David Corfield William Lawvere

“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.“

https://www.mta.ca/~cat-dist/archive/2000/00-3