From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8324 Path: news.gmane.org!not-for-mail From: "Eduardo J. Dubuc" Newsgroups: gmane.science.mathematics.categories Subject: Re: Re: looking for a reference... Date: Sun, 14 Sep 2014 11:41:09 -0300 Message-ID: References: Reply-To: "Eduardo J. Dubuc" NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=windows-1252; format=flowed Content-Transfer-Encoding: 8bit X-Trace: ger.gmane.org 1410822238 27751 80.91.229.3 (15 Sep 2014 23:03:58 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Mon, 15 Sep 2014 23:03:58 +0000 (UTC) Cc: Categories list , Alex Kruckman To: Dana Scott Original-X-From: majordomo@mlist.mta.ca Tue Sep 16 01:03:51 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 1XTfJF-0003JA-Hr for gsmc-categories@m.gmane.org; Tue, 16 Sep 2014 01:03:49 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:37871) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1XTfIE-00013l-Sh; Mon, 15 Sep 2014 20:02:46 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1XTfIE-0007Lw-Lh for categories-list@mlist.mta.ca; Mon, 15 Sep 2014 20:02:46 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8324 Archived-At: If you take C to be the dual of the category of models of a theory in universal algebra, and F to be the free functor (in this case F is covariant), then you will find a lot of familiar facts. I like the mathematics in your posting for its simplicity, and the idea of using contravariant functors very good. Whether it rings bells in some people or not, it is certainly a nice and worthwhile stuff just as it is, write it for the arXiv, and may be you can publish it somewhere afterwards. best e.d. On 13/09/14 21:28, Dana Scott wrote: > If you have comments/suggestions, please reply to Mr. Kruckman. Thanks. > > 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?ve >> 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(-) = 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 = signs here are natural isomorphisms): >> >> Hom_C(A,F(X)) = Hom_C(A,F(coprod_X 1)) = Hom_C(A,prod_X F(1)) = >> prod_X Hom_C(A,F(1)) = prod_X G(A) = 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 true of F(1). >> >> 3. Okay, let's say we have an axiomatization T for K. Then we have a ?representation >> 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))). >> >> 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. >> >> Thanks for taking the time to read this. Let me know if it rings a bell. >> >> -Alex > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]