Higher fundamental groupoids for spaces

Fundamental n-groupoids for topological spaces are introduced, by techniques based on Moore paths, similar to those used in [M. Grandis, Cahiers Topologie Géom. Différentielle Catégoriques 42 (2001) 101–136] for symmetric simplicial sets. Also the ‘directed case’ is treated, by means of a structure introduced in [M. Grandis, Cahiers Topologie Géom. Différentielle Catégoriques, to appear, http://www.dima.unige.it/~grandis/]: a directed topological space, where privileged directions are assigned and paths need not be reversible; such objects are provided with fundamental n-categories, as in the first cited paper, for simplicial sets. The present structures are compared with the previous ones, via a geometric realisation of symmetric and ordinary simplicial sets. The classical treatment of Kan complexes as non-directed structures is also considered.