FOM: rewards for metatheorems in category theory

[Stephen G Simpson]

Joe Shipman 05 May 1999 09:55:03 

 > is there a single application of, say, the Yoneda embedding theorem
 > which can't straightforwardly be redone in ZFC?

I strongly doubt it.

 > If not, why isn't there a metatheorem to this effect? 

That is a very good question.  

One aspect of this kind of question to keep in mind is: What would be
the *reward* for an f.o.m. researcher who took the trouble to
formulate and prove and publish such a metatheorem?

You have to consider the possibility that the ``reward'' might amount
to a kick in the teeth.  Mathematicians like Conway, Johnstone,
McLarty, Whiteley, Tragesser, et al could be expected to unjustly
dismiss the metatheorem as irrelevant, pedantic, not sufficiently
attuned to the ``mysterious dimension'', etc etc, yatta yatta yatta.
This expectation is based on their past behavior.

Don't forget that, according to Johnstone, Conway is the founder of
the Mathematician's Liberation Movement.  The tyrannical force that
Conway wants to liberate mathematicians from is none other than f.o.m.

A lot of category theorists are arrogantly dismissive of any and all
f.o.m. insights, even when those insights are directly relevant to
category theory.  Johnstone's preface is a good example of this
phenomenon.

I'd like to get to the bottom of why so many category theorists have
this anti-f.o.m. attitude.  Is it somehow related to the phony idea of
``categorical foundations''?

 > About the only thing McLarty and Simpson agreed on was that the
 > Tarski-Grothendieck Universes assumption could be eliminated from
 > applications of derived functor cohomology to number theory; 

Yes.

 > but nobody was willing to state a metatheorem,

I stated a metatheorem.  The metatheorem was: VNBG with global choice
is conservative over ZFC.  I also pointed out that Harthshorne's
textbook of algebraic geometry, including his exposition of derived
functor cohomology, appears to be directly formalizable in VNGB with
global choice.

Note McLarty's reaction to this metatheorem.  His reaction was to blow
more smoke, denounce me as ignorant of algebraic geometry, etc etc.

 > (**) Any statement about integers proved using the Yoneda lemma is a
 > theorem of ZFC
 > -- will category theorists reading this at least hazard an opinion on
 > whether (**) is true, or conjecture a precise form of (*)?

Probably not, because many category theorists seem to have an
anti-f.o.m. attitude which prevents their being interested in such
questions.

-- Steve