Combinatorial Group Theory: COMBINATORIAL GROUP THEORY

Annals of mathematics studies

This paper is a rather informal guide to some of the basic theory of 2-categories and bicategories, including notions of limit and colimit, 2-dimensional universal algebra, formal category theory, and nerves of bicategories.

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A 2-Categories Companion

This is an introductory survey of the passage from groups to groupoids and their higher dimensional versions, with most emphasis on calculations with crossed modules and the construction and use of homotopy double groupoids.

http://groupoids.org.uk/pdffiles/GRENoble09.pdf

Groupoids and crossed objects in algebraic topology

In Chap. 9, we studied in detail the operad Ass encoding the associative algebras. It is a paradigm for nonsymmetric operads, symmetric operads, cyclic operads. In this chapter we present several other examples of operads. First, we present the two other “graces”, the operads Com and Lie encoding respectively the commutative (meaning commutative and associative) algebras, and the Lie algebras. Second, we introduce more examples of binary quadratic operads: Poisson, Gerstenhaber, pre-Lie, Leibniz, Zinbiel, dendriform, magmatic, several variations like Jordan algebra, divided power algebra, Batalin–Vilkovisky algebra. Then we present various examples of operads involving higher ary-operations: homotopy algebras, infinite-magmatic, brace, multibrace, Jordan triples, Lie triples. The choice is dictated by their relevance in various parts of mathematics: differential geometry, noncommutative geometry, harmonic analysis, algebraic combinatorics, theoretical physics, computer science. Of course, this list does not exhaust the examples appearing in the existing literature. The reader may have a look at the cornucopia of types of algebras 2012 to find more examples.

Examples of Algebraic Operads

Decomposition spaces, incidence algebras and Möbius inversion I: Basic theory

This paper provides an explicit cobrant resolution of the operad encoding Batalin{Vilkovisky algebras. Thus it denes the notion of homotopy Batalin{Vilkovisky algebras with the required homotopy properties. To dene this resolution, we extend the theory of Koszul duality to operads and properads that are dened by quadratic and linear relations. The operad encoding Batalin{Vilkovisky algebras is shown to be Koszul in this sense. This allows us to prove a Poincar e{Birkho{Witt Theorem for such an operad and to give an explicit small quasi-free resolution for it. This particular resolution enables us to describe the deformation theory and homotopy theory of BV-algebras and of homotopy BV-algebras. We show that any topological conformal eld theory carries a homotopy BV- algebra structure which lifts the BV-algebra structure on homology. The same result is proved for the singular chain complex of the double loop space of a topological space endowed with an action of the circle. We also prove the cyclic Deligne conjecture with this cobrant resolution of the operad BV . We develop the general obstruction theory for algebras over the Koszul resolution of a properad and apply it to extend a conjecture of Lian{Zuckerman, showing that certain vertex algebras have an explicit homotopy BV-algebra structure.

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Homotopy Batalin–Vilkovisky algebras

This paper provides an explicit cofibrant resolution of the operad encoding Batalin-Vilkovisky algebras. Thus it defines the notion of homotopy Batalin-Vilkovisky algebras with the required homotopy properties. 
To define this resolution we extend the theory of Koszul duality to operads and properads that are defind by quadratic and linear relations. The operad encoding Batalin-Vilkovisky algebras is shown to be Koszul in this sense. This allows us to prove a Poincare-Birkhoff-Witt Theorem for such an operad and to give an explicit small quasi-free resolution for it. 
This particular resolution enables us to describe the deformation theory and homotopy theory of BV-algebras and of homotopy BV-algebras. We show that any topological conformal field theory carries a homotopy BV-algebra structure which lifts the BV-algebra structure on homology. The same result is proved for the singular chain complex of the double loop space of a topological space endowed with an action of the circle. We also prove the cyclic Deligne conjecture with this cofibrant resolution of the operad BV. We develop the general obstruction theory for algebras over the Koszul resolution of a properad and apply it to extend a conjecture of Lian-Zuckerman, showing that certain vertex algebras have an explicit homotopy BV-algebra structure.

Andrew Tonks

Papers

Decomposition spaces, incidence algebras and Möbius inversion I: Basic theory

Homotopy Batalin–Vilkovisky algebras

Homotopy Batalin-Vilkovisky algebras

Decomposition spaces, incidence algebras and Möbius inversion II: completeness, length filtration, and finiteness

A model structure à la Thomason on 2 - CAT