Higher fundamental groupoids for spaces

Fundamental n-groupoids for a topological space are introduced, by techniques based on Moore paths, similar to those used in [G3] for symmetric simplicial sets. Also the 'directed case' is treated, based on a structure introduced in [G6]: a directed topological space, where privileged directions are assigned and paths need not be reversible; such objects are provided with fundamental n-categories, as it was done for ordinary simplicial sets in [G3]. We end by comparing the present structures with the previous ones, via a geometric realisation of symmetric and ordinary simplicial sets, as spaces and directed spaces, respectively. All this essentially agrees also with the classical treatment of Kan complexes as non-directed structures. MSC: 55Pxx, 18G55, 18D05, 55U10.