University of Manchester

February 21st 2001

Titles and speakers are as follows.

- 11.30 am Tim Porter (Bangor)

``Simplicial higher category theory I.'' - 12.30 pm LUNCH

- 2.00 pm Mark Lawson (Bangor)

``Tilings and inverse semigroups.'' - 3.00 pm TEA
- 4.00 pm Tim Porter (Bangor) (Ronnie Brown is unfortunately unable
to give his talk)

``Simplicial higher category theory II: between algebraic homotopy and a pursuit of stacks"

Four pairs of facts: (i) Top is a simplicially enriched category. Any homotopy type can be modelled by a simplicially enriched groupoid. (ii) Top is groupoid enriched (Gabriel and Zisman). Groupoid enriched groupoids give crossed modules and these model 2-types. (iii) (Hardie, Kamps and Kieboom) Top is 2-groupoid enriched. A 2-groupoid enriched groupoid gives a 2-crossed module. (iv) (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 will attempt 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, will be discussed (briefly!).

Simplicial Higher Category II: between algebraic homotopy and a pursuit of stacks
This talk is related to (but not identical with)
the one announced earlier by Ronnie Brown, who is 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: (a)`decent' algebraic-categorical models
for n-types and an understanding of their automorphisms,and their
homotopy theory.
(b) 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 will give an introduction to the combinatorics of homotopy
coherence and will discuss the simplicially enriched category approach
including some of its deficiencies (if time permits).
The link between the two talks is that if (weak) n-categorical models of
n-types were better understood then we could hope for a theory that
would realise Grothendieck vision as it is described in his letter to
Breen.

Lunch will be taken locally (at "On the Eight Day", for example), and we expect to visit a local curry house (or similar) early in the evening.

Everyone who wishes to participate is welcome: we shall operate the usual arrangements for assistance with travel expenses. Please email Nige Ray nige@ma.man.ac.uk (or John Greenlees j.greenlees@sheffield.ac.uk, or John Hunton jrh7@mcs.le.ac.uk) if you are interested in attending, so that we can cater for appropriate numbers.

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

To TTT27

To TTT29