FOM: Re: What categories are good for

[Colin McLarty]

Joe Shipman wrote

>Colin argues persuasively that category theory can form a foundation for large
>areas of mathematics.  (For example, many of the "big" theorems of algebraic
>topology can be *rigorously* developed without ever talking about sets (or even
>thinking about them if one's training has been in the categorical tradition)).

which of course I agree with. He goes on:

>The big problem in "categorical
>foundations" is that "large areas of mathematics" is not ALL of mathematics.

       I agree with the claim about math too, but not that it is a problem
for categorical foundations. As a nuance I will insist on what Tait argued:
since ZF there can be no such thing as ALL of mathematics, since we know
that at least the ordinal chain can always be extended. Categorical
foundations give even more multifarious reasons why there can be no ALL. But
we must ask about very inclusive reaches of math.

>My impression is that the results achievable from categorical 
>foundations can be smoothly developed set-theoretically (i.e. as if 
>categories had never been invented)

        You can do it without categorical *foundations*. Much math today
would be humanly impossible without categorical *methods*.

>but that the converse is not true (that is, to *rigorously* develop,
>say, the theorems of DeBranges or Wiles or Freedman can be done from
>categorical foundations only by "inventing sets").

        Yes, when you want to invoke a lot of methods--say, geometric,
algebraic, cohomological, and arithmetic at once, you need a pretty broad
framework. One very long range project of many general category theorists
(Lawvere, Tierney, Joyal, Moerdijk, and others) is to see how far radical
use of Grothendieck's ideas can simplify all that work. But for the moment,
certainly the most natural approach is to assume some set theory.

        That set theory can easily be categorical. Categorical axioms versus
the ZF style of axioms (whether formalized or construed informally) make
absolutely no difference in the cardinals you can postulate or the
constructions you can perform. 

        In writing this I went back to see what Grothendieck used in his
appendix on set theoretic questions in topos theory (Se'minaire de
Ge'ome'trie Alge'brique IV, vol. 1). Of course the answer is that you can't
tell. You can rule out Lawvere's categorical set theory on historic grounds.
You would suspect Bourbaki's set theory since Bourbaki is given as author of
the appendix. But the semi-formal appendix can be perfectly easily cleaned
up in (extensions of) ZF or Lawvere's axioms, or Bourbaki's. The same is
true of the set theory chapters in most textbooks from calculus on up.
Kelley's TOPOLOGY is one of the rare texts outside set theory to give an
explicit foundation to its set theory.

        As I see it you can found "all" of math in membership-based set
theory (such as ZF or Feferman's theory of operations and collections). Or
you can give entirely categorical foundations. I prefer categorical
foundations for two reasons: They use more methods I need in other math
anyway, and categorical foundations are also available for more "large
areas" of math in their own terms.

Colin McLarty