The following preprint is available M. Grandis - L. Mauri, Cubical sets and their site, Dip. Mat. Univ. Genova, Preprint 452 (Jun 2002), 26 p. Abstract. Extended cubical sets (with connections and interchanges) are presheaves on a ground category, the extended cubical site K, corresponding to the (augmented) simplicial site, the category of finite ordinals. We prove here that K has characterisations similar to the classical ones for the simplicial analogue, by generators and relations, or by the existence of a universal symmetric cubical monoid; in fact, K is the classifying category of a monoidal algebraic theory of such monoids. Analogous results are given for the restricted cubical site I of ordinary cubical sets (just faces and degeneracies) and for the intermediate site J (including connections). We also consider briefly the reversible analogue, !K. Available at: ftp://www.dima.unige.it/Home/grandis/public/CubSet.dvi ftp://www.dima.unige.it/Home/grandis/public/CubSet.ps Marco Grandis 20-Jun-2002 12:20:43 -0300,1593;000000000000-00000000