FOM: 3 TOP problems, reply to Friedman

[Colin Mclarty]

        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).