hopf monoids to Hopf algebras: Examples 565618
- Adjoints of the Fock functors 581634
- Decorated Fock functors and creation-annihilation 599652
- Colored Fock functors 635688
- Appendices 655708
- Categorical preliminaries 657710
- Operads 669722
- Pseudomonoids and the looping principle 697750
- Monoids and the simplicial category 713766
- References 725778
- Bibliography 727780
- Notation index 741794
- Author index 763816
- Subject index 767820
- Back Cover Back Cover1842