As a small point, I'm quite sure Boardman and Vogt never used the term ``weak Kan complex'' that Andre rightly opposes: they referred to the ``restricted Kan condition'' (see SLN 347, p. 102). I have no idea who is guilty of ``weak'' in this context, but I think it was never in common use. In fact, Boardman and Vogt were right away sensitive to size issues, referring to ``simplicial classes'' rather than simplicial sets satisfying the inner horn condition. They did not think of them as special kinds of Kan complexes, which arguably they are not since Kan complexes are generally understood to be simplicial sets. Certainly that was how Kan complexes were understood when Boardman and Vogt were writing (published 1973, mostly written earlier). They already knew then that their notion gave them, in their words, ``good substitutes for categories''. I also like Andre's term ``quasi-categories'' and prefer it to the infinity alternatives, for the reasons he gives. Like Kan complex, it has a fixed and evocative definite meaning, unlikely to be confused with anything else. Now if only he would publish ... Responding to another part of this (endless) string, I also regret how words with a nice ring to them get so modified that they no longer have a clear precise meaning attached to them. Everyone can guess one word I'm thinking of (my mother was an opera singer). Peter May [For admin and other information see: http://www.mta.ca/~cat-dist/ ]