Ytukyg

In my favorite book Algebraic Operads by Loday and Vallette there is a theorem (2.3.1 in my book) which says that for a twisting morphism $alpha:Cto A$ from a connected wdga coalgebra to a connected wdga algebra the following are equivalent

1)The twisted tensor product $Cotimes_alpha A$ is acyclic.

2)The corresponding coalgebra morphism $f_alpha:Cto BA$ is a quasi isomorphism.

3)The corresponding algebra morphism $g_alpha:Omega Cto A$ is a quasi isomorphism.

The proof of this depends on a pretty technical lemma called the comparison lemma (2.5.1 in my book). The proof of this lemma starts with filtering the weight $n$ component of the twisted tensor product by
$$F_s((Cotimes_alpha A)^{(n)}):=bigoplus_{d+mleq s}C_{d}^{(m)}otimes A^{n-m}.$$
This is bounded below because $F_{-1}={0}$ they say but I think this is only true if $C$ is non-negatively graded (the lower grading).
I have two questions that I hope someone can help me with:

Is this lemma only true for non-negatively graded coalgebras?

In this case, what happen when $C_0neq mathbb{k}$ i.e. $C$ is not connected w.r.t. the homological grading? Then $Omega C=T(s^{-1}bar{C})$ is not concentrated in non-negative degrees so the lemma cannot be applied as in the proof of the fundamental theorem of twisting morphism.

Any help is much appreciated!

Edit: I see now that they do in fact, in the first chapter, state that homological grading is supposed to be non-negative throughout the book. But my two questions remain.

I'm not really fully aware of the whole picture here so take what I'm saying with a pinch of salt.

It is useful to think of a coalgebra $C$ as coming from a topological space $X$ by taking its chain coalgebra $C =C_*(X)$. If $X$ is simply connected, then we can take the chains which send the $1$-skeleton of any simplex to a chosen base-point of $X$, so that $C_0 = mathbb k$ and $C_1=0$. This has the effect that $Omega C$ is built from $s^{-1}overline{C}$, which lives in degrees $geqslant 1$, and then in particular our spectral sequence arguments work wonders. When this does not happen the arguments not
only do not work, but the claims we want are incorrect. Unfortunately, I don't have examples at hand.

However, to think why a counterexample should exist, consider that you want $A= C_*(Omega X)$, the Pontryagin algebra of $X$, $C = C_*(X)$ the chain coalgebra of $X$, and $tau : Cto A$ the map that sends a $p$-chain which collapses the $1$-skeleton to a point to the corresponding $(p-1)$-chain. Then $Cotimes_tau A$ wants to model the complex $C_*(LX)$, where $LX$ is free loops. The sequence of complexes $C_*(Omega X)to C_*(LX) to C_*(X)$ coming from the usual fibration then wants to be modeled by $Cto Cotimes_tau Ato A$, and the maps $Omega C_*(X)to C_*(Omega X)$ and $C_*(X) to BC_*(Omega X)$ want to be quasi-isomorphisms.

A theorem of Adams says that in the simply connected case, $Omega C$ is quasi-isomorphic to $C_*(Omega X)$, in fact. A theorem of Quillen shows that any simply connected dg coalgebra $C$ comes from a simply connected topological space, so this is not doing us any harm: the category of simply connected dg coalgebras is Quillen equivalent to that of simply onnected topologicl spaces. You can find more on this wonderful paper of his.

When a non-trivial $pi_1(X)$ appears, I think one needs to take into account the action of this on $Omega X$ and on $pi_*(X)$, for example, to start saying anything. I also think in this case you want hypotheses such as the action being nilpotent, or free. But at any rate all the rather linear homological algebraic results need to be modified to include 'non-linear' group actions, twisted modules and local coefficients, for example. I never really read much about this story, however.

Add: Relatively recently, Rivera and Zeinalian showed that the statement of Adams’ theorem holds for any connected space, without any assumptions on the fundamental group or its action; see arxiv.org/abs/1612.04801 (see comments)