(This is a reply to Jim's thought on H-spaces, not really to Mike's one on Malcev relations.) There would seem to be a need for a discussion of homotopy everything models of the theory of categories. The objects would have a composition that was homotopy associative up to arbitrary degree, would have homotopy identities, etc. Vogt back in 1974 provided an example of such a thing in the context of homotopy coherent diagams of spaces. The composition morphism needed to be chosen, but once chosen was homotopy associative to infinite degree. (Jean-Marc Cordier and I have examined this in the context of simplicially enriched categories.) It seems that a possible way out is to consider such an infinitely lax category as a simplicial class satisfying a weak Kan condition, that condition giving the composition. Grothendieck in his pursuit of stacks asked for infinitely lax categories or groupoids to model homotopy types, and recent work by Kapranov, Voevodsky, Steiner etc., as well as a mass of material originating "down-under" from Ross, and friends, suggest that the time is approaching when an attack on these problems can be made. Jean-Marc and I have a lot of information on doing homotopy coherence in simplicially enriched settings including homotopy coherent forms of the Yoneda lemma, the interchange law in this setting, and so on. Unfortunately it is not always obvious how to go beyond locally weakly Kan simplicial category, where at least the composition is defined, at least, so as to be able to handle weakly Kan simplicial classes. I would much appreciate peoples reactions to this view. Cheers, Tim. (Sender:Tim Porter: e-mail: mas013@uk.ac.bangor.vaxc School of Maths, University of Wales at Bangor, Dean Street, Bangor, Gwynedd, LL57 1UT. U.K. ==========================================================================