Dear Jean, Presheaves on all finite ordinals are called augmented simplicial sets. As to 'directed space' I use now this term in a non-technical sense, as referring to any topological structure suitable for directed algebraic topology, like: - preordered topological space, - d-space, i.e. space with distinguished paths, - c-space, i.e. space with distinguished cubes, - etc. See my book on 'Directed Algebraic Topology'. Thanks to Cambridge Un. Press, it can be freely (and legally) downloaded from: http://www.dima.unige.it/~grandis/BkDAT_page.html (When I first used the term 'directed space', it was equivalent to d- space; later I thought it was better to keep it for a more general meaning.) Best wishes Marco [For admin and other information see: http://www.mta.ca/~cat-dist/ ]