This talk is based on joint work with O. Caramello.

It examines the concrete construction of the new categories classified by fundamental groups as defined in the previous talk.

Many classical categories - such as the categories of finite groups or finite graphs and their embeddings or their surjective homomorphisms - naturally embed into larger categories classified by fundamental groups. The new ‘imaginary’ objects which have to be added to make these categories Galoisian can be described concretely.

These constructions allow one to associate new invariants - including cohomological invariants - to groups, graphs and many geometric objects.