From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5482 Path: news.gmane.org!not-for-mail From: pare@mathstat.dal.ca (Robert Pare) Newsgroups: gmane.science.mathematics.categories Subject: Small2 Date: Wed, 6 Jan 2010 10:30:15 -0400 (AST) Message-ID: <20100106143015.1C2D35C27D@chase.mathstat.dal.ca> References: 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 1262833867 31955 80.91.229.12 (7 Jan 2010 03:11:07 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Thu, 7 Jan 2010 03:11:07 +0000 (UTC) To: categories@mta.ca Original-X-From: categories@mta.ca Thu Jan 07 04:11:00 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 1NSim3-0005re-Q1 for gsmc-categories@m.gmane.org; Thu, 07 Jan 2010 04:11:00 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1NSiJX-0000SP-S5 for categories-list@mta.ca; Wed, 06 Jan 2010 22:41:32 -0400 In-Reply-To: Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5482 Archived-At: Thanks to all who replied to my posting either privately or on Categories. I'd like to clarify a few points. Ross's interpretation of what I meant, and which Vaughan agrees with, is not what I was trying to say. It is part of the picture but not the one I was promoting. Small categories play a different role in category theory than large categories. Small categories are used for indexing things. Large categories consist of the things we are indexing. Okay, syntactic was not the right word. Combinatorial might be better although that has finiteness overtones. Strict? Well, I'll just stick with small. Small is not the same as essentially small. As Jeff Egger pointed out, the category of finite sets is not small. It's not even clear what this category is. The ZFCists would say that for each set A we get a finite set {A} and another {{A}}, and Barwise might even wonder if these last two are distinct. But I'm digressing. Of course every small category can be considered as a large one (perhaps large isn't the right word either). Then two equivalent ones would be considered "the same". I don't think that this is how the working categorician works. Sometimes equivalent categories are "the same" and sometimes not. An equivalence relation is a category but we would lose something (everything?) if we identified it with equality in the quotient set. So I'm saying that, as a matter of "categorical hygiene", we could be a bit more explicit about what kind "equivalence" we are allowing. And without changing mathematical practice, I'm advocating that "small" should imply that it's okay to talk of equality of objects. As usual it helps put things in relief to generalize them. Consider category theory in a world based on a topos S. A small category is a category object in S. A large category like S or Group(S) is an indexed category, given by a pseudo-functor S^op -> Cat, or a fibration over S if you prefer. A small category C gives a large one by homming. But something is lost in the process. The object functor Cat -> Set is not a 2-functor so if you compose it with a pseudo-functor S^op -> Cat you get an assignment that doesn't preserve composition in any sense. So a general indexed category doesn't have a discrete category of objects. What you can do is take the groupoid of isomorphisms 2-functor Cat -> Gpd, so that a large has a groupoid of objects, thinking of a groupoid as a set with a 2-equality on it. An indexed category gotten from a small category gives an actual functor S^op -> Cat, so now you can compose with Ob : Cat -> Set so that a small category has a discrete groupoid of objects. And that's where the idea of considering categories with a specified "equality groupoid" of isomorphisms, and equivalences defined to have the isomorphisms from the corresponding groupoids. Small categories have only identities, so equivalence means isomorphism, and large categories have all isos. At the time there were no good examples of intermediate groupoids (just products of small and large categories and the like) but Hilbert spaces with unitary isos is a good one. Well this is my second posting in a week bringing my life-time total to three! Seems a good place to stop. Best wishes for the new year. May your happiness be large and your disappointments small. Bob [For admin and other information see: http://www.mta.ca/~cat-dist/ ]