Re: Can we ignore smallness?

[Dan Christensen <jdc@julian.uwo.ca>]

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