FOM: Re: Categorical foundations

[Colin Mclarty]

Reply to message from martin_schlottmann at math.ualberta.ca of Sat, 24 Jan
>
>
>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.


	Yes. And category theory was created to relate various
structures. The first major motive being to relate topological
spaces to groups in homology theory. Atiyah's K-theory relates
various categories of geometrical spaces to categories of linear
spaces. There is much more. Then people noticed they were able
to use more and more intrinsic descriptions of the various
structures, and the relations only got clearer as more ways
were found to ignore set theoretic details. Indeed people found
many of these particular structures could be entirely defined
by intrinsic means, in the language of category theory. Pursuing
those intrinsic definitions led to categorical foundations. 


	re translation between ZF and categorical set theory:

>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.

	Here you summarize the main theorems. But the text around 
them, and the proofs, give an algorithm translating any statement 
of ZF into a statement in categorical set theory. (In fact, a 
statement about Mostowski trees in the sense familiar from ZF.) 
So any extension of the weak set theory translates to an 
extension of categorical set theory. Translation the other way
is prefectly trivial: just read "arrow" as "function". Either
direction, the resulting theories are naively 
intertranslatable--at least if you consider Mostowski's idea 
of trees "naive".


>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.

	In my post I said we could compare "categorical set theory"
to ZF. And that is what the books do in the passages discussed.
This is not the same thing as "category theory". Neither is the
same thing as "categorical foundations for math". 

>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.

	The "category of categories" has been well enough
axiomatized to do this fine. See my book, Chapter 12. It
can avoid the devices you mention here. Detailed
questions about its strength are not yet much explored. It
seems to me a safe bet that if I were to explain it to
Harvey Friedman, he would say it "slavishly copies" Goedel
Bernays; and I would complain that he was missing its
attractive features.

>
>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.

	I have worked with this idea. I don't see it in my 
book just now, it's probably not there but see my historical
article "Defining sets as sets of points of spaces" in the 
JOUR. OF. PHILOSOPHICAL LOGIC 17 (1988) pages 75-90. So
far as I have seen, the "such that" clauses on the individual
categories generally demand much more logical strength than
the categorical connections among them. But I believe this is
just because we have not yet learned much about using the 
logic of categorical connections.

	
	I said:

>> Neither will name calling change the outcome by much.
>
>Certainly. I hope that this does not relate to one of
>my previous posts.

	No, I did not at all mean you. I should not have 
complained about other people while writing to you.

Colin