Subject: Re: comments on Blakers-Massey theorems
From: Bill Richter <richter@math.northwestern.edu>
Date: Mon, 24 Jul 2006 13:27:09 -0500

Ronnie, in his proof of the n-ad connectivity theorem, Tom credits you
with a proof, and I don't believe you ever published one.  Can you
clarify?  Back to you:

   I do not know if this algebraic information is used in or is
   relevant to the results of Tom's Calculus series of papers. It
   would be interesting if this strong algebraic information could be
   used in that sort of way.

I'm sure it's not used by Tom.  But yeah, it would be very interesting
to find some way to use it.
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Subject: Re: comments on Blakers-Massey theorems
From: "Ronnie Brown" <Ronnie@ll319dg.fsnet.co.uk>
Date: Mon, 24 Jul 2006 21:28:03 +0100

Bill,

The van Kampen theorem for diagrams of spaces in the paper with Loday 
states roughly that a colimit of connected n-cubes is connected and that 
\Pi (the fundamental cat^n-group functor) preserves those colimits. The 
connectivity is important, and is used in the inductive proof, and so is 
stated as a separate conclusion in the theorem. The n-ad connectivity 
theorem is an interpretation, for the case when the n-cube comes from an 
n-ad. Maybe this was not spelled out in the papers.

Even more interesting to me was to find that cat^n-groups were a suitable 
codomain for a GvKT, allowing something nonabelian to be calculated 
precisely, not up to extension. One idea not pursued was to have a 
spectral sequence whose terms were cat^n groups!!??? Could this be related 
to ideas of Michael Barratt's (which I never mastered)?

Ronnie