From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4616 Path: news.gmane.org!not-for-mail From: "Zinovy Diskin" Newsgroups: gmane.science.mathematics.categories Subject: Re: are sketches math objects? Date: Sat, 20 Sep 2008 13:34:01 -0400 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241020059 14051 80.91.229.2 (29 Apr 2009 15:47:39 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:47:39 +0000 (UTC) To: "Michael Barr" , categories@mta.ca Original-X-From: rrosebru@mta.ca Sun Sep 21 10:52:11 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 21 Sep 2008 10:52:11 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KhPGz-0006tf-0y for categories-list@mta.ca; Sun, 21 Sep 2008 10:46:49 -0300 Content-Disposition: inline Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 86 Original-Lines: 30 Xref: news.gmane.org gmane.science.mathematics.categories:4616 Archived-At: Okay, sketches are presentations of theories but Steve's claim was that they are not mathematical objects. Michael's and mine bewilderment is about why does the former imply the latter? (at least, why "of course" :) Zinovy On Fri, Sep 19, 2008 at 6:27 PM, wrote: > Dear Michael, > > Semantically, as Lawvere observed long ago, a monad gives rise not just to > a category of algebras but also to a forgetful functor into the category > on which the monad acts. For any category C the functor > > "semantics" : Mnd(C)^op --> CAT/C > > whose object map sends a monad on C to its associated forgetful functor is > full and faithful. Thus a pair of monads on C giving rise to isomorphic > forgetful functors must necessarily be isomorphic. So your observations > about different monads giving rise to the same algebras, while correct, do > not tell the whole story on the semantic side. > > The situation is of course different for sketches: they too give rise to > forgetful functors (into Set), but this does not suffice to determine a > given sketch up to isomorphism in the same way, and this justifies Steve > Lack's perspective of "sketches as presentations of theories". > > Mark Weber >