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categories: Topos theory and large cardinals

Andrej Bauer asked whether large cardinals other than inaccessible ones
have a natural definition in topos theory.  Indeed, like most questions
of set theory which have an objective content, this too is independent
of the a priori global inclusion and membership chains which are
characteristic of the Peano conception that ZF formalizes. Various kinds
of "measurable" cardinals arise as possible obstructions to simple
dualities of the type considered in algebraic geometry.  Actually, 
measurable cardinals are those which canNOT be measured by smaller ones,
because of the existence on them of a type of homomorphism which is
equivalent to the existence of a measure in the sense of Ulam.
Specifically, let V be a fixed object and let M denote the monoid object
of endomorphisms of V.  Then the contravariant functor ( )^V is actually
valued in the category of left M-actions and as such has an adjoint which 
is the enriched hom of any left M-set into V.  The issue is whether the
composite of these, the double dualization, is isomorphic to the identity
on the topos;  if so, one may say that all objects are measured by V, or
that there are no objects supporting non-trivial Ulam elements.  In any
case, the double dualization monad obtained by composing seems to add new
ideal Ulam elements to each object, i.e. elements which cannot be nailed
down by V-valued measurements.  Since fixed points for the monad are
special algebras, and since algebras are always closed under products
etc., it should be possible to devise a very natural proof based on monad
theory that the category of these non-Ulam objects is itself a topos and
even "inaccessible" relative to the ambient topos.
     Why is the above definition relevant?  The first example should be
the topos of finite sets with V a three-element set.  There the monad is
indeed the identity, as can be seen by adapting results of Stone and Post.
Extending the same monad to infinite sets, we obtain the Stone-Czech
compactification beta.
     The key example is a topos of sets in which we have V a fixed
infinite set.  As Isbell showed in 1960, the category contains no Ulam
cardinals in the usual sense if and only if the monad described above is
the identity.
     Further  examples involve the complex numbers as V, where actually M
can be taken to consist only of polynomials, with the same result;  this
example extends nicely from discrete sets to continuous sets, usually
discussed in the context of "real compactness".  Another kind of example
concerns bornological spaces.  The result always seems to be that the lack
of Ulam cardinals is equivalent to the exception-free validity of basic
space/quantity dualities.  
     Ulam (and other set theorists since) usually in effect phrase the
construction in terms of a two-element set V equipped however with
infinitary operations.  Isbell's remark shows that equivalently an
infinite set equipped with finitary (indeed only unary) operations can
discern the same distinctions between actual elements as values of the
Dirac-type adjunction map and ghostly Ulam elements on the other hand.

F. William Lawvere			
Mathematics Dept. SUNY Buffalo, Buffalo, NY 14214, USA
716-829-2144  ext. 117		   
HOMEPAGE:  http://www.acsu.buffalo.edu/~wlawvere