FOM: Categorical Foundations

[Martin Schlottmann]

I am Martin Schlottmann, sessional lecturer at the
Dept. of Math. Sciences of the Univerity of Alberta.
My research area is: discrete geometry, aperiodic long-range
order, quasicrystals, harmonic analysis. Therefore,
I am not professionally engaged in f.o.m., but I have
been very much interested in this field since a long time.


(The following is _not_ an attempt to define f.o.m.)
To my understanding, the first important goal of f.o.m. is to
provide a unified framework for mathematical reasoning which is
applicable to as large a part of mathematics as possible.
It is essential not only to determine
a language in which everything of importance can be expressed,
but also to set standards for mathematical constructions
and proofs. Such unified standards are of particular importance
when one wants to find cross-connections between different
areas of mathematical research and apply results of one field
in another without running into all sorts of contradictions.

Although far from representing the ideal solution to this
problem, ZFC seems to be a well-defined system in which
virtually all of mathematics (including category theory)
can be done (allowing suitable conservative extensions
of ZFC for at least practical purposes).

Now, several people have claimed that category theory, in
the form of topos theory, can achieve the same goal. More
precisely, it is claimed that a suitable extension by certain
axioms will suffice to provide a f.o.m., and although I
have not worked myself through all the details it seems
very probable to me that this claim is justified
(see e.g., Johnstone, P.T., "Topos Theory" and refs therin).

It is, furthermore, claimed that the categorical foundation
is simpler and more closely connected to the intuition
(of at least some people) than set theory. I have difficulties
to see this point and I would be glad if somebody could
clarify this a little bit.

The problem I have with the categorical approach is the
following: in order to carry out the foundational program
one has to give not only the basic axioms of a category
(or a topos) but also formulate principles for constructing
categories. Textbooks on category theory tend to be
systematically vague about this matter. For example, it
seems to be the common opinion that there is such a thing as
the (2-)categories of all categories. Now, for everybody
who has heard of the great foundational crisis at the beginning
of the present century this should trigger a red warning
light indicating possible contradictions. If I can do what
I want then I can freely construct the category of all
categories which are not objects of themselves (or something
along this line).

Of course, category theorists are aware of this danger and
try to escape by various precautions, e.g., the discrimination
between small and large categories, or the introduction of
a hierarchy of universes. In any case, there has to be _some_
restriction to the construction of categories, probably in
much the same way as it is necessary to have restrictions
for the constructions of sets. My guess is that, properly
formulated, category theory will be _at least_ as complicated
as ZFC. A theory of categories which is on one hand strong
enough to include all interesting mathematics and, on the
other hand, allow categories of categories or categories
of topoi or whatever will probably have a cumulative hierarchy
similar to ZFC.

I would like to challenge the proponents of category theory
to give (a pointer to) _the_ standard system of the theory
of categories which suffices for all interesting mathematics
(including the standard constructions in category theory like
categories of categories, functor categories, hom-functors
and so on) and to explain how this system is simpler or more
intuitive than ZFC.




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

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