categories: Can we ignore smallness?

[grandis@dima.unige.it (Marco Grandis)]

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/