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? >> > ____________________________________________________________ >> >> >>