Let $A$ be an augmented differential graded algebra over a field $k$. I will write $BA$ for its bar construction (whose homology is $Tor^A(k, k)$). This is a co-augmented differential graded coalgebra over $k$; write $\Omega BA$ for its cobar construction. There is a natural dga map $\Omega BA \to A$ which uses the fact that $\Omega C$ is the tensor algebra on a shift of $C$, and the projection map $BA \to A$.

I think I'm only going to highlight my ignorance here, but my question is: **what assumptions on $A$ imply that the map $\Omega BA \to A$ is an equivalence?**

If I am reading any of the following references correctly:

- Husemoller-Moore-Stasheff's "Differential homological algebra and homogeneous spaces" (section II.4)
- Felix-Halperin-Thomas "Rational homotopy theory" (exercise 2, section 19)
- Keller "A-infinity algebras, modules and functor categories" (Theorem 4.3, attributed to Lefevre),

then I think that the answer is supposed to be: "no assumptions are required at all."

I seem to have two possible counterexamples to this claim. But their being counterexamples seems to rely on the positive answer to another question: **when is $\Omega$ a homotopy-invariant functor?** That is: if $f: C \to D$ is an equivalence of dg coalgebras, is $\Omega f$ an equivalence of dga's?

Let me explain the (counter?)examples. Let $G$ be a finite group, $k[G]$ its group ring, and $k^G$ the dual Hopf algebra (the ring of functions on $G$ with values in $k$). Then both $k[G]$ and $k^G$ are augmented algebras, where the augmentation is the counit.

$B(k[G])$ is isomorphic to the simplicial chain complex of the bar construction $BG$ (with coefficients in $k$). If $k$ has characteristic zero, then this is is equivalent to $k$, via a transfer argument. If $\Omega$ is a homotopy invariant, then $\Omega B(k[G]) \simeq \Omega k = k$, which is certainly not $k[G]$ unless $G$ is trivial.

$k^G$ is isomorphic as a $k$-algebra to a product of fields $$k^G \cong \prod_{g \in G} k,$$ and the augmentation can be identified with the map which projects onto the factor corresponding to the identity $e \in G$. The kernel of the augmentation is a summand, and hence a projective $k^G$-module. From this, you can write down a very short resolution of $k$ over $k^G$ and compute $Tor^{k^G}(k, k) = k$, concentrated in $Tor_0$. So $B(k^G) \simeq k$. Again, if $\Omega$ is homotopy-invariant, then $\Omega B(k^G) \simeq \Omega k = k$, which is not equivalent to $k^G$.

So either I have misunderstood the references above, or $\Omega$ is not a homotopy invariant functor. I'm guessing it's the latter; can anyone point me to a reference which gives criteria for this to hold?