A theorem of Whitehead asserts that the topological 2-type of a (connected) space is uniquely characterised by the triple (pi_1, pi_2, k_3), where the pi_i, ileq 2 are the homotopy groups pi_i, ileq 2, k_3 is the Postnikov class in H^3(pi_1, pi_2), and, indeed all such triples may be realised. Such triples are synomous with a 2-group, Pi_2, i.e. a group `object’ in the category of categories, which plays the same role for 2-types as the fundamental group does for 1-types. In particular, there is a 2-Galois correspondence between the 2-category of champs which are etale fibrations over a space and Pi_2 equivariant groupoids generalising the usual 1-Galois correspondence between spaces which are etale fibrations over a given space and pi_1 equivariant sets.
The talk will explain the pro-finite analogue of this correspondence, so, albeit only for the 2-type, a much simpler and more generally valid description of the etale homotopy than that of Artin-Mazur.
All info is available on the Geometry and Topology webpage.