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Stream: theory: algebraic topology

Topic: Adams Cobar Construction

Patrick Nicodemus (Oct 18 2021 at 16:19):

I have been studying Adams' cobar construction for differentially graded algebras.
I would like to ask whether anyone has written down what this corresponds to in the simplicial case, if we work with simplicial Abelian groups or something like this. We could simply translate the result by Dold-Kan but I am not sure the raw data of the associated simplicial object would be very illuminating.

Patrick Nicodemus (Oct 18 2021 at 16:26):

Let me explain why I think this is slightly nontrivial. Here's the context I have seen for the cobar construction, in my own language.
Let $C$ be a differentially graded coalgebra. Usually texts say to assume that $C_0 = R$, where $R$ is the ground ring, and $C_1 = 0$, which is a connectedness assumption. I am assuming for simplicity that $C$ is concentrated in nonnegative degree.
$C$ is also coaugmented. To me this means that it acts on the unit object $1$ as a co-module for the comonoid, i.e. there is an action map $1\to C$ which is associative and co-unital.

Now we carry out the general "cobar" construction, which I now describe. For convenience we delete $C_0$ and add it back at the very end.
There is an adjunction between $C$-comodules (which are equivalently coalgebras in the comonad sense with respect to the comonad $C\otimes -$, the overloading of notation here is unfortunate) and objects on the underlying category, with the forgetful functor as left adjoint and the free functor as right adjoint. This gives rise to a monad on $C$-comodules, so that we can associate to each comodule a cosimplicial resolution, an augmented cosimplicial object.

Patrick Nicodemus (Oct 18 2021 at 16:27):

in the case we are talking about this looks a bit like
$1\to C\to C^{\otimes 2}\to C^{\otimes 3}\dots$
where the unit is in dimension $-1$ if you like and $X_n = C^{\otimes n+1}$.

Patrick Nicodemus (Oct 18 2021 at 16:29):

here the co-face maps are given by $\delta_n = C^{\otimes n}\otimes \eta$, the co-augmentation $\eta: 1\to C$, and $\delta_i : C^{\otimes i}\otimes \Delta\otimes C^{\otimes n-i}$, the comultiplication, for $i<n$.

Patrick Nicodemus (Oct 18 2021 at 16:34):

If we take the normalized Moore complex of this cosimplicial object we get a double complex. What confuses me is that this is is a mixed chain complex/cochain complex. Regarded as a bicomplex or double complex it is concentrated in the second quadrant of the plane.

Patrick Nicodemus (Oct 18 2021 at 16:35):

One can take the total complex of this double complex and then one has an ordinary chain complex.

Patrick Nicodemus (Oct 18 2021 at 16:37):

The assumptions in the beginning guarantee that the bicomplex is not just concentrated in the second complex but it's also concentrated above the line $(-k,k)$, i.e., everything nonzero in the bicomplex of bidegree $(i,j)$ where $i+j\geq 0$. As a consequence of this assumption, the total chain complex of the bicomplex is concentrated in nonnegative degree. This total chain complex is what we call the cobar construction.

Patrick Nicodemus (Oct 18 2021 at 16:39):

I hope this description is straightforward. What is odd about trying to adapt this to the case of simplicial Abelian groups is that if $X_{\bullet}$ is a simplicial Abelian group which is also an augmented coalgebra in that sense, it seems that this cobar construction should end up creating an augmented cosimplicial resolution of the unit object, which is a kind of "mixed bisimplicial Abelian group" $\Delta\times \Delta^{\rm op}\to \mathbf{Ab}$.

Patrick Nicodemus (Oct 18 2021 at 16:41):

I have never encountered these mixed bisimplicial objects before and I am at a loss because I do not know how to convert one of these to a simplicial Abelian group in general. This thing should be, I think, essentially equivalent to a bicomplex concentrated in the second quadrant, and then we could take its total complex, but then this total complex would be in general not concentrated in nonnegative degree, rather it would stretch off to negative infinity in general.

Patrick Nicodemus (Oct 18 2021 at 16:43):

I guess in the end my question is whether there is a straightforward way to describe Adams' cobar construction for simplicial Abelian groups without necessarily passing into the category of unbounded chain complexes. Under what assumptions on a simplicial Abelian group does this construction make sense?