Re: equivalent varieties

[F W Lawvere <wlawvere@buffalo.edu>]

An invariant way to see the particular equivalence discussed is to
note that the topos of presheaves on the category of finite sets
is the classifying topos for p-algebras for any given p > 1.
That is because any non-empty set is a retract of a finite power of
p and because left-exactness is equivalent to preserving finite
products in this particular case.

This representation suggests a different interpretation from the
usual "truth of properties" point of view concerning the essential
content of Boolean algebra. Namely, it concerns finite partitions
of a hypothetical whole and shuffling of these induced by arbitrary
maps between the index sets for the partitions, nothing more.

Coordinatizing the above shuffling of partitions using p = 3
has some advantages over p = 2, namely, the unary operations of the
theory suffice to characterize ultrafilters, i.e. to insure that
perceived points of a finite set are actually there; more formally,
the contravariant functor represented by 3 from finite sets to M-sets is
full where M is the 27-element monoid of these unary operations.



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F. William Lawvere
Mathematics Department, State University of New York
244 Mathematics Building, Buffalo, N.Y. 14260-2900 USA
Tel. 716-645-6284
HOMEPAGE:  http://www.acsu.buffalo.edu/~wlawvere
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