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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]. I will also mention the more general case of $A_\infty$ algebras; this case is discussed in this survey [2] or in Proute's thesis [3]. I posted a related question a few months ago in the mathoverflow post [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] 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] or [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]. 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? When is it surjective? 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.

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 quasi-isomorphism on the cokernels of the coaugmentations induced by the quasi-isomorphism of coaugmented dg-coalgebras $B\Omega C \to C$. In particular, $H(BA^+) \simeq H(IA)$.