# TTT66: Leicester

Department of Mathematics and Computer Science, University of Leicester.

## Date

3 November 2008

## Speakers

Nicola Gambino (Department of Computer Science, Leicester)

Homotopical structures in mathematical logic

James Cranch (Leicester/Sheffield)

Span diagrams in homotopy theory

Behrang Noohi (Kings College London)

String topology for stacks

## Abstracts

### Nicola Gambino: Homotopical structures in mathematical logic

The aim of the talk is to give an overview of the recently-discovered connections between abstract homotopy theory and mathematical logic. These connections concern Quillen's homotopical algebra on the one hand and Martin-Loef set theory on the other hand.

I will begin by giving an introduction to Martin-Loef set theory, assuming no prior knowledge of mathematical logic. Then, I will explain how the category of sets associated to Martin-Loef set theory admits a non-trivial weak factorisation system and relate this weak factorisation system to the natural Quillen model structure on the category of groupoids (joint work with Richard Garner).

Finally, I will explain how these results fit into a general program, aimed at extending the correspondence between mathematical logic and category theory to higher-dimensional categories.

### James Cranch: Span diagrams in homotopy theory

This talk will report on my PhD thesis in preparation under Neil Strickland. I will supply some propaganda for the theory of quasicategories (due to Joyal and Lurie). Then I will show how Lawvere's notion of an algebraic theory carries across to this context. Lastly I will describe a new approach to homotopy commutativity equivalent to, and in some senses simpler than, the classical language of E_infinity operads.

### Behrang Noohi : String topology for stacks

String topology (Chas-Sullivan, Cohen, Jones...) studies the homology of the loop space of a manifold by exploiting the so-called string operations. With the goal of producing an equivariant version of the theory, we formulate string topology for topological stacks and prove the existence of string operations under certain natural hypotheses.

As a consequence, we obtain equivariant string topology for compact Lie group actions on manifolds. This is a joint work with K. Behrend, G. Ginot, and P. Xu. Reference: arXiv:0712.3857v1 [math.AT]

## Further information

The TTT was partially supported by an LMS scheme 3 grant.