# TTT28: Manchester

## A Bangor Cat Day

Department of Mathematics, University of Manchester.

## Date

21 February 2001

## Speakers

Tim Porter (Bangor)

Simplicial higher category theory I

Mark Lawson (Bangor)

Tilings and inverse semigroups

Tim Porter (Bangor)

Simplicial higher category theory II: Between algebraic homotopy and a pursuit of stacks

Note: Ronnie Brown was unfortunately unable to give his talk

## Abstracts

### Simplicial higher category I

Four pairs of facts:

Top is a simplicially enriched category. Any homotopy type can be modelled by a simplicially enriched groupoid.

Top is groupoid enriched (Gabriel and Zisman). Groupoid enriched groupoids give crossed modules and these model 2-types.

(Hardie, Kamps and Kieboom) Top is 2-groupoid enriched. A 2-groupoid enriched groupoid gives a 2-crossed module.

(Brown and Higgins) Top can be enriched over crossed complexes. The `linear' part of arbitary homotopy types can be modelled by crossed complexes.

The talk attempted to put these results in a context of algebraic models for homotopy types and more generally homotopy theory, and of algebraic homotopy in the sense of J. H. C. Whitehead. The links between these ideas and the central importance of simplicial models, as a medium between the topology and the categorical models, was discussed.

### Simplicial higher category II: Between algebraic homotopy and a pursuit of stacks

This talk was related to (but not identical with) the one announced earlier by Ronnie Brown, who was unfortunately unable to give his talk.

Grothendieck in 1983 launched the pursuit of stacks. His 'vision' had been first expounded in a letter to Breen in about 1975. This required various prerequisites to work:

'Decent' algebraic-categorical models for n-types and an understanding of their automorphisms,and their homotopy theory.

A satisfactory theory of homotopy coherent category theory so as to be able to mimic the constructions and analyse the properties of stacks of n-types (generalising sheaves of sets).

The talk gave an introduction to the combinatorics of homotopy coherence and discussed the simplicially enriched category approach including some of its deficiencies. The link between the two talks was that if (weak) n-categorical models of n-types were better understood then we could hope for a theory that would realise Grothendieck's vision as it is described in his letter to Breen.

## Further information

The meeting was partially supported by a Scheme 3 grant from the London Mathematical Society.