From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6289 Path: news.gmane.org!not-for-mail From: "Fred E.J. Linton" Newsgroups: gmane.science.mathematics.categories Subject: Re: The omega-functor omega-category Date: Mon, 04 Oct 2010 17:00:17 -0400 Message-ID: Reply-To: "Fred E.J. Linton" NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1286280962 10001 80.91.229.12 (5 Oct 2010 12:16:02 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Tue, 5 Oct 2010 12:16:02 +0000 (UTC) Cc: Vaughan Pratt To: categories Original-X-From: majordomo@mlist.mta.ca Tue Oct 05 14:16:01 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.138]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1P36R6-0000c6-Ky for gsmc-categories@m.gmane.org; Tue, 05 Oct 2010 14:16:00 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:42015) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1P36QS-0001Xj-6b; Tue, 05 Oct 2010 09:15:20 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1P36QF-0000GJ-Hj for categories-list@mlist.mta.ca; Tue, 05 Oct 2010 09:15:07 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6289 Archived-At: In his message of Mon, 04 Oct 2010 08:16:25 AM EDT, Vaughan Pratt = quibbled with what on 10/2/2010 3:03 PM, = Michael Shulman had written: >> I personally prefer to say that "unique choice structure" is somethin= g >> "in between" property and structure. Kelly and Lack dubbed it >> "Property-like structure" in their paper with that title. The >> difference is exactly as you say: property-like structure is unique >> (up to unique isomorphism) when it exists, but is not necessarily >> "preserved" by all morphisms. > = > How should this terminology be applied when the property-like structure= > is necessarily preserved by all morphisms? > = > A group can be defined as a monoid with the property that all of its > elements have inverses. The inverse is preserved by all morphisms. A group can also be defined as a *semigroup* with that property. "The inverse" need no longer be "preserved by all morphisms." = > A Boolean algebra can be defined as a bounded distributive lattice with= > the property that all of its elements have complements. The complement= > is preserved by all morphisms. Depends what you take to be a bounded lattice. Do you specify *finitary* meets and joins, including the explicit empty ones that produce the bound= s? Or just *binary* ones, with the bounds *required* but not *specified*? In the former situation, yes, "the complement is preserved by all morphis= ms." In the latter situation, alas, no. > Are these merely "property-like structures," or are they actual > structures, despite being defined merely as properties? When such a "property-like structure" *is* preserved, it is perhaps = implicitly trying to behave like an "actual" structure, and could certainly be harmlessly added to the actual structural specifications, but, with so much riding on the *context* in which one is asking about that property-like structure, I'm not yet ready just to declare them, willy-nilly, to be "actual structures". Cheers, -- Fred [For admin and other information see: http://www.mta.ca/~cat-dist/ ]