FOM: Grothendieck universes

Stephen G Simpson simpson at
Fri Apr 16 20:04:02 EDT 1999

McLarty 16 Apr 1999 16:41:20

 > Derived functors between Abelian categories are themselves universe
 > sized sets. ...

Does this mean that within ZFC you can't even prove the existence of
*one* derived functor between Abelian categories?  Really?  I must be
confused.  Let's try to resolve this simple issue.  Could you please
outline the relevant parts of the definitions of Abelian category and
derived functor?

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''?  Are you saying that ZFC does not
prove the existence of any derived functors on that and similar-sized

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?

 > Certainly Friedman and Grothendieck would not agree on all instances
 > of "genuine mathematical meaning", ...

Why are you trying to be a spokesman for Friedman and Grothendieck?
Why don't you simply speak for yourself, i.e. make your own points in
your own voice?  It's very clear that Friedman totally rejects your
interpretation of his work.  I don't see any reason to assume that
your interpretation of Grothendieck's ideas is any more accurate.
That's why I'm asking for extended verbatim quotes from Grothendieck,
rather than your interpretation.

In order to force your Friedman/Grothendieck analogy (which Friedman
vehemently rejects), you play around with the phrase ``genuine
mathematical meaning''.  The disturbing thing is that you seem to be
using this phrase in two diametrically opposed senses.  In the
Friedman context, ``genuine mathematical'' refers to reliance on very
concrete, core mathematical concepts -- finite trees, etc.  But in the
Grothendieck context, ``genuine mathematical'' evidently refers to
reliance on Grothendieck universes -- a very abstract concept from
category theory.  What's going on here?  Your analogy seems to be
completely backwards.

 > I say the method has vital uses. But to use only that method, would
 > render the field meaningless. ...

This doesn't answer my question.  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?

-- Steve

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