From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5478 Path: news.gmane.org!not-for-mail From: F William Lawvere Newsgroups: gmane.science.mathematics.categories Subject: Re: Small is beautiful Date: Tue, 5 Jan 2010 12:31:31 -0500 Message-ID: Reply-To: F William Lawvere NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="utf-8" Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1262743180 31058 80.91.229.12 (6 Jan 2010 01:59:40 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 6 Jan 2010 01:59:40 +0000 (UTC) To: , Original-X-From: categories@mta.ca Wed Jan 06 02:59:32 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from [138.73.1.1] (helo=mailserv.mta.ca) by lo.gmane.org with esmtp (Exim 4.50) id 1NSLBM-0007AF-08 for gsmc-categories@m.gmane.org; Wed, 06 Jan 2010 02:59:32 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1NSKmN-0007cE-OC for categories-list@mta.ca; Tue, 05 Jan 2010 21:33:43 -0400 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5478 Archived-At: Bob Pare' made the excellent point that not only size but quality=20 is relevant. I definitely agree with the spirit of his remarks.=20 Bob happens to have used in passing the term 'syntactic'.=20 For clarity, the use of that term needs to be sharpened to avoid misunderstanding. =20 Actually, the term 'syntax' refers NOT to small categories such as algebraic theories or rings, but rather to their PRESENTATION by=20 signatures or by polynomial generators, et cetera. The process=20 of presentation is an adjoint pair quite distinct from the=20 semantical adjoint pair: both adjoint pairs have a category of=20 theories or of rings in common but are otherwise quite independent.=20 In particular, syntax is NOT the adjoint of semantics. Cratylus, Chomsky, and their 21st century followers can be refuted by looking soberly at the actual practice of mathematics (wherein the construction of sequences of words and of diagrams is pursued with great care for the purpose of communication. That syntax is only remotely dependent on the structure of the content that is to be communicated). Both of the functors ?--------------> theories -------------->Large categories Syntax Semantics are needed. The domain category of the first can be chosen=20 in various useful ways: sketches or diagrams of signatures et cetera.=20 Happy new year! =20 Bill On Fri 01/01/10 9:48 AM , pare@mathstat.dal.ca (Robert Pare) sent: >=20 > I would like to add a few thoughts to the "evil" discussion. >=20 > My 30+ years involvement with indexed categories have led me > to the following understanding. There are two kinds of categories, > small and large (surprise!). But the difference is not mainly one > of size. Rather it's how well we can pin down the objects. The > distinction between sets and classes is often thought of in terms > of size but Russell's problem with the set of all sets was not one of > size but rather of the nature of sets. Once you think you have the set > of all sets, you can construct another set which you had missed. > I.e. the notion is changing, slippery. There are set theories where > you can have a subclass of a set which is not a set (c.f. Vopenka, > e.g.)Smallness is more a question of representability: a functor may fail= to > be representable because it's too big (no solution set) or, more often, > because it's badly behaved (doesn't preserve products, say). > Subfunctorsof representables are not usually representable. >=20 > In our work on indexed categories, Schumacher and I had tried to treat > this question by considering categories equipped with a groupoid of > isomorphisms, which we called *canonical*, and then consider functors > defined up to canonical isomorphism. In small categories only > identitieswere canonical whereas in large categories, all isomorphisms we= re > canonical.Our ideas were a bit naive and not well developed and earned us= some > ridicule,so we quietly stopped talking about it. Recently, Makkai develop= ed > an extensive theory of functors defined up to isomorphisms, FOLDS, but > did not consider the possibility of specifying which isomorphisms ahead > of time, so small categories were not included. >=20 > When I used to teach category theory, before Dalhousie made me chuck my > chalk chuck, I would tell students there were two kinds of categories > inpractice. Large ones which are categories of structures, corresponding > tovarious branches of mathematics we wished to study. These categories > supported various universal constructions, all defined up to > isomorphism.Two large categories are considered to be the same if they ar= e > equivalent.It was considered impolite to ask if two objects were equal. T= hen > there are the small categories which are used to study the large ones. > These are syntactic in nature. For these, one can't expect the kinds of > universal constructions that large categories have, but now it's okay, > even necessary, to consider equality between objects. I went on to say > that there were then four kinds of functors. Functors between large > categorieswere to be thought of as constructions of one structure from an= other, > e.g.the group ring. Functors between small categories were interpretation= s > ofone theory in another or reindexing or rearranging. Functors from small > to large categories were models or diagrams in the large one. These > kindsof functors are perhaps the most important of the four, although thi= s > maybe debatable. The fourth kind, from large to small are rarer. They can > be thought of as gradings or partitions of the large category. >=20 > Well, after these ramblings, perhaps my message is lost. So here it is: > Small categories -> equality of objects okay > Large categories -> equality of objects not okay > Small is beautiful, not evil. >=20 > Bob >=20 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]