FOM: Thinking with universes

[Colin Mclarty]

    	The FOM discussion of Grothendieck universes may go
better with a little more indication of how universes come into
cohomological number theory. It is not the way that a set
theorist's intuition might easily suggest. As always, recall 
that we all agree universes are formally avoidable, but let me
sketch how they are actually used.
    
    	For example, Wiles cites Altman and Kleiman's proof
of the "Grothendieck duality theorem for fields". That proof
uses universes. Friedman and Shipman have both expressed the
suspicion that universes may be attractive for very large
fields in general but not for specific finite (or countable,
or otherwise very small) fields. But actually the size and 
'concreteness' of the field has nothing to do with it. It is 
the basic nature of derived functor cohomology so let me say 
what that is. The simplest account on this level uses 
Grothendieck's own perspective (i.e. his perspective from 
at least 1970 on), so I will do that.
    
    	This is only a sketch. Were it an actual 
definition you would be reading Gelfand and Manin METHODS 
OF HOMOLOGICAL ALGEBRA or other standard references. For 
a fuller yet concise account see Tamme INTRODUCTION TO 
ETALE COHOMOLOGY. 
    
    	Eilenberg, Cartan, Serre, Grothendieck and others
developed a uniform approach to cohomology for many kinds
of structures. 
    
    	For any topological space E, a "sheaf on E"
is a local homeomorphism h:T-->E. That is continuous 
function h such that each point x of T has some open 
neighborhood which h maps one-one onto an open set in E.
Given sheaves h:T-->E and h':T'-->E on E, a sheaf map
f:h--h' is just a continuous function f:T-->T' such that
the composite h'f:T-->T'-->E equals h. The category of 
sheaves on E is a Grothendieck topos Sh(E) so we can 
interpret the usual constructions of mathematics in a 
natural way in Sh(E). In particular an "Abelian group
sheaf" on E is just an Abelian group in the topos Sh(E).
Let AbSh(E) name the category of Abelian group sheaves on 
E. 
    
    	The Cech cohomology of E has been around since
the 1930s but it has a very neat definition in these terms:
It is "the derived functor of the global section
functor from AbSh(E) to Abelian groups" which Grothendieck
defines by its universal property among "delta functors". 
For a full account see Tennison SHEAF THEORY. A "delta
functor" is actually a certain kind of family of functors
H^n from AbSh(E) to Abelian groups, one for each natural
number n, and definition of the derived functor quantifies
over delta functors.
    
    	This definition may look brutal if you are not
familiar with it. Perhaps like Friedman's recent
parody definition of complex algebraic numbers. But in
fact it is already useful. More importantly it generalizes
to other cohomology theories that were only invented by
its help. For many of these theories this framework is
used for proving the general theorems, even though it 
is formally dispensible, and even though one or another 
particular calculation of cohomology can be made with far 
less apparatus.
    
    	A key FOM moral is at hand: No matter how small 
and concrete the space E may be, the category AbSh(E) is
very large. It is a proper class in the most naive
terms, and so is any functor from it. The definition 
of cohomology quantifies over these functors. By using 
Grothendieck universes we can make everything here a 
set--specifically, if E lies in any universe U
then we can make AbSh(E) a set in any universe that
contains U.

	All that follows is some further examples that
I may want to refer to on FOM. Group cohomology predates
this derived functor approach, and Galois cohomology
predates the full Grothendieck conception. But both
commonly draw on this approach for perfectly practical
reasons. The approach was invented as a step in creating
etale cohomology and many others used in number theory.  
    
    	Consider any group G (an ordinary group, a set
with multiplication et c.). Define a "sheaf on G" to
be any set S with an action of G on it. That is for
each g in G and each x in S we define g(x) in such a
way that given also g' in G we have g'(g(s)) equal to
(g'g)(s), and for e the unit element of G we have e(x)=x.
A sheaf map from S to a sheaf S' is just an equivariant
function f:S-->S', that is a function such that for each
g in G and x in S we have f(g(x))=g(f(x)). These sheaves
and maps form a Grothendieck topos Sh(G), and AbSh(G)
is the category of Abelian groups in that topos.

	The group cohomology of G is the derived
functor of the global section functor from
AbSh(G) to Abelian groups. See Brown COHOMOLOGY OF
GROUPS. Again, note that no matter how small G is,
even if it is the trivial one element group {e},
the category AbSh(G) is very large.

	Let K/k be any field extension. That is, K
is a field and k a subfield of it. The Galois group
Gal(K/k) is a 'profinite group' which means a group
with a certain topology on it. Define a "sheaf on
K/k" to be a sheaf on the group Gal(K/k) as defined
above, except also satisfying a certain continuity
condition. The same approach as before gives the
Galois cohomology of K/k. See Cassels and Frohlich 
ALGEBRAIC NUMBER THEORY--but compare Washington's 
article in MODULAR FORMS AND FERMAT'S LAST THEOREM 
for a "utilitarian approach" to calculating many 
cohomologies with much less apparatus.
    
	A scheme is a kind of algebraic space, the
space defined by some 'polynomial equations' in a
very general sense which naturally generalizes the
classical sense, see the introduction to Grothendieck
and Dieudonne ELEMENTS DE GEOMETRIE ALGEBRIQUE. For
any scheme X we can define sheaves on X, specifically
the "etale sheaves". These form a topos Sh(X). The 
Abelian groups in that topos form a catgegory 
AbSh(X), and the derived functor of the global section 
functor from AbSh(X) to Abelian groups, is the etale 
cohomology of X.