On the exact sequence associated to a fibration of 2-groupoids
by
Enrico Vitale
Université catholique de Louvain
Coauthors: Rudger Kieboom (Vrije Universiteit Brussel)

One of the basic construction in algebraic topology is the homotopy long exact sequence of groups and pointed sets associated, for each point of the domain, to a Hurewicz fibration between two topological spaces. Recently, Hardie, Kamps and Kieboom have shown that, up to the dimension 2, this exact sequence is a special case of a 9-term exact sequence associated to a fibration of bigroupoids. In fact a fibration of bigroupoids is locally a fibration of groupoids and it also induces a fibration on the classifying (Poincaré) groupoids, so that they can use the construction, due to R. Brown, of a 6-term exact sequence associated to a fibration of groupoids. The aim of this talk is to obtain the Hardie-Kamps-Kieboom exact sequence in a way which seems to us more natural ; for the sake of clarity, we work with 2-groupoids instead of the more general notion of bigroupoids.

In associating groups and pointed sets to a fibration of 2-groupoids, something is missed : a kind of 2-dimensional exactness. Roughly speaking, the idea is the following : Brown associates a 6-term exact sequence of groups and pointed sets to a fibration of groupoids ; since a 2-groupoid is to be thought as a higher dimensional groupoid, we have to associate a sequence of "higher dimensional" groups and pointed sets to a fibration of 2-groupoids. And in fact we show that from a fibration of 2-groupoids (in fact, from any 2-functor between two 2-groupoids) it is possible to obtain a 6-term "2-exact" sequence of categorica groups and pointed groupoids. Then, applying the functors \pi₀ (= isomorphism classes of objects) and \pi₁ (= automorphisms of the base object) we obtain two 6-term exact sequences of groups and pointed sets which can be pasted together to give the Hardie-Kamps-Kieboom exact sequence.

Date received: May 28, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caeq-15.