Date: Tue, 15 Feb 2000 17:49:26 +0100 (MET)
From: Donald Stanley <stanley@mpim-bonn.mpg.de>
Subject: Response on coalgebras (fwd)

Let C be the category of differential graded coalgabras over a
ring with the underlying modules flat.

First of all I think there needs to be some restriction
on the ring. If we take then ring Z/4 then I think we don't
even have pushouts.
So lets let the ring be Z (but a field should be ok too).

Then C is cocomplete with colimits being the
colimit in all coalgebras modulo torsion
(this may seem strange but consider the category of
Hausdorff spaces which is cocomplete as well)
I am assuming C is complete.
(or at least small complete but I didn't check)

If we take the same cofibrations and weak equivalences
as Paul suggested (ie such that the underlying map
of chain complexes is a cofibration or weak equivalence)
then we can take a set of generating cofibrations to be the
set of all maps

f:A ---> B

where f is a cofibration of chain complexes and
A and B are countable type.

This data should determine a closed model category if and only if
for any map the second map in the factorization that we get from the
infinite gluing construction using the generating cofibrations
is a weak equivalence.

This is always true if we restrict to connected coalgebras.
The second map is always an H_* surjection so the
only required thing is to be able to kill elements in the kernel.


Don Stanley

>
> As of some notes from 1996, it was an open question as
> to how to make a closed model category from coalgebras
> which as dg modules are flat over the gorund ring
>
> one problem: the naive cokernel is no longer flat
>
> .oooO   Jim Stasheff            jds@math.unc.edu
> (UNC)   Math-UNC                (919)-962-9607
>  \ (    Chapel Hill NC          FAX:(919)-962-2568
>   \*)   27599-3250


---------- Forwarded message ----------
Date: Mon, 14 Feb 2000 11:54:45 EST
From: DON DAVIS <dmd1@Lehigh.EDU>
To: toplist <Distribution.List@lehigh.edu>
Subject: Response on coalgebras

Date: Mon, 14 Feb 2000 09:59:57 -0600 (CST)
From: Paul Goerss <pgoerss@steenrod.math.nwu.edu>
Subject: Re: 2 questions

There is a partial answer to the question about coalgebras. While it is
not specified, I take coalgebras to be coassociative, but not necessarily
cocommutative. None of the remarks below apply to the cocommutative case.

Let us assume that the model category structure would be the one
"inherited" from dg modules; that is, a morphism of dg coalgebras would be
a cofibration or weak equivalence if and only if it is as dg modules.
Then, one would have to prove, at the very least, that the ``cofree''
coalgebra functor S preserves acyclic fibrations. Actually, it is
sufficient to prove something a little different: for an arbitrary dg
coalgebra A, the functor

V --> A \tensor S(V)

preserves weak equivalences for certain types of Vs. As far as I know,
this hasn't been done in the generality specified. But one can say the
following:

1.) If the dg modules lie in positive degrees, with differential of degree
-1, then S(V) is a product of tensor powers of V, and in any given degree
that product is finite. So the result should hold.

2.) By imposing certain types of filtrations on the coalgebra, one can
make the calculations, at least in principle. This is work of Hinich: ``DG
coalgebras as formal stacks'' is the preprint I have. He works over a
field.

3.) If the ground ring is a field, it is possible to say a lot more. Then
the dual of a dg coalgebra is a profinite dg algebra, and by carefully
controlling completions, one can make the necessary calculations for S(V).
Ezra Getzler and I worked this out for V in non-negative degrees. This is
a part of a larger project, but this much has been written out in detail.
For all I know, others have done this, too. Thomas Kahl, in Belgium, has
worked on such things, for example.

Regards,

Paul

On Sat, 12 Feb 2000, DON DAVIS wrote: