FOM: Categorical Foundations

[Martin Schlottmann]

The CAT/SET debate tends to peter out but, nevertheless,
I would like to invite discussion on one point which
has not yet been adressed appropriately (if people
are not all too tired of this matter).

I am concerned about possible motivation of rewriting
all the foundation of mathematics in category language.
If it is correct that CAT-foundations and SET-foundations
are intertranslatable, and this has been demonstrated
to a certain extend in various posts by references
and actual writing-out axioms, I cannot understand,
on one hand, why "category theorists" insist on redoing
all the work of foundation which has been done since the
beginning of the century in a new language, and on the
other hand, the vigor with which "set theorists" deny
that this new language is suitable for foundations.
Probably in every foundational system, there will be
a thick layer of technical definitions between the
axioms and everyday mathematics, and those definitions
are unlikely to provide any deep insight ( <x,y>:={{x},{x,y}}
does not reveal anything profound on the nature of the
ordered pair), and if the foundations are done in the
proper way, they should be invisible to the everyday level
anyway.

That there must be more to this question may be seen
from the following citation of S. Mac Lane from
Reports of the Midwest Category Seminar III, p. 192,
Lecure Notes in Mathematics 106, Springer, Berlin 1969
(thanks to Solomon Feferman who drew attention to this
reference in his FOM post of Jan 25).

"The development of category theory has posed problems
for the set theoretic foundations of mathematics. These
problems arise in the use of collections of _all_ sets,
_all_ groups, or of _all_ topological spaces. It is
the intent of category theory that this `all' be taken
seriously. [...]
Indeed, Lawvere [...] has suggested that a foundation
might be based upon an axiomatization not of sets, but of
the category of all categories. This attractive possibility
is not yet fully developed [...]" (original emphasis)

One could add that, consequently, also talking about
the category of _all_ categories should be taken
seriously.

Now, there would be a point in categorical foundation
if the promise implied in the above quote could be
fulfilled. It seems to be not unfair to say that
one of the original motivations of categorical
foundations was that category theory was in need
of foundations in the first place!

As there have been almost three decades since, I would
expect a lot of progress on this issue, so I would like
to invite the "categorists" to comment on this. Is
categorically founded category theory able to speak
about "all" groups etc? Are these _really_ all groups etc
which exist in ZFC? (For that matter, the translation of
set theory into well-pointed topoi with choice does _not_
seem to suffice, as there are severe restrictions on
separation; please correct me if I am wrong here.)

-- 
Martin Schlottmann <martin_schlottmann at math.ualberta.ca>

Department of Mathematical Sciences, CAB 583
University of Alberta, Edmonton AB T6G 2G1, Canada