From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5451 Path: news.gmane.org!not-for-mail From: pare@mathstat.dal.ca (Robert Pare) Newsgroups: gmane.science.mathematics.categories Subject: Small is beautiful Date: Fri, 1 Jan 2010 10:48:26 -0400 (AST) Message-ID: Reply-To: pare@mathstat.dal.ca (Robert Pare) NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1262377722 26442 80.91.229.12 (1 Jan 2010 20:28:42 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Fri, 1 Jan 2010 20:28:42 +0000 (UTC) To: categories@mta.ca Original-X-From: categories@mta.ca Fri Jan 01 21:28:34 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from mailserv.mta.ca ([138.73.1.1]) by lo.gmane.org with esmtp (Exim 4.50) id 1NQo6q-0006m2-4j for gsmc-categories@m.gmane.org; Fri, 01 Jan 2010 21:28:32 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1NQnf2-0007V3-Rg for categories-list@mta.ca; Fri, 01 Jan 2010 15:59:48 -0400 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5451 Archived-At: I would like to add a few thoughts to the "evil" discussion. 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). Subfunctors of representables are not usually representable. 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 identities were canonical whereas in large categories, all isomorphisms were 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 developed 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. 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 in practice. Large ones which are categories of structures, corresponding to various 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 are equivalent. It was considered impolite to ask if two objects were equal. Then 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 categories were to be thought of as constructions of one structure from another, e.g. the group ring. Functors between small categories were interpretations of one theory in another or reindexing or rearranging. Functors from small to large categories were models or diagrams in the large one. These kinds of functors are perhaps the most important of the four, although this may be debatable. The fourth kind, from large to small are rarer. They can be thought of as gradings or partitions of the large category. 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. Bob [For admin and other information see: http://www.mta.ca/~cat-dist/ ]