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:

> Two postings here: a question about H-derivations and a question
> about coalgebras.....DMD
> __________________________________________
>
> ______________________________________________________ Date: Sat, 12
> Feb 2000 12:23:31 -0500 (EST) From: James Stasheff <jds@math.unc.edu>
> Subject: question about coalgebras
>
> 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
>
>         http://www.math.unc.edu/Faculty/jds
>
>