Subject: categories: Can we ignore smallness?

From: [email-masked] (Marco Grandis)
Date: Wed, 6 Dec 2000 16:56:07 +0100

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: [email-masked]
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/

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From: Dan Christensen <[email-masked]>
Date: 06 Dec 2000 16:49:35 -0500

I agree with what Marco Grandis wrote, suggesting that sometimes it is
important to know that the hom sets in a category are small, and want
to just supplement what he said with some examples from topology.  In
topology, one often wants to use a generalized homology or cohomology
theory E to compute something, and it can be useful to "localize" 
a space with respect to this (co)homology theory.  The localization 
X --> L_E X can be characterized as the terminal map from X which
induces an isomorphism under E.  The existence of such localizations
for all X is equivalent to the category Top[(E-isomorphisms)^{-1}]
having small hom sets, and so knowing that the latter is true means
that one has an important tool for practical computations.

The paper by Bousfield

    Bousfield, A. K. 
    The localization of spaces with respect to homology. 
    Topology 14 (1975), 133--150.

is considered quite important because it showed that for any
generalized homology theory E, localizations exist, and these
localizations now play a central role in homotopy theory.  Bousfield
proved the existence by showing that the category of fractions above
has small hom sets.  And he did that by showing that there is a model
structure on the category Top with the E-isomorphisms as the weak
equivalences.

Note that it is still an open question as to whether *co*homological
localizations exist for every cohomology theory E!  Casacuberta,
Scevenels and Jeff Smith have recently shown that they exist if you
assume Vopenka's principle, but if anyone can prove this in general or
show it is independent of ZFC, that would be considered very
interesting.

Dan