From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6286 Path: news.gmane.org!not-for-mail From: Michael Shulman Newsgroups: gmane.science.mathematics.categories Subject: Re: The omega-functor omega-category Date: Mon, 4 Oct 2010 11:41:55 -0700 Message-ID: References: Reply-To: Michael Shulman 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 1286280867 9534 80.91.229.12 (5 Oct 2010 12:14:27 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Tue, 5 Oct 2010 12:14:27 +0000 (UTC) Cc: categories To: Vaughan Pratt Original-X-From: majordomo@mlist.mta.ca Tue Oct 05 14:14:22 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpy.mta.ca ([138.73.1.139]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1P36PP-000099-Sm for gsmc-categories@m.gmane.org; Tue, 05 Oct 2010 14:14:16 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:57576) by smtpy.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1P36OZ-0007RA-KT; Tue, 05 Oct 2010 09:13:23 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1P36OT-0000Cw-R5 for categories-list@mlist.mta.ca; Tue, 05 Oct 2010 09:13:18 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6286 Archived-At: By definition (at least according to the usage under discussion), something necessarily preserved by all morphisms is a "property," although it can also be regarded as a particular degenerate case of a structure and, I guess, also a degenerate case of a property-like structure. property =3D forgetful functor is full and faithful structure =3D forgetful functor is faithful property-like structure =3D forgetful functor is pseudomonic http://ncatlab.org/nlab/show/stuff%2C+structure%2C+property Mike On Mon, Oct 4, 2010 at 12:52 AM, Vaughan Pratt wrot= e: > > On 10/2/2010 3:03 PM, Michael Shulman wrote: >> >> I personally prefer to say that "unique choice structure" is something >> "in between" property and structure. =A0Kelly and Lack dubbed it >> "Property-like structure" in their paper with that title. =A0The >> 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. =A0The inverse is 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. =A0The complement > is preserved by all morphisms. > > Are these merely "property-like structures," or are they actual > structures, despite being defined merely as properties? > > Vaughan > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]