The following preprint is available: M. Grandis, Lax 2-categories and directed homotopy, Dip. Mat. Univ. Genova, Preprint 530 (2005), 18 p. Abstract. Directed Algebraic Topology is a recent field, where a 'directed space' X, e.g. an ordered topological space, has directed paths (which are generally not reversible) and a *fundamental category*, replacing the fundamental groupoid of the classical case. In dimension 2, the directed 2-dimensional paths of X naturally produce a fundamental *lax 2-category*. This is an extension of a bicategory where comparison cells are not assumed to be invertible, and some choice for their direction is needed. Our geometric guideline gives a choice which is different from the ones previously considered by Burroni, Borceux, Leinster. Available in pdf at: http://www.dima.unige.it/~grandis/LCat.pdf Marco Grandis Dipartimento di Matematica Universita` di Genova via Dodecaneso 35 16146 GENOVA, Italy e-mail: grandis@dima.unige.it tel: +39.010.353 6805 fax: +39.010.353 6752 http://www.dima.unige.it/~grandis/ 24-Jun-2005 07:20:14 -0300,1223;000000000000-00000019