Vincent's paper vs27@mcs.le.ac.uk wrote:

Hi Walter, let me advert my paper on a similar and related subject http://arxiv.org/abs/math/0602463 "Flatness, preorders and generalized metric spaces" that treats completions of non symmetric spaces. Cheers, V.

reminds me of a question I've been meaning to ask for several years, in fact since my CT'04 talk on communes over bimodules, but wasn't quite sure how to formulate. In any setting, ordinary or enriched, it is possible to introduce presheaves immediately after defining "category," even before defining "functor." Ordinarily one does not do so because functors are more fundamental to category theory than presheaves, being an essential stepping stone to the notion of natural transformation, Mac Lane's motivating entity for the whole CT enterprise. But just as dessert tends to lose its appeal when complete demolition of the main course is a prerequisite, so are applications of CT most effective for a foreign (non-CT) audience when they don't assume that the whole CT enchilada has been digested. For applications of presheaves it is helpful to know what is the absolute minimum of CT required by the audience. Just as it is not necessary to understand the principle of the internal combustion engine when getting one's driver's license by showing that one can control such an engine, it should not be necessary to know what a functor, natural transformation, adjunction, or colimit is to freely construct a presheaf on a small category J as a colimit. The following construction should suffice for those who know nothing more about CT than the definition of category. Grow a presheaf category C starting with C = J (with Set^{J^op} as the unstated secret goal) as follows. Independently adjoin objects x to C. For each such x and each object j in J, further adjoin morphisms from j to x (more generally in the V-enriched case, assign an object of V to C(j,x)), with composites of the morphisms of C(j,x) with those of J chosen subject only to the requirement that C remain a category. For any objects x,y of C, with x not in J (y in J is ok), populate C(x,y) with as many morphisms f,g,... as possible (in the V-enriched case, a suitable limit), again choosing composites with morphisms from any j to x arbitrarily, subject to the requirements that (i) if for all j and all morphisms a: j --> x, fa = ga, then f = g, and (ii) again that C remain a category (which then determines all remaining composites x --> y --> z). A pre-question here is, did I inadvertently leave anything out? My main question is, is there a reference for this process that I can cite? Any such reference must make the point that the prerequisites for this process include categories but exclude the rest of CT (as prerequisites---obviously some additional parts of CT are directly derivable, the point is that they're not prerequisites for the student). Ordinarily one reason for not bothering with such a thing would be that one can avoid even the categories by talking about equational theories with only unary operations. My application however is to communes, which are trickier to describe from a purely algebraic perspective (they're chupological rather than coalgebraic), but very natural from the above colimit-based perspective. Vaughan