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/