... about adding slogans: "there is no mathematics without structures" and "there are no structures without transformations (between structures)" In stating this, I would go back to (Gauss and) Reimann (and Klein). This was a turning point of last century mathematics. Geometry is the analysis of (possibly) curved space and the unity of geometries is found on the notion of transformation (over manifolds, say: continuous, differentiable ...). Mathematics is no more found (only) on "quantities", since ratios of length and of angles, at the heart of Euclidean geometry, are not preserved in non-euclidean frames (their group of automorphisms are not closed under omotheties). Category Theory is the theory which inherited this fantastic broadening of perspective. --Giuseppe Longo Lab. "Jacques Herbrand" CNRS et Ecole Normale Superieure (Postal addr.: LIENS 45, Rue D'Ulm 75005 Paris (France) ) http://www.dmi.ens.fr/users/longo e-mail: longo@di.ens.fr (tel. ++33-1-4432-3328, FAX 4432-2080) Upon kind permission of the M.I.T. Press, the book below is currently downloadable from Longo's web page above (its n-th edition is out of print...): Andrea Asperti and Giuseppe Longo. Categories, Types and Structures: an introduction to Category Theory for the working computer scientist. M.I.T.- Press, 1991. (pp. 1--300).