From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8325 Path: news.gmane.org!not-for-mail From: "Fred E.J. Linton" Newsgroups: gmane.science.mathematics.categories Subject: Re: looking for a reference... Date: Sun, 14 Sep 2014 15:10:24 -0400 Message-ID: Reply-To: "Fred E.J. Linton" NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=windows-1252 Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1410822314 28482 80.91.229.3 (15 Sep 2014 23:05:14 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Mon, 15 Sep 2014 23:05:14 +0000 (UTC) Cc: Alex Kruckman To: Dana Scott , Categories list Original-X-From: majordomo@mlist.mta.ca Tue Sep 16 01:05:08 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 1XTfKW-0003mQ-5l for gsmc-categories@m.gmane.org; Tue, 16 Sep 2014 01:05:08 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:37877) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1XTfJH-00017u-Sy; Mon, 15 Sep 2014 20:03:51 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1XTfJH-0007N5-Qh for categories-list@mlist.mta.ca; Mon, 15 Sep 2014 20:03:51 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8325 Archived-At: Referring to Alex's question in Dana's original post (below): Sorry, but I'm not seeing how this is all that different from just picking an object A in C, contemplating all powers of A in C, and asking about the full subcategory K (of C) of all subobjects = of those powers? In case C is a variety, surely that's a well-understood sort of class of algebras (closed under products and subalgebras), n'est-ce pas? Cheers, -- Fred --- ------ Original Message ------ Received: Sun, 14 Sep 2014 09:28:51 AM EDT From: Dana Scott To: Categories list Cc: Alex Kruckman = Subject: categories: Re: looking for a reference... > If you have comments/suggestions, please reply to Mr. Kruckman. Thanks= =2E > = > On Sep 13, 2014, at 10:02 AM, Alex Kruckman wrote:= > = >> Professor Scott, >> = >> 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? >> = >> Here it is: >> = >> 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 e= lement >> 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 i= s 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)= : >> = >> 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)) >> = >> 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 tr= ue of F(1). >> = >> 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 alge= bra. Well, >> there's a canonical such embedding, given by the unit of the adjuncti= on A -> F(G(A)). >> That is, A -> F(Hom_C(A,F(1))). = >> = >> Examples of these observations include all the constructions of algeb= ras from >> sets by powerset - the Stone representation theorem for Boolean algeb= ras (minus the >> topology, of course), but also the representation theorems for lattic= es, semilattices, etc. >> = >> Thanks for taking the time to read this. Let me know if it rings a be= ll. >> = >> -Alex > = [For admin and other information see: http://www.mta.ca/~cat-dist/ ]