It seems that the following message I sent last March 16 never has been lost somewhere during the trip: ******************************************************* To: CATEGORIES@mta.ca Subject: Re: regepis, stablefunctors... Status: RO Answer to Michael Barr. Sorry for having written the other way round what I intended to say in my second paragraph (as remarked by Paul Taylor). Then the algebraically closed fields are a counterexample to your "having a multi- initial object is having an initial object in each slice" precisely because they have the SECOND property but NOT THE FIRST ONE. I suppose you meant "COMPONENT" and not "slice". I think the rest is correct: I said that " in AXIOMATIC categories of finitary models, the existence and preservation by the forg.funct. of pullbacks imply the one of wide pullbacks (and filtered colimits) ". (I meant the theory was also finitary first-order...). In fact it is a classical result of Richter that filtered colimits in such categories are always preserved by the forgetful functor (of course as a result of the existence of the (usual) ultraproducts and elementary substructures). This result was refined by Volger and then (with a result of Pare)we get that the existence of pullbacks and their preservation implies the existence of filtered colimits and also the existence and preservation of cofiltered limits (and hence of "wide pullbacks": those are limits of diagrams with a terminal object). Michel Hebert ===================================================================