Titles and speakers are as follows.
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 firstname.lastname@example.org (or John Greenlees email@example.com, or John Hunton firstname.lastname@example.org) if you are interested in attending, so that we can cater for appropriate numbers.