Dear all, I am pleased to advert a paper that has been for (too)long in limbo. I eventually put the paper on the math archive in march http://arXiv.org/abs/math.CT/0403164 I acknowledge F. Borceux that suggested me a major improvement. This work was first presented in Amiens in Nov. 03. and at the last PSSL in Cambridge (Thank you all for your feedback). Here is a short abstract. The notion of P-flatness and Q-acessibility are introduced where the prameters P and Q stand for family of indexes in the sense of Borceux-Kelly. The following points are proved. Fixing a family P of indexes and Q denoting the family of P-flat indexes: - for a small category A, a presheaf on A is P-flat if and only if it is a Q-colimit of representables. - The full subcategory of the category of presheaves on a small A generated by P-flat presheaves is the free Q-cocompletion of A. Conversely Q-accessible categories occur as categories of P-flat presheaves on small A's. This correspondence yields meaningful internal description for free-Q-cocomplete objects for suitable Q's. For instance Lawvere's Cauchy-complete categories are exactly the full subcategories of P0-flat presheaves on small A's for P0 the family of ALL indexes. Remarkably this theory of accessibility extends to the enriched context (the only constraint on the base V is that it should be sym. & closed). Actually it is very likely that the whole theory might be developed in a 2-category with a Yoneda structure. Other completions by means of P-flat preshaves may be considered. For instance one obtains this way completions of metric spaces in terms of asymmetric Cauchy-filters. One also retrieves the algebraic completions of partial orders. Best regards to you all, Vincent.