Monoidal functors, species and Hopf algebras / Marcelo Aguiar, Swapneel Mahajan.

Format:

Book

Author/Creator:

Aguiar, Marcelo, 1968-

Contributor:

Mahajan, Swapneel Arvind, 1974-

Series:

CRM monograph series ; v. 29.

CRM monograph series / Centre de recherches mathématiques, Montréal ; v. 29

Language:

English

Subjects (All):

Hopf algebras.

Combinatorial analysis.

Categories (Mathematics).

Symmetry groups.

Quantum groups.

Physical Description:

li, 784 pages : illustrations ; 27 cm.

Place of Publication:

Providence, R.I. : American Mathematical Society, [2010]

Summary:

"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."--Publisher's description.

Contents:

Monoidal categories

Graded vector spaces

Monoidal functors

Operad lax monoidal functors

Bilax monoidal functors in homological algebra

2-monoidal categories

Higher monoidal categories

Monoidal structures on species

Deformations of Hopf monoids

The Coxeter comolex of type A

Universal constructions of Hopf monoids

Hopf monoids from geometry

Hopf monoids from combinatorics

Hopf monoids in colored species

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

Appendix A: Categorical preliminaries

Appendix B: Operads

Appendix C: Pseudomonoids and the looping principle

Appendix D: Monoids and the simplicial category.

Notes:

Includes bibliographical references and indexes.

ISBN:

9780821847763

0821847767

OCLC:

645708042