$\require{AMScd}$

**Background:** This question is about the bar and cobar constructions, and their relationship with the indecomposables of a dg-algebra. A brief summary of the bar and cobar constructions on ncatlab can be found at [\[1\]][1]. I will also mention the more general case of $A_\infty$ algebras; this case is discussed in this survey [\[2\]][2] or in Proute's thesis [\[3\]][3]. I posted a related question a few months ago in the mathoverflow post [\[4\]][4]. There I called the dg-module of indecomposables the ``linearized chain complex.'' It seems that the former is more standard terminology so I've switched for the time being.

**Setup:** Let $(A,\partial,\epsilon)$ be an augmented dg-algebra over a field $k$, i.e. a dg-algebra $(A,\partial)$ along with dg-algebra map $\epsilon:A \to k$. Here $k$ is a dga with trivial grading and differential. 

The **module of indecomposables** $(IA,\partial)$ of $(A,\partial,\epsilon)$ is a dg-module whose underlying graded $k$ vector-space is $IA = \ker(\epsilon)/\ker(\epsilon)^2$. The differential $\partial$ on $A$ descends to a differential $\partial$ on $IA$. Furthermore, the correspondence $(A,\partial,\epsilon) \mapsto (IA,\partial)$ is functorial. That is, given a map: $$f:(A,\partial,\delta) \to (B,\partial,\epsilon)$$ of augmented dg-algebras, i.e. a dg-algebra map with $\epsilon \circ f = \delta$, we get a map of dg-modules:
$$If:(IA,\partial) \to (IB,\partial)$$

The **bar complex** $BA$ of $(A,\partial,\epsilon)$ to a dg-coalgebra whose underlying graded vector-space is:
$$
BA := k \oplus BA^+ \qquad BA^+ := \bigoplus_{i=1}^\infty \ker(\epsilon)[1]^{\otimes i}
$$
The differential combines multiplication and differentiation from $(A,\partial)$ (see [\[1\]][1] for an actual definition) and the coaugmentation $\iota:k \to BA$ is just the inclusion. Similarly, the **cobar complex** $\Omega C$ of an augmented dg-coalgebra $C$ is an augmented dg-algebra admitting an analogous description to that of $BA$, as a certain tensor algebra. Both constructions are functorial, and there are generalizations of to $A_\infty$ algebras and coalgebras (see [\[2\]][2] or [\[3\]][3]). 

The functors $B$ and $\Omega$ form an adjoint pair ($B$ is right adjoint to $\Omega$) and we have natural adjoint maps $\Omega B A \to A$ and $C \to B \Omega C$, which are quasi-isomorphisms of augmented dg-algebras and coaugmented dg-coalgebras, respectively. There is a reference for this in the post [\[5\]][5]. In particular, we get a natural map of dg-modules between the indecomposables:
$$
I\Omega B A \to IA
$$
I believe that it is true that $I\Omega C \simeq \text{coker}(\kappa)$ for any coaugmented dg-coalgebra $(C,\partial,\kappa)$ (an *actual* isomorphism of chain groups, not just a quasi-isomorphism) so that the above map yields a canonical map of dg-modules and a corresponding map on homology:
$$
BA^+ \to IA \qquad H(BA^+) \to H(IA)
$$

>**Question:** Under what circumstances is the natural map $I\Omega B A \to IA$ an isomorphism on homology? When is it surjective on homology? I'm particularly interested in the case where the underlying graded algebra of $(A,\partial,\epsilon)$ is the free graded commutative algebra over some generators. Any partial answers are much appreciated.

**Examples:** In some cases this map is simple to understand. For instance, when $A$ is an algebra with the trivial differential and grading, then $H(IA) = IA$ and I believe that the map on homology $H(BA^+) \to H(IA)$ is just projection onto $H^1(BA^+) \simeq IA$. 

Another context where something similar happens is when $A = \Omega C$ where $(C,\partial,\kappa)$ is a coaugmented dg-coalgebra (or even an $A_\infty$ coalgebra). Then the map:
$$\text{coker}(\iota) \simeq BA^+ \to IA \simeq \text{coker}(\kappa)$$
is just the $A_\infty$ quasi-isomorphism on the cokernels of the coaugmentations induced by the quasi-isomorphism of $A_\infty$ coalgebras of $B\Omega C \to C$, itself induced by the adjunction map $\Omega B \Omega C \to \Omega C$. Since $A_\infty$ quasi-isomorphisms descend to isomorphisms on homology, we know in particular $H(BA^+) \simeq H(IA)$ under the map of interest.

  [1]: https://ncatlab.org/nlab/show/bar+and+cobar+construction
  [2]: https://arxiv.org/abs/math/9910179
  [3]: http://www.logique.jussieu.fr/~alp/these_A_Proute-TAC.pdf
  [4]: https://mathoverflow.net/questions/299047/correct-notion-of-chain-homotopy-for-linearized-homology-of-augmented-dgas
  [5]: https://mathoverflow.net/questions/241579/when-is-bar-cobar-duality-an-equivalence