The following preprint is available: M. Grandis, The shape of a category up to directed homotopy, Dip. Mat. Univ. Genova, Preprint 509 (May 2004), 44 p. Abstract. This work is a contribution to a recent field, Directed Algebraic Topology. Categories which appear as fundamental categories of 'directed structures', e.g. ordered topological spaces, have to be studied up to appropriate notions of directed homotopy equivalence, wider than ordinary equivalence of categories. Here we introduce 'past' and 'future equivalences' of categories - sort of symmetric versions of an adjunction - and use them and their combinations to get 'directed models' of a category; in the simplest case, these are the join of the least full reflective and the least full coreflective subcategory. MSC: 55Pxx, 18A40, 68Q85. Keywords: homotopy theory, adjunctions, reflective subcategories, directed algebraic topology, fundamental category, concurrent processes . Available as pdf or ps: http://www.dima.unige.it/~grandis/Shp.pdf http://www.dima.unige.it/~grandis/Shp.ps With best regards 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/