The following preprint is available in pdf: M. Grandis, A symmetric cubical category associated to a directed space. Dip. Mat. Univ. Genova, Preprint 590 (2010). Available at: http://www.dima.unige.it/~grandis/Fnd.pdf Abstract. The recent domain of directed algebraic topology studies 'directed spaces', where paths and homotopies cannot generally be reversed. The general aim is modelling non reversible phenomena, but the present applications are mostly concerned with the theories of concurrent processes and rewrite systems. At the place of the classical fundamental groupoid of a topological space, a directed space has a fundamental category, whose applications to concurrency have already been studied in many papers. Here, we want to study an infinite dimensional version of the fundamental category of a directed space, of a cubical type and more precisely a symmetric cubical one, because transposition symmetries occur naturally and simplify the coherence properties. We introduce a 'Moore' strict symmetric cubical category of a directed space X, with concatenation laws in the various directions and transpositions (which permute variables). On the other hand, standard cubes give a lax cubical structure, where concatenations are associative up to invertible reparametrisation but degeneracies are only lax-unital. With best regards Marco Grandis PS. I have been informed by R. Brown of a recent preprint of his on Moore n-paths for a topological space and their cubical structure, see: arXiv: 0909.2212v2 Sorry of missing that. I will add something about it, in my work. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]