Dualization monads on sets

Dualization monads on sets

[wlawvere]

As discussed his week  by Oswald Wyler, Vaughan Pratt, and Ernie
Manes, the opposite of the category of small sets is equivalent to
the algebras over the monad T obtained by dualizing into Z   for
Z>1 ; there is really no difference which Z is used. But that
changes if we consider a truncation of T, i.e., the part supported by
arity I for some I (the truncated monad has value at X equal to the
union of the images of  T(I)--> T(X) over all I-->X). There is still the
possibility that some subcategory  of sets will be full in the
opposite of the category of algebras. For example, as Vaughan
points out, if Z=3, I=1, at least the finite sets can be so captured.

An important way to capture all small sets is to take Z=2 and I=
countable (Ulam) or Z=countable and I=1  (Isbell in 1960 showed
that these are equivalent).  Taking unary operations (I=1) is always
possible by enlarging Z. (For some reason Ulam spoke of
measures, which is really quite misleading because measures
are additive but here the condition that they be multiplicative is at
least as important; likewise the set -theorists' terminology
"measurable" (for sets too big to be captured) is misleading
because "to be capturable" is intuitively to be measurable by
procedures valued in Z.)

I advocate that an additional axiom on small sets is that all such
monads (Z infinite, I=1) should be the identity, because all known
constructions of geometry and analysis preserve capturability and
moreover all mathematical situations where one can reasonably
expect a space/quantity duality are spoiled without this axiom on
the underlying discrete sets. It is well known that the negation of
the axiom, as applied to all sets, implies its consistency.
Recognizing a category of small sets, so defined, as an ordinary
object in the cartesian-closed category of categories of course in
no way prevents the consideration of possible much larger sets or
categories in the same system, no more than does recognizing
the category of finite sets.