FOM: faking Grothendieck universes

[Stephen G Simpson]

To back up his point about the alleged need for Grothendieck
universes, McLarty 15 Apr 1999 12:27:06 claimed that Hartshorne's
textbook on algebraic geometry employs them.  This claim seemed highly
suspicious, but unfortunately I don't have access to my copy of
Hartshorne this semester, so I couldn't easily look it up.

Now I've got Hartshorne from the library, so I can straighten this
out.  In the USA, sports analogies are very popular, so I will explain
this in terms of a baseball analogy.

Hartshorne says in his preface that his book is an introductory
textbook, complete with more than 400 exercises, and the only
prerequisites are some results from commutative algebra and elementary
topology -- no complex analysis or differential geometry or anything
else.  So far as I can see by thumbing through the book, the concepts
``universe'' and ``Grothendieck universe'' are never mentioned.  There
is a 19-page subject index, in fine print.  There is no index entry
for ``universe'' or ``Grothendieck universe''.

Strike one.

McLarty says:
 > Now turn to Chapter 3 of Hartshorne. This is no "few remarks at the
 > beginning". It is 80 pages in the middle and is used for key proofs
 > in the rest.
 ...
 > The key theorem ineliminably quantifies over universe sized sets.
 ...

Hartshorne's chapter III begins by defining an Abelian category in the
standard way, and giving seven examples.  Example 1 is the category of
Abelian groups.  All of the examples are like this one, in that they
are proper classes.  (In set theory, a proper class is defined to be a
class which is not a set.)

[ The most obvious candidate for a place to literally formalize
Hartshorne's book is VNBG set/class theory, perhaps with global
choice.  This is conservative over ZFC.  Of course the logical
strength is much less than this. ]

MacLane's textbook ``Categories for the Working Mathematician'' (page
22) defines a universe, U, to be a certain kind of set. (It is a
transitive set of the form V_kappa where kappa is an inaccessible
cardinal.  Although MacLane does not use the words ``transitive'' or
``inaccessible'', he does use the word ``set''.)  Thus, according to
MacLane, a universe is definitely *not* a proper class.  

The theorems in Hartshorne's review of homological algebra, including
what McLarty calls ``the key theorem'', quantify over functors on
Abelian categories.  This includes universe size sets if and only if
we assume the existence of universe size sets.  Hartshorne never
assumes the existence of universe size sets.

Strike two.

McLarty on Hartshorne:
 > The key theorem ineliminably quantifies over universe sized sets.

The most charitable way to understand this statement of McLarty is
that, *if* Hartshorne had followed Grothendieck's original treatment,
then Hartshorne might have assumed the existence of a universe U and
set things up in such a way that all Abelian categories are, by
definition, of cardinality less than or equal to card(U).  In other
words, replace ``set'' by ``set of size less than U'', and ``proper
class'' by ``set of size U''.  *If* Hartshorne had done things in this
way, then he would have been quantifying over aggregates which include
sets of size U, i.e. universe size sets.  (Editorial comment: This is
what Hartshorne might have done if he were an f.o.m. amateur.)

OK.  This isn't what Hartshorne did, but let's temporarily go along
with McLarty and *pretend* it's what Hartshorne did.

However, even with this charitable interpretation, in order to
evaluate McLarty's term ``ineliminably'', we still need to ask:

  Are the ``universe size sets'' (i.e. sets of size card(U) in the
  Grothendieck treatment, or proper classes in Hartshorne's treatment)
  *eliminable*?

The answer is that they are straightforwardly, trivially, eliminable.

Let me explain.  For example, take what McLarty calls ``the key
theorem'', i.e. the existence and universal properties of derived
functors.  Hartshorne states this for functors on Abelian categories
satisfying a mild condition, the existence of ``enough injectives'',
i.e. each object is a subobject of an injective object.  This is the
case for Hartshorne's examples of Abelian categories, but it is also
the case if we cut down in obvious ways.  For example, it is the case
for the category of countable Abelian groups.  It is also the case for
the category of Abelian groups of cardinality less than or equal to
lambda, for any fixed infinite cardinal lambda.  Thus the theorems on
pages 204-206 apply to functors on these categories as well.  These
are ``small'' (i.e. set size) Abelian categories.  Furthermore, if
these theorems were stated *only* for ``small'' (i.e. set size)
Abelian categories, or *only* for Abelian categories of size strictly
less than card(U), respectively (depending on whether we are following
the Hartshorne approach or the Grothendieck approach), then no content
would be lost, and all the applications in Hartshorne's book would go
through with very minor, trivial changes.  I'm sure Hartshorne and
everyone else in the field is well aware of this.

Strike three.  

McLarty, you have struck out.  Go to the showers.

McLarty:
 > Hartshorne quantifies over many other universe sized sets, of
 > course, already in chapter 2.

Could you please cite where in chapter II Hartshorne does this?

McLarty:
 > Is this "excessive generality"? In a proof theoretic sense, if you
 > only want the theorems in number theory, it sure is.

About ``excessive generality'', Hartshorne says in his preface:

  The methods of schemes and cohomology are developed in Chapters II
  and III, with emphasis on applications rather than excessive
  generality.

I wonder what kind of ``excessive generality'' Hartshorne is referring
to.  Could he be referring to Grothendieck universes?  If so, this
appears to be the only reference to Grothendieck universes in the
entire book.

McLarty:
 > In a practical sense, no one would pursue a career in algebraic
 > number theory without studying Hartshorne.

It's now clear that McLarty has been misreading Hartshorne, pretending
to see Grothendieck universes where there are none, and piously
misusing the authority of Hartshorne in order to puff up the
importance of Grothendieck universes.  It's reasonable to assume that
McLarty has also been misinterpreting other authors in this field.  I
don't know whether this is unintentional, proceeding from confusion on
McLarty's part, or deliberate, proceeding from a desire to deceive.

-- Steve