# TTT96: Leicester

Supported by the London Mathematical Society.

## Date

3 June 2015

## Speakers

### Imma Gálvez Carrillo (UPC Barcelona)

Incidence algebras with Möbius inversion from decomposition spaces

Incidence algebras with Möbius inversion from decomposition spaces

Abstract: Decomposition spaces are simplicial (infinity-)groupoids that satisfy a certain exactness condition (encoding decomposition), strictly weaker than the usual Segal condition (encoding composition).

A decomposition space comes with an associated incidence coalgebra, whose dual gives rise to an incidence algebra, with Möbius inversion. The coefficients involved in this 'algebra' are infinity-groupoids, but under suitable finiteness conditions homotopy cardinality can be taken and we are back in the realm of classical incidence algebras.

One important example of a decomposition space is the Waldhausen S-construction of an abelian (or stable infinity-) category, whose incidence algebra is a derived Hall algebra, and many other convolution algebras that arise classically as ad hoc quotients of incidence algebras of posets can be obtained more canonically from decomposition spaces.

Joint work with J Kock (Universitat Autònoma de Barcelona) and A Tonks (University of Leicester).

Reference: arXiv:1404.3202

### Mariam Pirashvili (University of Leicester)

Second cohomotopy and nonabelian cohomology

Second cohomotopy and nonabelian cohomology

Abstract: The main difficulty in the theory of non-abelian cohomology is that for cosimplicial groups only zero-th and first dimensional cohomotopy are known. In this talk we introduce a new class of cosimplicial groups, called centralised cosimplicial groups, for which we are able to define a second cohomotopy, with all expected properties. The main examples of such cosimplicial groups come from 2-categories.

### Joe Palacios Baldeon (University of Liverpool)

Symmetric powers of motivic spaces

Symmetric powers of motivic spaces

Abstract: Motivic spaces depend of two coordinates, simplicial and geometric, where the geometric coordinate means, namely, the category of quasi-projective schemes over a field. Geometric symmetric powers of motivic spaces are left Kan extensions of the abstract, or categoric, symmetric powers defined on the geometric coordinate.

In my talk, I will explain how they provide a Lambda structure on the unstable motivic homotopy category. I will also sketch a comparison of four kinds of symmetric powers in the stable motivic homotopy category.

### Behrang Noohi (Queen Mary University of London)

Singular chains on topological stacks

Singular chains on topological stacks

Abstract: I will discuss ongoing work (with Thomas Coyne) on the construction of singular chains on topological stacks and explain how it fits with the existing approaches to defining homotopy types of topological stacks.

Reference: arXiv:1502.04995

## Further information

Wine reception as part of the 150th Anniversary of the London Mathematical Society (LMS).

This event was in conjunction with the LMS sponsored workshop on 'Cluster algebras and finite dimensional algebras'.