Subject: Re: questions about DGCAs over Z
From: Justin Smith <jsmith@drexel.edu>
Date: Wed, 29 Nov 2006 11:25:48 -0500

Replying to:
>> Subject: differential graded coalgebras over Z
>> From: "John R. Klein" <nielkj@gmail.com>
>> Date: Mon, 27 Nov 2006 11:48:08 -0500
>> 
>> Consider
>> 
>> 1) the category of  differential graded coalgebras (dgca).
>> 
>> 2) For fixed dgca C, the category of its dg-comodules.
>> 
>> Question A: Using quasi-isomorphism for "weak equivalence"
>> Is it known whether (1) and (2)  admit Quillen model structures ?
>> 
>> 3) Assuming (1) and (2) admit such a structure, consider the category
>> of "spectra" formed
>> from from objects of (1).
>> 

I have a paper that develops a Quillen model structure on coalgebras
over operads (which certainly would include dcga's) but weak equivalence
is chain-homotopy equivalence rather than quasi-isomorphism. I must also
admit non-Z-free underlying modules, since they arise naturally in the
constructions.

Essentially, all underlying modules must be "nearly free" in the sense
that every countable submodule is free (the constructions I use
naturally lead to underlying modules that look like the Bayer-Specker
group Z^aleph_0, which is well-known to not be free).

See

http://vorpal.math.drexel.edu/research/current.html

for a pdf file of the paper.

The use of non-Z-free underlying modules makes the "cellular gambit"
impossible, so that I must strengthen the definition of weak
equivalence.

In the special case of dcga coalgebras, it might be possible to rework
my theory to eliminate non-free modules, since I think cofree coalgebras
in this case can be constructed to be Z-free.

Best,
Justin Smith
_____________________________________________________________

Subject: Re: questions about DGCAs over Z
From: Aniceto <aniceto@agt.cie.uma.es>
Date: Thu, 30 Nov 2006 12:02:29 +0100

This is just a very partial answer to the question A above which I assume 
it
is requested over the integers as ground ring:

In their paper (Homotopie moderee et temperee avec les coalgebras...., 
Arch.
Math. (59), 130-145, 1992) Daniel Tanre and Hans Scheerer showed that the
category of differential graded cocommutatives and coassociatives 
coalgebras
(over the  integers) is a cofibration category (so you still have the
corresponding homotopy caegory) considering:

Cofibrations: injective morphisms with free coker.

Weak equivalences: morphisms which are taken to weak equivalence by the
Quillen functor (from coalgebras to Lie algebras).

Hope it is useful

Aniceto