From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4611 Path: news.gmane.org!not-for-mail From: Mark.Weber@pps.jussieu.fr Newsgroups: gmane.science.mathematics.categories Subject: Re: Bourbaki and Categories Date: Sat, 20 Sep 2008 00:27:41 +0200 (CEST) Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain;charset=iso-8859-1 Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1241020056 14037 80.91.229.2 (29 Apr 2009 15:47:36 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:47:36 +0000 (UTC) To: "Michael Barr" , categories@mta.ca Original-X-From: rrosebru@mta.ca Sat Sep 20 10:15:22 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 20 Sep 2008 10:15:22 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Kh2EK-0006UJ-Fl for categories-list@mta.ca; Sat, 20 Sep 2008 10:10:32 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 81 Original-Lines: 42 Xref: news.gmane.org gmane.science.mathematics.categories:4611 Archived-At: Dear Michael, Semantically, as Lawvere observed long ago, a monad gives rise not just t= o 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 i= s 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, d= o 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 > Previously, Michael Barr wrote: > > Of course sketches are mathematical objects in their own right. Of > course, the functor that assigns to each sketch the corresponding theor= y > is not full or faithful. But the definition is precise, the notion of > model is also precise, so I have no idea what, if any, content there is= in > the claim. Incidentally, you might with equal justice claim that tripl= es > are not mathematical objects since two distinct triples can have > isomorphic categories of Eilenberg-Moore algebras. In fact there are > triples (or theories) on Set that have infinitary operations, yet whose > category of models is isomorphic to Set. > > Michael >