Subject: Re: comments & clarification
Date: Thu, 9 May 2002 09:29:58 -0500 (CDT)
From: Brayton Gray <brayton@math.uic.edu>

With rergard to inverse limits in the category of spaces, it is clear that
the inverse limit of CW complexes does not have to be the homotopy type of
a CW complex. Consider a countable product of copies of S^2. The homology
is not finitely generated, but if this were homotopy equivalent to a CW
complex, the image of such an equivalence would lie in a finite
subcomplex, being compact. One of the advantages of the category of
simplicial sets is that you avoid this problem.

>
> Subject: clarification
> Date: Thu, 9 May 2002 09:25:38 +0100 (GMT Daylight Time)
> From: Andrey Lazarev <A.Lazarev@bristol.ac.uk>
>
> Perhaps I should clarify my question. Let D:I-->Top
> be a diagram of topological spaces (say, pointed, compactly
> generated, weakly Hausdorff). Here the indexing category I is not
> necessarily a directed set. Then we could form its homotopy (inverse)
> limit holim(D). I insist that the diagram D consist of CW-complexes.
>
> Question: is it true that holim(D) has homotopy type of a CW-complex?
>
> Note that it is important that we take the inverse limit. In the case
> of the homotopy direct limit the answer is positive and is a more or
> less a model category theory result. (hocolim of the object-wise
> cofibrant diagram is cofibrant.)
>
____________________________________________

Subject: Lazarev's question
Date: Thu, 9 May 2002 13:27:04 -0400 (Eastern Daylight Time)
From: "Nicholas J. Kuhn" <njk4x@tetra.mail.virginia.edu>

If X is a space, the infinite product
X x X x X x ...,
with the usual product topology, is the limit of the tower of fibrations
X <-- X x X <-- X x X x X <--
and is thus homotopy equivalent to the holimit of this tower.

If X is also a finite CW complex then the infinite product is compact,
(and thus already has the compactly generated topology).

However, if X is not contractible, then X will NOT have the homotopy
type of a CW complex.  The simplest case would be X = two points.

The notion of a CW complex dates from a 1949 article by JHC Whitehead,
and examples like this seem to have been known from the beginning.
Milnor's 1959 TAMS article "On spaces having the homotopy type of a CW
complex" is a fine starting place for searching (backwards) through the
literature.

Nick Kuhn