Subject: correction
From: Tom Goodwillie
Date: Wed, 19 Jul 2006 09:54:02 -0400
As so often happens, I wish I had reread my post one more time before
posting because I have not quite said what I meant.
>
>
> For a cosimplicial space X consider, for each n>0, the n-dimensional
> cubical diagram which I will call X[n]. It has X^n as "first" space
> and X0 as "last". Each map in it is a codegeneracy map.
> (To be more precise, it is induced by the inclusion of the poset of
> subsets of an n-element set into the category of all nonempty finite
ordered sets.)
I should have said, "the inclusion of the poset of subsets of an n-element
ordered set into
{finite ordered sets}, followed by a standard *contravariant* functor from
{finite ordered sets} to
{nonempty finite ordered sets} that increases cardinality by one".
TG
Subject:
Re: response re totalization
From:
"Michael J. Hopkins"
Date:
Wed, 19 Jul 2006 09:55:12 -0400 (EDT)
To:
dmd1@lehigh.edu (Don Davis)
I first learned about results of this kind from Bousfield (in a
handwritten letter). He published them in
On the homology spectral sequence of a cosimplicial space.
Amer. J. Math. 109 (1987), no. 2, 361--394,
though they date from much earlier.
Michael Hopkins
>> One posting today: a response to yesterday's question.............DMD
>> _______________________________________________________________
>>
>> Subject: Re: three postings
>> From: Tom Goodwillie
>> Date: Tue, 18 Jul 2006 10:42:57 -0400
>>
>> The proof that I sketched the other day for Victor Turchin
>> (who must be the inquiring party here) assumed a connectivity
>> hypothesis, which I will now explain.
>>
>> For a cosimplicial space X consider, for each n>0, the n-dimensional
>> cubical diagram which I will call X[n]. It has X^n as "first" space
>> and X0 as "last". Each map in it is a codegeneracy map.
>> (To be more precise, it is induced by the inclusion of the poset of
>> subsets of an n-element set into the category of all nonempty finite
>> ordered sets.)
>> The hypothesis is that this diagram is (an+b)-cartesian, meaning that
>> the canonical
>> map from the first space to the holim of the others is (at least)
>> (an+b)-connected.
>> (This basically means that the nondegenerate part of X^n is
>> (an+b-1)-connected.
>> Modulo the usual need to fuss about pi_0 and pi_1, it is saying that
the
>> homotopy group
>> spectral sequence vanishes above a certain line.)
>>
>> If a>1 and b is nonnegative, then it's not at all hard to show (using
>> what I call the
>> "higher Blakers-Massey theorem and its dual" in my "Calculus 2") that -
>>
>> (1) the tower of partial totalizations {Tot^s} converges in the sense
>> that the connectivity
>> of the map Tot-->Tot^s tends to infinity with s,
>>
>> (2) the same is true for the (levelwise) suspension SigmaX of the
>> cosimplicial space X,
>>
>> (3) the canonical map from Sigma(Tot^s X) to Tot^s(Sigma X) also has a
>> connnectivity tending
>> to infinity,
>>
>> and therefore
>>
>> (4) in the limit you get a weak equivalence SigmaTotX --> TotSigmaX.
>>
>>
>> It seems to me that the condition b\geq 0 can be relaxed somewhat,
>> something like b>-a.
>>
>> Of course you need some hypothesis to get (4). But it may be that there
>> is some sneaky
>> approach I am unaware of that makes (4) true even in some cases when
(1)
>> is not. I'd be curious.
>>
>>
>> I have not thought about what this same argument gives for more general
>> hocolim instead of suspension.
>>
>> Tom Goodwillie
>>
>> >
>> >
>> > From: "vitia-t"
>> > Date: Tue, 18 Jul 2006 11:27:33 +0400 (MSD)
>> >
>> > Is it well known that the homotopy totalization of a cosimplicial
>> space commutes with the suspension
>> > (under some mild conditions)? I have a sketch of the proof from Tom
>> Goodwillie, but I wonder whether
>> > it was already known.
>> > Probably in general the homotopy totalization commutes with homotopy
>> colimits?
>> > ____________________________________________________________
>>
>>
>>