Date: Wed, 11 Sep 1996 11:54:25 -0300 (ADT) Subject: Co-well-poweredness of varieties of algebras Date: Tue, 10 Sep 1996 21:18:11 -0700 From: William H. Rowan If we are given an algebra A in a variety of algebras, then the set of onto homomorphisms with domain A is obviously a small set. Is the same true of the set of epimorphisms in the variety, with domain A? I believe this is what is meant by asking, "Is the variety co-well-powered?" In any case, I think the answer must be known, and I think the answer is yes, but I don't have any reference. So, could someone help me with this? Bill Rowan Date: Wed, 11 Sep 1996 13:15:58 -0300 (ADT) Subject: Re: Co-well-poweredness of varieties of algebras Date: Wed, 11 Sep 1996 11:48:31 -0400 (EDT) From: Peter Freyd William Rowan asks if equational varieties, when viewed as categories, are well-co-powered. Yes. A method of proof appears in my 1964 "Abelian Categories" on pages 91-93. The context of the method is much more general: it holds for any category whose objects are defined as the models of a given set of elementary sentences and whose maps are defined as the functions that preserve a given set of elementary formulae. Ditto for well-poweredness. I believe, also, that that book is the initial appearence of "co-well-powered". It should, of course, have been "well-co-powered". Date: Wed, 11 Sep 1996 13:22:00 -0300 (ADT) Subject: Re: Co-well-poweredness of varieties of algebras Date: Wed, 11 Sep 1996 12:13:07 -0500 (EST) From: MTHISBEL@ubvms.cc.buffalo.edu Co-well-poweredness is long known; I think it was an exercise in Peter Freyd's book {\it Abelian Categories}. I think I gave a smaller upper bound for the number of epi-quotients in my La Jolla conference (1965) paper "Epimor- phisms and dominions". As I recall, my argument obviously extends to infinitary algebras but Peter's doesn't. That is, infinitary algebras with rank. With no rank, it was unknown for a couple of years in the mid-sixties whether co-well-poweredness still holds; but it turned out there is a very natural counterexample, the variety of frames (dual of locales). John Isbell Date: Wed, 11 Sep 1996 21:24:07 -0300 (ADT) Subject: Re: Co-well-poweredness of varieties of algebras Date: Wed, 11 Sep 1996 12:55:51 -0400 (EDT) From: Peter Freyd Minor point. John mentions that his argument for algebras "obviously extends to infinitary algebras but Peter's doesn't." But it does. I guess it's a matter of what one means by "obviously". Date: Mon, 16 Sep 1996 10:35:23 -0300 (ADT) Subject: Re: Co-well-poweredness of varieties of algebras Date: Mon, 16 Sep 1996 14:22:06 +0200 (MET DST) From: Jiri Rosicky The fact that varieties are co-well-powered follows from much more general results proved in M.Makkai and R.Pare, Accessible categories: the foundation of categorical model theory, Cont. Math. 104, AMS 1989 (e.g., any accessible category with pushouts is co-well-powered). For locally presentable categories, which cover varieties, it is shown in P.Gabriel and F.Ulmer, Lokal Presentierbare Kategorien, L.N. in Math. 221 (1971). These results are also contained a recent book Locally Presentable and Accessible Categories, Cambridge Univ. Press 1994 written by J.Adamek and me. Jiri Rosicky