FOM: Explaining Simpson and McLarty to each other

[Stephen G Simpson]

JSHIPMAN at bloomberg.net writes:
 > Colin, I think Steve really wants to know the following: If Set
 > Theory did not exist, would it be necessary to invent it?  That is,
 > can you come up with an alternative Analysis 101 curriculum, in
 > which you prove useful theorems like Fubini's theorem, Stokes's
 > theorem (bridge to topology), some useful formulation of Fourier
 > Analysis (bridge to engineering), basic properties of the Riemann
 > zeta function (bridge to Number Theory), Chebyshev's inequalities
 > (bridge to probability and statistics), and so on, starting from
 > categories or toposes rather than set theory?

Yes, that's what I wanted to know.  That is my first question about
topos-theoretic foundations.  Naturally I want the answer to pay
attention to what axioms are appropriate for this, in addition to the
elementary topos axioms.

 > If all you need are natural number objects and right inverses those
 > are not hard to motivate,

That was my second question.  I think that the right inverses may not
be so easy to motivate in terms of what I understand to be the
original motivation of the topos axioms.  McLarty says that right
inverses don't exist in categories of sheaves, except in trivial
cases.  On the other hand, Awodey says they do exist in categories of
sheaves over complete Boolean algebras. (Is that the same as
Scott-Solovay Boolean-valued models of set theory?)  I say these guys
need to get together and compare notes.

In the meantime, McLarty has backed away from the right inverses.
Maybe he thinks they are not all that important for real analysis, or
maybe he sees a difficulty in motivating them independently of ZFC.
In any case, the theory that's on the table now is topos theory plus
the natural number object -- no right inverses.

-- Steve