Subject: Re: two more on cofibrant
Date: Thu, 24 Apr 2003 21:35:14 -0500
From: Bill Richter
To: dmd1@lehigh.edu
Mark Johnson's post made me realize one of my favorite model
categories has a noncofibrant terminal object.
Mark took a model category C, and an object B, & made the new model
category C / B of maps p_X: X ---> B, morphisms being maps f: X ---> Y
s.t. p_Y . f = p_X. Dwyer & Spalinski, generalizing an example of
Quillen's, showed C / B is a model category, with all 3 classes of
morphisms created by the forgetful functor C / B ---> C. Mark's
counterexample: suppose B isn't cofibrant in C. Nice work!
Now by dualizing Dwyer & Spalinski, we have a model category A \ C / B
for any fixed morphism f: A ---> B, and we have Tom Goodwillie's
original counterexample, whenever f isn't a cofibration:
If C is a model category and f:A->B is a morphism of C then
consider the category of "objects between A and B": Objects are
factorizations A->X->B of f, and maps (A->X->B)->(A->Y->B) are
commutative diagrams
A -> X -> B
1 | | | 1
V V V
A -> Y -> B
Until Mark posted, I didn't realize I "understood" Tom's category.
Now let C = Top, with the Strom structure (see Dwyer & Spalinski), and
pick a space B, and then my favorite model category is
B coprod B \ Top / B
due to John Klein, who calls it the category of charged spaces over
B. Clearly the terminal object, the `unreduced' fold map
B coprod B ---> B
is not a cofibration, hence a non-cofibrant terminal object. Thanks to
Tom and Mark for explaining this (new!) fact to me.
\begin{shaggy-dog-story}
John's model category of charged spaces over B is important in
Poincare embedding. John used it to prove a fiberwise Freudenthal
theorem, using the adjoint functors
Sigma_B: Top / B <===> B coprod B \ Top / B : Omega_B
which are obviously relevant to Poincare embeddings, as Shmuel
Weinberger observed long ago: given an embedding M >---> X, with
complement W, then the `unreduced' fiberwise suspension
Sigma_B (X) = B cup W x I cup B
is the complement of the embedding M >---> X x I, and as John
observed, a charged space. There's a respectable fiberwise homotopy
theory literature (not from the Dwyer-S model cat pov), but it's all
about pointed and unpointed fiberwise spaces, i.e.
Top / B and B \ Top / B
But as John observed, his new category of charged spaces is the one
that's relevant to compressing an embedding M >---> X x I into X.
Then I took John's category and fiberwised my Duke paper to give the
first clean homotopy theory proof of Poincare surgery, which is
basically to prove the Whitney embedding theorem in the Poincare
category. That's a test case to decide whether the surgery machine
(which reduces manifolds to homotopy theory + K theory). Basically my
contribution was to add EHP sequences to John's Freudenthal theorem.
Later John shocked me by giving a non-fiberwise proof of my result,
which I called the fiberwise unstable normal invariant theorem, and it
gives reason to suspect that there's a clean non-fiberwise homotopy
proof of Poincare surgery.
I think, however, that my original proof is worth writing up, since it
involves writing a book about fiberwise homotopy theory, which I
imagine would be useful even if Poincare surgery doesn't require it.
But no interest has been shown in this writing project, and so I put
it off in favor of supposedly more lucrative ventures :) I think what
happened is that the surgeons would be interested in a nice homotopy
theory proof of Poincare surgery, but they maybe won't be competent to
referee it, so they need a community of homotopists who's interested
enough in Poincare complexes to produce/referee such a proof. That's
fine. But there's a bootstrapping problem, in that the surgeons don't
seem to want to admit there's a Poincare surgery issue prior to the
triumphant proof. The homotopists might get interested in problems
their surgery colleagues bring to their attention. It worked for me.
I made my peace with this bootstrapping problem years ago, but now I'm
getting hammered the other way! Mark Mahowald's Memoir has good
applications to the surgery machine: Mark can classify 3 cell Poincare
complexes, and calculate their surgery obstructions. I'm trying to
write it up as the sorceror's apprentice and sell it to the surgeons.
Unfortunately, my sorceror's apprenticeship involves fixing a number
of bugs/documentation-errors in Mark's work. But in order to sell it
to the surgeons, I've gotta publish it through the homotopists. I'm
getting the impression that I'm only allowed to fix Mahowald bugs if I
can score, ahead of time, an audience of surgeons who need to read it.
It's the dual bootstrapping problem: the surgeons might get interested
in problems their homotopy colleagues bring to their attention.
\end{shaggy-dog-story}