I don't quite understand this question. I was interested since I am looking at the plus construction as part of my work at the moment.

Let P be a presheaf on the site (C,J) and consider the "classical" plus construction P^+(c) = colim_{R \in J(c)}Match(R,P) where Match(R,P) is the set of matching families for the cover R \in J(c) and the colimit is taken over J(c) ordered by reverse inclusion (cf. McLane & Moerdijk) . This is a nice filtered colimit so x \in P^+(c) can be expressed as an equivalence class of matching families. Suppose now that J is given by a basis K. It is not immediately clear (at least not for me) what happens in a variant of the above where Match(R,P) is taken as the set of matching families for the K-cover R. Indeed, the notion of "common refinement" for K-covers is not as handy as the one for J-covers for the task at hand since op-ordering K-covers will not necessarily give a filtered category. The other obvious candidate for a "category K(c)" where a factorisation witnessing a refinement (of K-covers) is a morphism (in the opposite category) will probably fail to be filtered as well, so I wondered if anybody allready looked at such things. I agree that a one-sentence prose might have been a bit messy... Cheers Krzysztof