Subject: Re: followup & conference
Date: Sat, 6 Apr 2002 17:00:22 -0500
From: Tom Goodwillie
>Recall that the question was whether the functor F(?,X) converts
>"finite" holimits to homotopy hocolimits in the category of spectra.
>
>Or, more or less equivalently, when are "finite" holimits hocolimits?
>
>However it is unclear is what "finite" means. I thought at first that
>the indexing category should be finite with finitely many morphisms.
>However it does not seem to be finite enough as actions of finite
>groups make clear.
>
>I now believe that the correct notion of finiteness here is that the
>indexing category I is very small in the sense of Dwyer-Spalinsky.
>That is, not only the set of morphisms and objects is finite, but
>also the all morphisms go in the same direction, that is a long
>enough string of morphisms must contain an identity morphism.
>
Yes, that kind if finiteness (in other words, the nerve of I is a
finite simplicial set -- finitely many nondegenerate simplices) is
enough. I wonder what other small categories have this property
regarding holim and hocolim.
Although finite groups do not, groups G such that BG has
the homotopy type of a finite complex do. It seems to me that
one can generalize this last class of examples and talk about small
categories I such that the constant functor S:I -> Spectra given by
the sphere spectrum has what you might call a finite projective
resolution. These also have that property. But that class of
examples doesn't include those "very small" categories. I
wonder if there is a common generalization.
By the way, the nerve of I must be at least stably homotopy equivalent
to a finite complex if hocolim F(S,X) is going to coincide with
F(holim S,X) where S is that constant functor again.
I don't know any references. Which reminds me, does anybody have
a reference for the fact that holim over a very small category
commutes (up to weak equivalence) with hocolim over a filtering
category (or a directed set)?
Tom Goodwillie
Subject: Re: followup & conference
Date: Sun, 7 Apr 2002 15:30:34 -0500
From: Tom Goodwillie
A further thought:
If J -> I is a functor satisfying Bousfield and Kan's
"left cofinality" condition, which insures that any
holim over I is equivalent to the associated holim
over J, then (since the associated functor J^{op} -> I^{op}
satisfies right cofinality, which implies the same thing
for hocolim) we can say that
if J has the property under discussion (hocolim of dual is
dual of holim), then so does I. So this gives more examples.
And in fact that class of examples I mentioned (groups with homotopy-
finite classifying spaces) can be obtained in that way from the
other class of examples (very small categories). If X is a finite
connected simplicial complex with contractible universal covering space,
then the barycentric subdivision of X yields a poset that has a
left cofinal functor to the fundamental group of X.
Tom Goodwillie