RE: question on finiteness in toposes

[categories <cat-dist@mta.ca>]

Date: Tue, 14 Jan 1997 10:52:38 -0500 (EST)
From: F William Lawvere <wlawvere@ACSU.Buffalo.EDU>
 

Sorry, I used K/S for an abbreviation of what was called
Kuratowski until someone pointed out that it was due to
Sierpinski :an object whose mark belongs to the smallest
sub-semilattice of its power set which contains the
singleton map, or in case there is an NNO an object
which in a suitable sense is locally enumerable by
the segment under a section of the NNO .

While the K/S definition is right for the construction of
the object classifier over an arbitrary base topos (as Gavin
showed) and hence for classifiers for various kinds of
finitary algebras over an arbitrary base topos, still
the theory of it in the last 25 years of topos theory seems
to mainly be justified by formal analogy and/or independence
relative to abstract set theory (=topos with choice).
However there are important uses of "finiteness" in
algebraic geometry and differential topology (where topos theory
after all started)   :
Consider a ringed topos E,R . For example, the sheaves on an
algebraic variety  or on a Cinfty manifold. Within the abelian
category of R-modules in E, we need to single out two important
subcategories
FAC (Serre 1955)=coherent sheaves..these tend to be an abelian
subcategory and tend to vary covariantly as one E,R is mapped to
another E',R' (thus give rise to an extensive K-homology)
and vector bundles , which one thinks of as a finite-dimensional
vector space varying smoothly over the base space of E ,so
they cry out for internalization ; in algebraic geometry these
are identified with locally FINITELY free R-modules... they
vary contravariantly with E,R  (so give rise to K-cohomolgy
rigs which act on the FACs,ie intensives acting as densities on
the extensives; with further conditions on E,R one can at the
level of the riNgs generated by these rigs define a sort of
Radon/Nikodym derivative via an alterating sum of Tors , but
in general the covariant abelian category FAC and the
contravariant tensored category Vect are distinct...The
"derived category" of E,R (now allegedly replacing homological
algebra in complex analysis and C*-algebra theory)  should
be the derived category of one of these two linear categories
(here I mean dc in the linear sense..nonlinear "derived categories"
are more like the stable homotopy of E))

Already the intuitionists speculated about (in effect) subobjects
of K/S objects, and  it seems we need something of the sort
perhaps a category of finites closed under subquotient in
order to define the notion of eg finitely-generated R-module
in a way which not merely mimics abstract set theory but actually
captures the vector bundles .

Perhaps it will be easier if E itself satisfies a noetherian
condition.

It would be best if the desired content could be entirely int-
ernalized to E,R but perhaps it is really relative to a base
S,K..but perhaps without restriction on S ??

I hope this clarifies the problem.
Sincerely
Bill