From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2270 Path: news.gmane.org!not-for-mail From: Vaughan Pratt Newsgroups: gmane.science.mathematics.categories Subject: re: Query (Q-algebras) Date: Wed, 07 May 2003 11:55:08 -0700 Message-ID: <200305071855.LAA22674@coraki.Stanford.EDU> References: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii X-Trace: ger.gmane.org 1241018540 3341 80.91.229.2 (29 Apr 2009 15:22:20 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:22:20 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Thu May 8 13:25:52 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 08 May 2003 13:25:52 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 19DoCJ-0003rf-00 for categories-list@mta.ca; Thu, 08 May 2003 13:24:43 -0300 X-Mailer: exmh version 2.1.1 10/15/1999 In-Reply-To: Message from Oswald Wyler of "Mon, 05 May 2003 13:46:43 EDT." X-Scanner: exiscan for exim4 *19DUBp-0003yO-00*288I3uw4n6c* Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 5 Original-Lines: 28 Xref: news.gmane.org gmane.science.mathematics.categories:2270 Archived-At: The operations of Q-alg are the functions between the powers of the set Z, forming the (Lawvere) theory T of Q-algebras, which are product-preserving functors from T to Set. (So if Z is 3 then there are 27 = 3^3 "Boolean" operations in place of the familiar 4 = 2^2.) With all powers Q-algebras are equivalent to CABAs, with only finite powers and finite-product-preserving functors they are equivalent to Boolean algebras. A natural next question would be, what is obtained when T is taken to be Set itself, in each of the cases when the functors T->Set are required to preserve all limits, and just the discrete ones? Vaughan Pratt >From: Oswald Wyler >For every set Z, there is a self-adjoint contravariant functor Q=ENS(--,Z), >with unit/counit h:Id-->Q^op Q given by (h_X)(x)(f)=f(x). Let Q-alg denote >the category of algebras for the monad induced by this self-adjunction. >If Z is not empty or a singleton, then the comparison functor ENS^op-->Q-alg >is an equivalence by results of M. Sobral. If Z has two members, then Q-alg >is isomorphic to CaBool, the category of complete atomic Boolean algebras. >What is known about Q-alg if Z has more than two members (beyond the fact >that Q-alg and CaBool are equivalent)?