# Abstracts for Conference for Young researchers in homotopy theory and categorical structures, February 13-15, 2017

Alternatively have a look at the program.

## Homotopical Morita theory for corings

(Joint work with Alexander Berglund)

## Galois extensions in motivic homotopy theory

Galois extensions of ring spectra in classical homotopy theory were introduced by Rognes.

In this talk I will discuss a general formal framework to study homotopical Galois extensions and

concentrate on the applications to motivic homotopy theory. I will discuss several examples of

homotopical Galois extensions in motivic setting comparing them to the ones known from classical

homotopy theory.

This is a joint project with Agnès Beaudry, Kathryn Hess, Mona Merling and Vesna Stojanoska.

## Higher enveloping algebras and configuration spaces of manifolds

I will describe a construction providing Lie algebras with enveloping algebras over the operad of little $n$-dimensional

disks for any $n$. These algebras enjoy a combination of good formal properties and computability, the latter afforded

by a Poincaré-Birkhoff-Witt type result. The main application pairs this theory with the theory of factorization homology

in a study of the rational homology of configuration spaces, leading to a wealth of computations, improvements of classical results, and a combinatorial proof of homological stability.

## The structure of the category of strickt n-categories

The category n-Cat of strict n-categories and strict n-functors is endowed with

a rich structure that is only partially understood. First, it is endowed with a tensor

product, generalizing Gray's tensor product of 2-categories, first defined by Al-Alg

and Steiner in the early 90's. This implies that n-Cat, endowed with lax transformations

and higher lax transformations, form an n-Gray-category in some appropriate sense.

Second, it is endowed with a join construction, generalizing the join construction of

## Cubical $(\omega,p)$-categories

Handling higher structures such as higher categories usually involves conceiving them as conglomerates of

cells of certain shape. Such shapes include simplices, globes, or cubes. The aim of this talk is to bridge the

gap between two such results:

The first one is the equivalence between cubical and globular $\omega$-groupoids. Although this equivalence

is useful in theory, in practice it is complicated to make explicit the functors composing this equivalence. This

## Gong Show, I (Conference for Young researchers in homotopy theory and categorical structures)

## Clone of Gong Show, II (Conference for Young researchers in homotopy theory and categorical structures)

Speakers, titles and abstracts, see this PDF.

## Open discussion (Conference for Young researchers in homotopy theory and categorical structures)

## 2-Segal sets and the Waldhausen construction

$2$-Segal objects, which are a generalisation of ordinary Segal objects, were introduced and studied by

Dyckerhoff-Kapranov and Gálvez-Kock-Tonks. An important example of a $2$-Segal object is the Waldhausen

construction of an exact category. The Waldhausen construction makes sense for a more general input, and the

goal of the talk is to explain that, in the discrete setting, the Waldhausen construction is in fact quite exhaustive.

More precisely, it induces an equivalence between the category of stable pointed double categories and the

## On the category of Poisson Hopf algebras

A Poisson Hopf algebra is both a Poisson algebra and a Hopf algebra such that the comultiplication and

the counit are morphisms of Poisson algebras. Such objects are situated at the border between Poisson

geometry and quantum groups. We investigate the category of Poisson Hopf algebras with respect to

properties such as (co)completeness or existence of (co)free objects. For instance, we prove that there

exists a free Poisson Hopf algebra on any coalgebra or, equivalently that the forgetful functor from the