Alternatively have a look at the program.

## Higher para-toposes

A para-topos is a cartesian closed (locally) presentable category.

A higher para-topos is defined to be a cartesian closed (locally) presentable infty-category.

If $\mathcal{E}$ is a higher para-topos, then so is the $\infty$-category $Cat(\mathcal{E})$ of complete Segal spaces in $\mathcal{E}$.

The construction $\mathcal{E} \mapsto Cat(\mathcal{E})$ can be iterated and it has fixed points.

## Strict n-categories as models for homotopy types (joint work with Dimitri Ara)

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 defined 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 defined a Quillen model structure

on Cat and a Quillen equivalence of this structure with the

## The multitopic universe

My talk will focus on the contents of the manuscript

entitled "The multitopic omega-category of all multitopic omega-categories"

posted on my website in 1999; the version posted in

2004 is a minor corrected variant. The proposal is for a notion

of weak omega-dimensional (or infinity) category, called multitopic

category, together with the definition of the structure that (small)

multitopic categories form, which structure is itself a (large) multitopic

category, here called the multitopic universe. The structure

## Hochschild homology, lax codescent, and duplicial structure

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.

## Weakly globular n-fold categories

In this talk I introduce a new model of weak n-categories,

called weakly globular n-fold categories. This is based on the notion

of iterated internal category, satisfying additional conditions. It

develops a new paradigm to weaken higher categorical structures,

leading to potential applications. I will illustrate how this model

compares to another approach to weak higher categories due to

Tamsamani and Simpson.

## Commutativity conditions for monoidal 3-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 $K(A, n)$.

## Equivariant bordism from the global perspective

Global homotopy theory is, informally speaking, equivariant

homotopy theory in which all compact Lie groups acts at

once on a space or a spectrum, in a compatible way. In this talk I

will advertise a rigorous and reasonably simple formalism to make

this precise, using orthogonal spectra. I will then illustrate the formalism

by a geometrically motivated example, namely equivariant

bordism of smooth manifolds.

## On model-comparison results for $\infty$-categories

The introduction of each new definition of a weak higher

category poses an accompanying comparison problem, the challenge

being to convert between higher categories defined via the

new model and higher categories defined 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

## Comparing (co)localizations across Quillen adjunctions

The talk is based on joint work with Oriol Raventós and Andy

Tonks. We show that several apparently unrelated formulas involving

localizations or cellularizations in homotopy theory arise

from comparison maps associated with pairs of adjoint functors.

Such comparison maps are used to study liftings of homotopical

(co)localizations to categories of (co)algebras over (co)monads in

suitable model categories. We discuss, with different methods, the

case of strict algebras and the case of homotopy algebras. Warnings

## Higher homotopy structures-then and now (teleconference talk)

Looking back over 55 years of higher homotopy structures, I will

reminisce as I recall the early days and ponder how I now see them.

From the history of $A_\infty$-structures and later of $L_\infty$-structures, I

will sketch how they morphed into the topic of this Program on

Higher Structures in Geometry and Physics.