FOM: 3 TOP problems, reply to Friedman
Colin Mclarty
cxm7 at po.cwru.edu
Tue Mar 10 10:12:17 EST 1998
Reply to message from friedman at math.ohio-state.edu of Thu, 05 Mar
>One new way I thought of that might help clarify the situation is the
>following question. What are the major open problems in "autonomous,
>coherent, and comprehensive" f.o.m. from your point of view?
Below I describe three open problems.
>And what has
>been acheived in "autonomous, coherent, and comprehensive" f.o.m. from your
>point of view that is comparable to the work of Aristotle, Frege, Cantor,
>Cauchy, Zermelo, Hilbert, Russell, Godel, and Cohen? The answer "copying
>them" is not satisfactory.
Here is a point of kinship between categorical foundations
and ZF (or ZF descendents such as Feferman's theory): Neither of us
need claim credit for Aristotle, Frege, Cantor, Cauchy, Hilbert, or
Russell. We are both heirs to all of them--though I'd have included
Dedekind in that list and might not have thought of Russell. I will
mention that Aristotle, Cauchy, and Hilbert are all notoriously
"list 2" types. Goedel's incompleteness theorems too have no closer
tie with set theory than with any other first order foundations.
Beyond that the game of "my daddy can beat your daddy" is better
played in person over beer, especially as nearly everyone who ever
worked on categorical foundations is still around and active.
THREE OPEN PROBLEMS IN CATEGORICAL FOUNDATIONS:
The most prestigious is on-going axiomatizations in
homology. This includes describing "motives", a kind of universal
(co-)homology which involves hard technical problems. See the
AMS volume titled MOTIVES (I don't have the reference available,
I believe it is a two volume set) or look for "motives" or
"Voevodsky" on the web. It also includes axiomatizing "triangulated
categories" and their relation to Abelian categories--that is the
relation between linear equations and linear transforms as ways
of specifying linear spaces. Gelfand and Manin's book METHODS OF
HOMOLOGICAL ALGEBRA gives one answer to this, probably not final.
These axioms are *likely* to be foundational in the sense of
autonomous well motivated first order axioms which deliver the
theorems of vast reaches of homology; they are *certain* to
be useful in practice even if you prefer official foundations
in ZF.
More traditional foundations: Is there a reasonable
"category of all categories" which includes itself among its
objects? Every set theory with a universal set gives categories
which include themseves as objects, by the usual definition of
categories and functors in terms of sets. But these categories
are never cartesian closed and so are not reasonable versions of
a "category of all categories". And this set theoretic approach
construes "object of" in an unnecessarily narrow way.
Categorically, we merely want a category C which
has an object encoding the same functorial relations as C. Can
we take good advantage of the extra freedom? Probably the best
approach would use Benabou's theory of fibrations and
representability ("Fibred categories and the foundations of
naive category theory" JSL 50, 1985, 10-37). Exploring the
problem will certainly throw light on the relation between
large and small categories.
Third (the most clearly manageable): Can the adjoint
lifting theorem for triposes (from Andy Pitts's dissertation)
be gracefully transported to regular categories (presumably
with some supplemental conditions)? If so, the theory of
realizability toposes could be radically simplified,
among other things (see Japp van Oosten's dissertation).
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