FOM: Re: Categorical foundations

[Martin Schlottmann]

Related to the post of Friedman, 1/23/98 19:31
I would like to remark that I do not consider myself
on one particular side maintaining pressure on the
other side.
Rather, I want to seize the opportunity of the present
discussion to get some things straight, at least for
myself. Of course, at the moment I may be biased to
prefer set theory language to category language but
I am rather keen not to miss a single piece of interesting
mathematics only out of bad habits.

Therefore, I am embarking on a rather detailled
discussion here. If this is not suitable for FOM,
let me know.


In reply to McLarty, 1/23/98 15:25

Colin McLarty wrote:
> [...]
>         Cateporical foundations as a whole is about exploring foundations
> for a great many different, and mutually inconsistent, theories: a category
> of linear spaces and transformations cannot also be a category of
> didfferentiable spaces, and neither can be a category of recursive sets.
> [...]


This is probably not meant to imply that the theories of linear
spaces, differentiable spaces, and recursive sets are mutually
inconsistent. They are certainly compatible inside ZFC, and
I see this as an important feature of set theory that it
allows to mix structures from formerly widely seperated
fields. The theory of von Neumann algebras may serve as
an example of a beautiful interplay between algebra, topology
and order. I think any foundational framework should enable
an easy synthesis of whatever topics which seems fruitful.


> [...]
>         The translations are familiar. A warning here: Do not attempt to
> understand the translation between categorical set theory and set theories
> related to ZF without first understanding each of categorical set theory and
> ZF. The translations can be found in Johnstone TOPOS THEORY or Mac Lane and
> Moerdijk SHEAVES IN GEOMETRY AND LOGIC.
> [...]


Both books give a restricted theory of sets inside
topos theory language. One may discuss the necessity of
replacement, but seperation only for bounded formulae
seems rather weak.

There is no rigorous backwards translation of category theory
into set theory in either book. The problem is that both
books do not specify the general theory of categories precisely
enough. This seems to be not a rarely occuring feature in
this field.

MacLane's book:
"In general, we shall not be very explicit about set-theoretical
foudations, and we will tacitly assume we are working in some
fixed universe U of sets." (p.12)
"Ignoring set-theoretical difficulties, there is thus a
 'category of categories'." (p.13)

Johnstone's book:
"The phrase 'Grothendieck universe' does not appear anywhere
in the [Johnstone's] book. This is intentional; I have
deliberately been as vague as possible (except in $9.3)
about the features of the set theory I am using, since
it really doesn't matter." (p.xix; remark: in $9.3, he
does not specify the set theory which is used for the
presentation of category theory but the weak set theory
he is going to reconstruct in a topos.)

The only rigorous method of presentation of category
theory in the literature I am aware of which is sufficient
for pertinent constructions like category of sets, functor
categories, hom functors etc, uses Grothendieck universes
and, therefore, inaccessible cardinals. This I see as a
major drawback of the approach, to have these sneaking
in right from the beginning.

It would be better to have an approach which gives, at
the beginning, a couple of axioms like "There is a category
of sets such that...", "For any two categories such that...,
there is a categories of the functors between them", "For
every category such that..., there is a (hom) bifunctor
into a category of sets such that..." and so on. Then,
one could consider the question into which extension of
ZFC this can be translated.

> [...]
> Neither will name calling change the outcome by much.
> [...]

Certainly. I hope that this does not relate to one of
my previous posts. I am interested in a serious diskussion
and not in demonstratively being right. Unfortunately,
both moderators are themselves engaged in this discussion
and, therefore, not permanently able to calm down the waves
a little bit.

> [...]
>         Your proposal is fine. And it has been done.
> [...]

Maybe, but certainly not in the books cited above. I
would be thankful for every reference which clarifies
this issue.

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

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