TITLE: Seifert--van Kampen for the loop space dga
QUESTION [6 upvotes]: I recently noticed that I could mostly prove a special case of the following statement. I think it's true in general, though perhaps only for nice spaces.

Let $X$ be a topological space, and suppose $X=X_0\cup X_1$, where $X_0$, $X_1$, and $Y:=X_0\cap X_1$ are all nonempty and path-connected. Then
  $$
C_*(\Omega X)\cong C_*(\Omega X_0)\otimes_{C_*(\Omega Y)}C_*(\Omega X_1)
$$
  where $\otimes$ is the homotopy pushout of dg-algebras and $\cong$ is $A_\infty$ quasi-isomorphism.

For disconnected spaces, it should still be true, where one replaces $C_*(\Omega X)$ with the dg-category of chains on the path space.
This seems like a natural mostly-extension of the classical Seifert--van Kampen theorem, which makes me think it exists somewhere in the literature, but I couldn't find it. Does anyone know where to find this statement?
Thanks!

REPLY [5 votes]: Sketch of proof. I will use the following ingredients
0) the category of spaces will be the category of simplicial sets. All the computation are in the derived sense. 
1) use the Quillen adjunction $$F: sSet^{\otimes}\longleftrightarrow sMod_{k}^{\otimes}: U$$ 
between the category of monoids in simplicial sets and  monoids in simplicial $k$-modules .
2) the category of simplicial $k$-algebras is Quillen equivalent to the category of differential graded $k$-algebras (i.e. monoids in the category of chain complexes of $k$-modules in positive degree) 
$$N: sMod_{k}^{\otimes}\longleftrightarrow Ch^{\otimes}_{k}: S$$ 
Now suppose that $T$ is  a homotopy push out $X\leftarrow Y\rightarrow Z$ of connected pointed simplicial sets (Kan complexes), then $\Omega T$ is the homotopy pushout of  $\Omega  X\leftarrow \Omega Y\rightarrow \Omega Z$ in the category in $sSet^{\otimes}$ (notice that the model that I'm using for loop space gives me a honest simplicial monoid). The functor $F$ commutes with homotopy push out, it means that that $F(\Omega T)$ is a homotopy push out of $F(\Omega  X)\leftarrow F(\Omega Y)\rightarrow F(\Omega Z)$ in $sMod_{k}^{\otimes}$, therefor the $NF(\Omega  X)\leftarrow NF(\Omega Y)\rightarrow NF(\Omega Z)$ is a homotopy push out in the category $Ch^{\otimes}_{k}$ and the functor $NF$ can be identified with $C_{\ast}(-,k)$.
It means that $$C_{\ast}(\Omega T)\simeq C_{\ast}(\Omega X)\sqcup_{C_{\ast}(\Omega Y)}^{h}C_{\ast}(\Omega Z)$$ as differential graded algebras, where $\sqcup$ is the coproduct in the category of $DGA$. This statement is true for any pointed connected spaces $X, Y$ and $Z$.