Two responses to the cosimplicial question, which is repeated at the bottom.........DMD ____________________________________ Subject: Re: question & 2 conferences Date: Thu, 5 Apr 2001 17:17:08 +0200 (IST) From: Assaf Libman 1. All you need can be found in Bousfield and Kan's book or a paper of the same authors in Topology 11. Not explicitly, but you must be able to make the required changes. You only need to recall that the normalized complex is chain homotopic to the non normalized one. The E^1 term is precisely what you want. Convergence is clearly a different issue (and usually difficult). There are several approaches. No clear cut answer (after all this is a second quadrant SS). I am not sure about spectra, what is the "total object" here ? 2. Of course. The reduced cosimplicial category is cofinal in the cosimplicial category. Assaf Libman, Dept. of Mathematics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel ___________________________________________________ Subject: Re: question & 2 conferences Date: Thu, 5 Apr 2001 14:25:03 -0400 (EDT) From: Martin Bendersky Here is a reply to the cosimplicial question. Dror and Dwyer "A long homology localization tower" Comment. Math Helv 1977 study such diagrams and prove that the totatlization without codegeneracies is homotopic to tot with degeneracies for a cosimplicial space. In order to show that there that E_1 of the Bousfield Kan spectral sequence is the cobar complex it seems that you might need codegeneracies up to homotopy (at least that is what John Hunton and I had to assume). --Martin Bendersky > Date: Thu, 5 Apr 2001 09:53:36 +0100 (GMT Daylight Time) > From: Andrey Lazarev > > Does anyone know a reference for cosimplicial objects without > codegeneracies? > > Specifically, I would like to have a result of the following sort: > > 1.Let X=(X_i) be a cosimplicial space or spectrum without > codegeneracies that is fibrant in a suitable sense. Then there exists > a spectral sequence converging to the homotopy of the totalization of > X (built using cofaces only) whose E_1-term is \pi_i(X_j) and the > differential is the alternated sum of cofaces. > > 2.If X actually does have codegeneracies i.e. is a usual cosimplicial > guy then the two totalizations (one built with codegeneracies, the > other without) are weakly equivalent. > > Andrey Lazarev > >Subject: question for the discussion group >Date: Thu, 5 Apr 2001 09:53:36 +0100 (GMT Daylight Time) >From: Andrey Lazarev > >Does anyone know a reference for cosimplicial objects without >codegeneracies? > >Specifically, I would like to have a result of the following sort: > >1.Let X=(X_i) be a cosimplicial space or spectrum without >codegeneracies that is fibrant in a suitable sense. Then there exists >a spectral sequence converging to the homotopy of the totalization of >X (built using cofaces only) whose E_1-term is \pi_i(X_j) and the >differential is the alternated sum of cofaces. > >2.If X actually does have codegeneracies i.e. is a usual cosimplicial >guy then the two totalizations (one built with codegeneracies, the >other without) are weakly equivalent. > >Andrey Lazarev >A.Lazarev@bristol.ac.uk >Phone +44 117 928 7997 >School of Mathematics >University of Bristol >Bristol BS8 1TW UK >---------------------------------------