From: grandis@dima.unige.it (Marco Grandis)
To: categories@mta.ca
Subject: Can we ignore smallness?
Date: Wed, 6 Dec 2000 16:56:07 +0100 [thread overview]
Message-ID: <v02140b05b653fa5e73a2@[130.251.167.61]> (raw)
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/
next reply other threads:[~2000-12-06 15:56 UTC|newest]
Thread overview: 2+ messages / expand[flat|nested] mbox.gz Atom feed top
2000-12-06 15:56 Marco Grandis [this message]
2000-12-06 21:49 ` Dan Christensen
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