Subject: Garbled posting
Date: Wed, 13 Aug 2003 12:46:28 -0500 (CDT)
From: Peter May
To: dmd1@lehigh.edu
At some point along the line, the cutting and pasting failed,
so that my posting is garbled by omissions. Would you mind
reposting it? Peter
Model structures
SInce there has been some confusion in this thread
and I happen to have just written a relevant paper,
let me answer Gaucher's question and advertise some
work of Mike Cole and of Johann Sigurdsson and myself.
There is a relevant general theorem, unpublished, due
to Mike Cole:
Theorem (Cole)
If C is a topologically bicomplete category such that
mapping path fibrations commute with colimits of
inclusions, then the homotopy equivalences, Hurewicz
fibrations (homotopy lifting property) and implied
cofibrations (left lifting property with respect to
Hurewicz fibrations) specify a model structure on C.
Here C is a topological category with limits and colimits
and with tensors and cotensors with spaces (that is the
meaning of ``topologically bicomplete'').
The mapping path fibration hypothesis holds if you
work with compactly generated spaces (weak Hausdorff
k-spaces). It seems not to hold if you work with general
spaces or with k-spaces. Strom proves the conclusion
of the theorem for general spaces, and his argument
works for k-spaces, but the details of his proof must
of course differ from Cole's. One point in Strom's work
is a proof that, in spaces, closed Hurewicz cofibrations
(homotopy extension property) are exactly the implied
cofibrations of the theorem. In compactly generated
spaces, you can omit the word ``closed'' and have the
same conclusion. This is where what James calls Strom
structures enter into the picture (there is no mention
of model categories in James's work).
However, there is another quite subtle point in Strom's
proof of the model axioms. The factorization of a map as
the composite of a cofibration and an acyclic fibration makes
use of the fiberwise join (or generalized Whitney sum), and a
key point is an old observation (of somebody named Hall) that
the fiberwise join of (Hurewicz) fibrations is a fibration.
There is another, and amazing, relevant general theorem,
also unpublished and also due to Mike Cole.
Theorem (Cole)
Let (W_h,Fib_h,Cof_h)$ and (W_q,Fib_q,Cof_q) be model
structures on the same category. Suppose that W_h is
contained in W_q and Fib_h is contained in Fib_q.
Then there is a third ``mixed'' model structure
(W_q, Fib_h, Cof_m). The ``mixed'' cofibrations Cof_m
are the maps in Cof_h that factor as the composite of
a map in W_h and a map in Cof_q. An object is m-cofibrant
if and only it is h-cofibrant and of the h-homotopy type of
a q-cofibrant object.
Think of compactly generated spaces, say. The h-model
structure is the one of the first theorem: homotopy
equivalences and Hurewicz fibrations. The q-model
structure (q for Quillen of course) uses weak homotopy
equivalences and Serre fibrations; the q-cofibrations
are the retracts of the cell complexes. The mixed model
structure uses weak homotopy equivalences and Hurewicz
fibrations. Its cofibrant objects are the spaces of the
homotopy types of CW-complexes.
Anybody interested in these matters is referred to my
forthcoming paper: ``Model categories and ex-spaces''.
It will be posted here in a week or two. It gives
a careful overview of the relevant model structures
and their parametrized generalizations, which turn out
to behave far more subtly than we topologists are
used to. One requires base change adjunctions in
homotopy categories. Pullback along a map of base spaces
has both a left and a right adjoint, and model category
theory cannot give you that both of these adjunctions
descend to homotopy categories, no matter what model
structures you use. Precisely, if both adjunctions are
Quillen adjunctions and either takes homotopy equivalences
of base spaces to Quillen equivalences, then the homotopy
categories are trivial: one object and one morphism.
The spectrum level analogue of this theory is in a joint
paper with Johann Sigurdsson, ``Parametrized equivariant
stable homotopy theory'', which should also be posted in
a few weeks. As the title of the second paper indicates,
both papers are written equivariantly.
Peter May