Abstracts for Opening Conference for the Program on Higher Structures in Geometry and Physics

Quillen realized in the sixties that small categories modelize
homotopy types. More precisely, he proved that the Gabriel-
Zisman localization of the category Cat of small categories by the
weak equivalences defined by the Grothendieck nerve is equivalent
to the homotopy category of simplicial sets. He also proved some
important properties of weak equivalences in Cat known as theorem
A and theorem B. Later, Thomason defined a Quillen model structure
on Cat and a Quillen equivalence of this structure with the

The duplicial objects of Dwyer and Kan are a generalization of the cyclic
objects of Connes. I will describe these duplicial objects in terms of the decalage
comonads, and give a conceptual account of a construction of duplicial objects due
to Böhm and  Stefan. This is done in terms of a 2-categorical generalization of
Hochschild homology. If time permits, I will also discuss duplicial structure on nerves
of categories, bicategories, and monoidal categories.

After recalling the various commutativity conditions which
it is natural to impose on the multiplication law in a monoidal 2-
groupoid, I will examine the corresponding question for 3-groupoids.
I will then explain the relation between this question and the occurrence
of certain 3-torsion elements in the integral homology of
the Eilenberg-Mac Lane spaces $K(A, n)$.

The introduction of each new definition of a weak higher
category poses an accompanying comparison problem, the challenge
being to convert between higher categories defined via the
new model and higher categories defined via pre-existing models.
A secondary question is whether proofs involving a particular model
of higher categories have any implications for other related models.
This talk will give a preliminary report on work in progress with
Dominic Verity that develops a framework in which categorical constructions