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I think that explaining where this comes from (namely, Cisinski's work, right?) and giving a reference could help the non-expert but interested readers and thus improve the question (just my two cents).
Dear Minhyong: I don't know much of the underlying Mathematics either but I enjoyed it too, thanks a lot for sharing your experience. Now I've got two answers quite different in nature and in tone and since I'm equally pleased with both of them perhaps I should just let them benefit the community and don't decide to officially accept one of them.
Actually, your « Swiss cheese memory » seems to be a misnomer, as was pointed out by Tom Leinster in his book « Higher Operads, Higher Categories » (he acknowledges the help of Paul-André Melliès for that). See bottom of p. 63 of arxiv.org/pdf/math/0305049.
As far as Ladegaillerie's work is concerned, you certainly remember the part(s) of R&S devoted to it and its fate, namely that it had not been published until then (Grothendieck mentions a "forthcoming paper", though). Malgoire's name is mentionned only once in R&S (he had a joint article with Christine Voisin published). Whether something happened after R&S was written I don't know. As regards their works I'll try to find some piece of information and let you know if I learn anything.
Dear Leo: Thank you very much for this informative answer, which makes clear that some mathematicians have studied the mathematical part in R&S. Given that there are some texts developing these ideas, do you know if any of them explicitly mentions R&S as a mathematical source? As regards the development of homotopical algebra, I'd rather incline to think that it stems from "Pursuing Stacks" and "Les Dérivateurs" rather than R&S, since I've never come across R&S as a reference in texts developing homotopical algebra à la Grothendieck. But I've come across a very small portion of them only.
Dear jc: To answer this question, one would have to have read not only "Pursuing Stacks" and "Les Dérivateurs", but also the letters Grothendieck have sent to various mathematicians and which may have influenced them. Therefore I thought I had better ask the question the way I did, but sure enough I wish someone could answer yours.
