FOM: Grothendieck universes

[Colin Mclarty]

Reply to message from simpson at math.psu.edu of Fri, 16 Apr


>For instance, what about the category of countable Abelian groups?  Is
>that an Abelian category?  What is the ``size'' of that category?  Is
>that category ``universe-sized''?

	We need to be precise here. Since countable sets come in
every rank, obviously that category has a proper class of objects. 
You must have some further restriction in mind, say only one
representative of each isomorphism class, or some restiction on
rank. Whatever restriction you place you must make sure it is
compatible with all the constructions you want to do.

	I have mentioned a restriction that will work for all
of Grothendieck's purposes, namely to V(w+w) but that is already
huge. Much smaller restrictions will work for Wiles's proof 
and perhaps someone can tell just what they would be. But I 
would be very surprised if it is published anywhere.

>It sounds to me as if you are playing with definitions in order to
>avoid the genuine question that I raised.  The genuine question was,
>hasn't anyone ever explained the idea of derived functor cohomology in
>a way that didn't refer to Grothendieck universes?

	Without referring to them explicitly yes, all the time.
Hartshorne does that. But when he says which categories he will 
use, they are all universe sized. The simplest is the category
of (all) Abelian groups. He never discusses cardinality 
restrictions or other devices to keep the categories smaller.
He shows how to get down to really small structures for
the particular calculations in Cech cohomology but in his 
version the Cech calculations draw on the general theorems
of the derived functor definition.

	Every treatment of derived functor cohomology that I have
seen does this. I haven't read them all, but I have looked at
the standard references. I can tell you no other treatment is as 
widely read as Hartshorne.

>I asked you to resolve the apparent
>contradiction between your statements ``the proofs which eliminate
>universes have no genuine mathematical meaning'' and ``the methods are
>*quite* meaningful mathematically''.  You ducked the question.  Care
>to try again?

	To give all the proofs, from the beginning, with no 
reference to universes would produce a jumble with no 
mathematical meaning. Knowing how to get down from universes, 
to a few smaller structures on appropriate occasions, is vital. 
You need general theorems and particular calculations.