From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6495 Path: news.gmane.org!not-for-mail From: "F. William Lawvere" Newsgroups: gmane.science.mathematics.categories Subject: Re: Stone duality for generalized Boolean algebras Date: Wed, 26 Jan 2011 12:19:54 -0500 Message-ID: References: , Reply-To: "F. William Lawvere" NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="Windows-1252" Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1296092068 6126 80.91.229.12 (27 Jan 2011 01:34:28 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Thu, 27 Jan 2011 01:34:28 +0000 (UTC) To: , , categories Original-X-From: majordomo@mlist.mta.ca Thu Jan 27 02:34:23 2011 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.114]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1PiGkg-0001h0-U4 for gsmc-categories@m.gmane.org; Thu, 27 Jan 2011 02:34:23 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:42017) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1PiGkH-0007Gf-OW; Wed, 26 Jan 2011 21:33:57 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1PiGkE-0000X1-CJ for categories-list@mlist.mta.ca; Wed, 26 Jan 2011 21:33:54 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6495 Archived-At: Dear George You ask would any categorically thinking mathematician say that the category of pointed sets needs further description as the category of sets= and partial maps? =20 In 1969-1970=2C recalling a) the preorigins of sheaf theory a hundred years ago in the still-non-trivial problem of extending of partial maps in analysis and topology=2C and b) desirous of an instrument for describing sheafication in finitely algebraic terms =20 Myles and I proposed Ytilda->Omega as one of the two axioms for an elementary theory of toposes (the other being the Pi right adjoint to pullback=3B applying Grothendieck=92s method of relativiza= tion using any given model U of those axioms=2C the 2-category of U-Toposes was obtained thus capturing precisely the original SGA4 notion by choice of U).= Of course these axioms were soon shown to be deducible from special cases=2C but the importance of classifying partial maps X..->Y remains. =20 The fact that this construction reduces to Y+1->1+1 in sets misled some recursion theorists to try to represent partia= l recursive maps that way=2C but the categorically thinking mathematician noted that in their category=2C the complement of the domain is typically not a subobject of X. As Phil Mulry showed with his Recursive Topos=2C subobjects= of Omega provide a precise specification of degrees of complication for the inclusion of the domain of definition by pulling back along X->Omega. (Although it would seen that Hilbert schemes as subobjects of Omega might provide similar representability=2C that apparently has not been pursued). =20 In the Boolean case Y+1 can be viewed as an action of the two-element monoid of idempotents (the instrument for analysis of objects in in a protomodular c= ategory)=2C in other words the category of partial maps can be embedded in a topos.Over= a general topos=2C that can be replaced by actions of Omega as a multiplicative monoid . =20 Of course partial maps are special binary relations=2C but of a qualitatively special= kind that requires its own status. In Cat=2C if replace subobjects by discrete opfibrations=2C the analogous =93partial maps=94 (=94machines=94) turn out= to be representable but give rise analogously to special distributors. =20 Peter Freyd=92s dictum has a dialectical companion. Category theory can sometimes discern the germ of nontrivial in the trivial. =20 > From: janelg@telkomsa.net > To: fejlinton@usa.net=3B categories@mta.ca > Subject: categories: Re: Stone duality for generalized Boolean algebras > Date: Mon=2C 24 Jan 2011 23:15:17 +0200 >=20 > Dear Fred=2C >=20 > Please forgive me=2C but let us distinguish between serious questions and > trivialities: >=20 > Andrej Bauer asked: >=20 > "...How exactly does this extend to generalized Boolean algebras?..." >=20 > And the answer is trivial (without quotation marks): The category GBA of > what he called generalized Boolean algebras is dually equivalent to the > category 1\STONE of pointed Stone spaces. This follows from Stone duality > (since GBA is equivalent to BA/2)=2C but also extends it: just as BA is a > non-full subcategory of GBA=2C STONE can be considered as a non-full > subcategory of 1\STONE via the functor that adds base points. And this wa= y > the dual equivalence between GBA and 1\STONE indeed extends the Stone > duality. >=20 > Although your first message about BA/2 was written after mine=2C I am sur= e you > know these things (you probably knew them before I knew the definition of= a > category...) >=20 ... [For admin and other information see: http://www.mta.ca/~cat-dist/ ]