From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8323 Path: news.gmane.org!not-for-mail From: Dana Scott Newsgroups: gmane.science.mathematics.categories Subject: Re: looking for a reference... Date: Sat, 13 Sep 2014 17:28:10 -0700 Message-ID: Reply-To: Dana Scott NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 (Mac OS X Mail 7.3 \(1878.6\)) Content-Type: text/plain; charset=windows-1252 Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1410701136 17690 80.91.229.3 (14 Sep 2014 13:25:36 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Sun, 14 Sep 2014 13:25:36 +0000 (UTC) Cc: Alex Kruckman To: Categories list Original-X-From: majordomo@mlist.mta.ca Sun Sep 14 15:25:30 2014 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp3.mta.ca ([138.73.1.127]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1XT9o1-0005Ge-Lh for gsmc-categories@m.gmane.org; Sun, 14 Sep 2014 15:25:29 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:54173) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1XT9nE-00045C-Ny; Sun, 14 Sep 2014 10:24:40 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1XT9nD-0002Hn-F1 for categories-list@mlist.mta.ca; Sun, 14 Sep 2014 10:24:39 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8323 Archived-At: If you have comments/suggestions, please reply to Mr. Kruckman. Thanks. On Sep 13, 2014, at 10:02 AM, Alex Kruckman wrote: > Professor Scott, >=20 > In writing up some work I did with another graduate student, we=92ve > noticed that one argument is really a special case of a very general > fact. It's easy to prove, and it's quite nice, but I've never seen it > explicitly noted. Have you? >=20 > Here it is: >=20 > 1. Suppose we have a contravariant functor F from Sets to some other > category C which turns coproducts into products. This functor = automatically > has an adjoint, given by G(-) =3D Hom_C(-,F(1)), where 1 is the one = element > set. If you like, the existence of G is an instance of the special = adjoint > functor theorem, but it's also easy to check by hand. The key thing is = that > every set X can be expressed as the X-indexed coproduct of copies of = the one > element set, so we have (the =3D signs here are natural isomorphisms): >=20 > Hom_C(A,F(X)) =3D Hom_C(A,F(coprod_X 1)) =3D Hom_C(A,prod_X F(1)) =3D > prod_X Hom_C(A,F(1)) =3D prod_X G(A) =3D Hom_Set(X,G(A)) >=20 > 2. Now let's say the category C is the category of algebras in some = signature. > Let's call algebras in the image of F "full", and let's say we're = interested > in the class K of subalgebras of full algebras. This class is closed = under > products and subalgebras, so if it's elementary, then it has an = axiomatization > by universal Horn sentences (i.e. it's a quasivariety), and moreover = every algebra > in the class is a subalgebra of a product of copies of F(1), so a = universal Horn > sentence is true of every algebra in the class if and only if it's = true of F(1). >=20 > 3. Okay, let's say we have an axiomatization T for K. Then we have a = =93representation > problem": given an algebra A satisfying T, embed it in some full = algebra. Well, > there's a canonical such embedding, given by the unit of the = adjunction A -> F(G(A)). > That is, A -> F(Hom_C(A,F(1))).=20 >=20 > Examples of these observations include all the constructions of = algebras from > sets by powerset - the Stone representation theorem for Boolean = algebras (minus the > topology, of course), but also the representation theorems for = lattices, semilattices, etc. >=20 > Thanks for taking the time to read this. Let me know if it rings a = bell. >=20 > -Alex [For admin and other information see: http://www.mta.ca/~cat-dist/ ]