From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6276 Path: news.gmane.org!not-for-mail From: Michael Shulman Newsgroups: gmane.science.mathematics.categories Subject: Re: The omega-functor omega-category Date: Sat, 2 Oct 2010 15:03:20 -0700 Message-ID: References: Reply-To: Michael Shulman NNTP-Posting-Host: lo.gmane.org Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1286111622 23277 80.91.229.12 (3 Oct 2010 13:13:42 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sun, 3 Oct 2010 13:13:42 +0000 (UTC) Cc: John Baez , categories To: Steve Vickers Original-X-From: majordomo@mlist.mta.ca Sun Oct 03 15:13:40 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 1P2ONn-0005ii-5W for gsmc-categories@m.gmane.org; Sun, 03 Oct 2010 15:13:39 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:45776) by smtpy.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1P2OMu-0006C9-C5; Sun, 03 Oct 2010 10:12:44 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1P2OMq-0002pn-Hd for categories-list@mlist.mta.ca; Sun, 03 Oct 2010 10:12:40 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6276 Archived-At: I personally prefer to say that "unique choice structure" is something "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. In terms of forgetful functors, property-like structure corresponds to a functor which is *pseudomonic*, i.e. faithful, and full-on-isomorphisms. Another 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. Mike On Fri, Oct 1, 2010 at 7:22 AM, Steve Vickers w= rote: > Dear John, > > There are respects in which properties are not exactly equivalent to > degenerate, "unique choice" cases of structure. It can make a difference > whether you consider something as property or structure, and one > situation where the difference enters is when you consider > homomorphisms, i.e. structure-preserving functions. > > For example, finiteness of sets looks like a property, but it can also > be expressed as structure. The finiteness of a set X is, as structure, > an element T of the finite powerset of X (i.e its free semilattice) such > that x in T for all x in X. The structure, if it exists at all, is > unique: T is the whole of X. > > If f: X -> Y is a function between finite sets X and Y then for f to be > a homomorphism of finite sets, i.e. for it to preserve finiteness as a > structure, means that the direct image of T_X is T_Y, i.e. f is onto. > > This may look artificial, but in fact it is exactly what you are forced > to do if you wish to express finiteness in a geometric theory, as when > presenting classifying toposes. The problem is that geometric theories > are rather restricted in what properties they can express, so a frequent > solution is to convert properties into structure. > > Another example is for decidable sets, i.e. those for which equality has > a Boolean complement - an inequality relation. (We are talking about > non-classical logics here.) A homomorphism then has to preserve > inequality as well as equality, and so be 1-1. > > This is comparable with what you say in your paper with Shulman, if you > replace categories with classifying toposes. (After all, you use > topological ideas in your paper, and geometric logic is well adapted to > topology.) For the classifying toposes, the difference between > properties and structure is that properties correspond to subtoposes. A > subtopos inclusion is a geometric morphism that, at a first level of > approximation that ignores deeper topology, is full and faithful on > points. This matches your classification for forgetting at most > properties. But the thing about the geometric theories is that they > oblige you to work with the category of finite sets _and surjections_, > and this is what stops the functor FinSets -> Sets from being full. It > is only faithful and so forgets at most structure. > > Regards, > > Steve Vickers. > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]