From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6280 Path: news.gmane.org!not-for-mail From: Michael Shulman Newsgroups: gmane.science.mathematics.categories Subject: Re: The omega-functor omega-category Date: Sun, 3 Oct 2010 15:11:39 -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 1286194137 25018 80.91.229.12 (4 Oct 2010 12:08:57 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Mon, 4 Oct 2010 12:08:57 +0000 (UTC) Cc: categories To: Colin McLarty Original-X-From: majordomo@mlist.mta.ca Mon Oct 04 14:08:56 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 1P2jqf-0008Tn-OR for gsmc-categories@m.gmane.org; Mon, 04 Oct 2010 14:08:54 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:58708) by smtpy.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1P2jpw-0004zB-RE; Mon, 04 Oct 2010 09:08:08 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1P2jpp-0001ga-9n for categories-list@mlist.mta.ca; Mon, 04 Oct 2010 09:08:01 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6280 Archived-At: Rereading my message, I realized I should perhaps clarify that not just the name, but (as far as I know) the concept itself was originated by Kelly and Lack. The example of semigroups is also in their paper, which is concerned mainly with the case when the forgetful functor is also 2-monadic. The resulting "property-like 2-monads" generalize both "lax-idempotent" (=3D "Kock-Zoberlein") 2-monads, such as those which assign colimits, and the dual "colax-idempotent" 2-monads, such as those which assign limits. But they are strictly more general than either: for instance, a 2-monad which assigns both limits and colimits is property-like, but not lax- or colax-idempotent. Mike On Sun, Oct 3, 2010 at 6:32 AM, Colin McLarty wrot= e: > I like this discussion by Mike Shulman. =A0And a propos of the related > discussion of terminology I note the terms here describe mathematical > features (very well, I think) rather than focusing on whether one > *likes* the features. > > 2010/10/2 Michael Shulman : > >> 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. =A0In terms of forgetful functors, >> property-like structure corresponds to a functor which is >> *pseudomonic*, i.e. faithful, and full-on-isomorphisms. =A0Another nice >> example is that being a monoid is a "property" of a semigroup, i.e. a >> semigroup can have at most one identity element, but a semigroup >> homomorphism between monoids need not be a monoid homomorphism. > > best, Colin > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]