Please change the line that says that in a Mal'cev category, every relation is Mal'cev to one that says that if an equational category has a Mal'cev operation, then every relation is Mal'cev. Thanks, Mike ==========================================================================
The observations by Martin Hoffman are interesting. In the interests of clarity, I do object to his calling the category of pullback squares the category of Mal'cev relations. One is an existential structure, while the other builds the thing that exists into the structure. It makes it into an entirely different category. Not less interesting, but different. It is (somewhat) like the difference between the categories of complete lattices and complete semilattices. One interesting (and trivial) observation is that if Mal'cev relations are effective, then a quotient of a subobject of a quotient. This is (like everything I've said, but forget to mention) in the presence of sufficient limits and colimits. Now if one could show that the amalgamation property held, one would have that a pushout of a mono is mono in that case. Michael ==========================================================================
the difference beteen the two categories triggers memories of discussions with John Moore about the non-category of H-spaces the problem is with the morphisms either an H-map is one that respects the two multiplications up to some unspecified homotopy OR an H-map is a pair consisting of a map say f:X --> Y AND a specific homotopy h_t:X x X --> Y either way there are problems thoughts? jim ==========================================================================
(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. ==========================================================================
participants (3)
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barr＠triples.Math.McGill.CA -
jds＠rademacher.math.upenn.edu -
MAS013＠BANGOR.AC.UK