From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/1739 Path: news.gmane.org!not-for-mail From: grandis@dima.unige.it (Marco Grandis) Newsgroups: gmane.science.mathematics.categories Subject: Can we ignore smallness? Date: Wed, 6 Dec 2000 16:56:07 +0100 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" X-Trace: ger.gmane.org 1241018057 32680 80.91.229.2 (29 Apr 2009 15:14:17 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:14:17 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Wed Dec 6 16:21:21 2000 -0400 Return-Path: Original-Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id eB6Jos225630 for categories-list; Wed, 6 Dec 2000 15:50:54 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 12 Original-Lines: 64 Xref: news.gmane.org gmane.science.mathematics.categories:1739 Archived-At: Dear categorists, in the last week there were some messages about categories of fractions and the smallness of their hom-sets, set forth by a question of Ph. Gaucher (Subject: category of fraction and set-theoretic problem; 30 Nov). I was puzzled by this sentence, in M. Barr's reply (30 Nov): > ... "But first, I might ask why it matters. Gabriel-Zisman ignores the >question and I think they are right to. Every category is small in >another universe." ... The reason why I think it matters should be clear from this example. U is a universe and Set is the category of U-small sets. Set has U-small hom-sets and is U-complete (has all limits based on U-small categories); it is not U-small. Of course it is V-small for every universe V to which U belongs; but then, it is not V-complete. The relevant fact, here, should be: - to have U-small hom sets and U-small limits for the SAME universe, i.e., a balance between a property (small hom-sets) which automatically extends to larger universes and another (small completeness) which automatically extends the other way, to smaller ones. Similar balances arise, less trivially, in categories of fractions. I think that the interest of proving they have small hom-sets (when possible) is related to other properties of such categories, holding for the same universe but not in larger ones. Thus: HoTop (the homotopy category of U-small topological spaces) has U-small hom-sets and U-small products. (It lacks equalisers; but it has weak equalisers, whence U-small weak limits.) [HoTop is the category of fractions of Top with respect to homotopy equivalences. One proves that it has U-small hom sets by realising it as the quotient of Top modulo the homotopy congruence. U-small products (as well as U-small sums) are inherited from Top, because they are "2-products" there, i.e. satisfy the universal property also for homotopies. Weak equalisers are provided by homotopy equalisers in Top.] With best regards Marco Grandis Dipartimento di Matematica Universita' di Genova via Dodecaneso 35 16146 GENOVA, Italy e-mail: grandis@dima.unige.it tel: +39.010.353 6805 fax: +39.010.353 6752 http://www.dima.unige.it/STAFF/GRANDIS/ ftp://pitagora.dima.unige.it/WWW/FTP/GRANDIS/