"Universes" within ZFC

"Universes" within ZFC

[Colin McLarty]

        The following improves on remarks I made to some people at the
recent Buffalo meeting, and on the FOM e-mail list.

   	This is a step towards showing Grothendieck's method of universes for
algebraic geometry can be formalized nearly unchanged in ZFC, without the
further axiom asserting existence of Grothendieck universes. To any set of
sites (or ringed sites) we associate a "generating cardinal" k and consider
as universes the sets V(i) where i is a limit ordinal larger than k and k
not cofinal in i. ZFC proves every set is a member of such a "universe".
These universes are at least models of Zermelo set theory plus countable
replacement. So, for example, for any set S and finitary operation on S in
the universe, the closure of S under that operation is also in the universe.
This obviously suffices for most of Grothendieck's theorems in TOHOKU and
SGA. The only theorem that stands out as needing more is existence of
injectives in AB5 categories with generators. I will show these universes
suffice for that theorem, for the AB5 categories associated to those
(ringed) sites. The problem is simply to find bounds on the transfinite
inductions we use. But these are not as simple as I had thought.    

        For any site C we can construct a generator U for the category of
Abelian sheaves on C, which works no matter which universe of sets we use to
define the topos on C (assuming, of course, that C is in the universe). Any
of the standard constructions have this property. And U has a fixed poset of
subsheaves (up to equivalence as monics to U) also independent of the
universe of sets. The "generating cardinal" for C is the first infinite
cardinal greater than the number of subobjects of U. Actually we can make it
smaller by a slightly more complicated definition: We can take the first
infinite cardinal greater than the length of  any chain of subobjects of U
well ordered by inclusion.

        If we use a ringed site and modules over that  ring, then U depends
on both the site and the sheaf of rings, but it still does not depend on the
universe of sets.

	The generating cardinal for a set of (ringed) sites is the supremum of the
generating cardinals for each one.

        Suppose we have a set of sites, and it has generating cardinal k.
Let i be any limit ordinal larger than k and k not cofinal in i. We will use
"small" to mean "a member of V(i)." So "small" here does not mean "of small
enough cardinality". It means "of low enough rank". A small set can be
isomorphic to a non-small one.

	The key fact is that the axiom of replacement holds within V(i) for any
countable family of sets and for any family of sets indexed by k. That is,
any set of small sets which is the range of a function defined on k is
itself a small set. The reason is this: Consider any function f:k-->S where
S is a subset of V(i) but we do not assume f is small. Compose f with the
rank function from S to the ordinals, taking each element of S to its rank.
This composite cannot be unbounded in V(i), by assumption, so its image has
a supremum m inside of V(i), and so f is a subset of V(m) and so is small. 

        We use locally small categories in V(i): any subset of V(i) may be
objects, but for any objects A and B there is a small set of morphisms
hom(A,B) from A to B. An AB5 category is an Abelian category with all small
colimits, such that suprema of small filtered sets of subobjects (of a given
object B) are preserved by intersection (with any subobject of B). It
follows that every small filtered set of subobjects is a stably effective
epimorphic family over its supremum.

	When we speak of a well-powered category we assume each object A has a
selected small set Sub(A) of one representative of each equivalence class of
monics to A. By a "subobject" of A we mean one of these selected
representatives.

The following lemmas and theorems hold for any well-powered AB5 category
with generator U.

Lemma 1: Given any objects C and D, there is a small set isomorphic to the
set of all partial morphisms from C to D, that is all morphisms f:V-->D from
subobjects V>->C.
PROOF: There is a small set of subobjects Sub(C+D). A subset of C+D whose
projection onto C is monic, is the graph of a partial morphism, and vice versa. 

LEMMA 2: An object M is injective iff it looks injective to U. That is, iff
every morphism h:V-->M from a subobject V>->U extends to some g:U-->M.
PROOF: Given any subobject B>->A and morphism f:B-->M consider the obvious
partially ordered set E of pairs <B',f'> where B' is a subobject of A
containing B, and f' extends f. By Lemma 1 this is isomorphic to a small
poset, and so every linearly ordered subset of it is small. So AB5 and
Zorn's lemma show it has maximal elements.  The assumption on M implies that
any maximal element of E must have B'=A.

LEMMA 3: Each object A has a monic A>->M1 such that every morphism f:V-->A
from a subobject V>->U extends to a morphism from U to M1. 
PROOF: Consider the set I of all morphisms f:V-->A from any subobject V of
U, to A. For each such f:V-->A we will paste a copy of U onto A by
identifying V with its image under f. The result of all that pasting is M1.
It is the cokernel of the natural morphism from (the coproduct of all
domains of morphisms in I) to  A+(the I-fold copower of U). By Lemma 1, the
set I is isomorphic to a small set so that the coproducts do exist. 

THEOREM: Every object a has a monic A>->M where M is injective. 
PROOF: We iterate the construction from Lemma 2 transfinitely, for each Mj
give a monic Mi>- >M(j+1), continuing to the first cardinal larger than
Sub(U). Let the colimit be M. By AB5, each Mj has a monic Mj>->M. Now take
any subobject V>->U and any f:V-->M. If we knew that f factored as V-->Mj
for some ordinal j we would be done, since by construction f would extend to
a morphism U-->M(j+1). So consider the subobjects Vj>->V gotten by pulling
back Mj>->M along f. This is an increasing chain of subjects,  and Sub(U)
cannot be cofinal in this chain, so the chain is eventually constant. Say it
is constant from Vs onwards. By AB5 the supremum of this chain is V, and so
V=Vs and f factors through Ms. That proves the theorem. Notice we could have
just iterated the construction to the first cardinal larger than any chain
of subobjects of U well ordered by inclusion.