Seifert--van Kampen for the loop space dga 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!
 A: 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 $Z$.   
A: Zack, the coproduct in the category of associative algebras is quite different from the tensor product. For instance, the coproduct of two free associative algebras on one generator, $k\langle X\rangle$ and $k\langle Y\rangle$, is free on two generators, $k\langle X\rangle\amalg k\langle Y\rangle=k\langle X,Y\rangle$, while the tensor product is not $k\langle X\rangle\otimes k\langle Y\rangle=k[X,Y]$ since it is the commutative algebra of polynomials in two variables.
If you replace tensor product with homotopy push-out in your equation, and if all spaces are simply connected, then the result is true. In this case, the chain coalgebra $C_*(X)$ is the push-out $C_*(X_0)\oplus_{C_*(Y)}C_*(X_1)$ (this push-out is the same in the category of chain complexes and in the category of coalgebras). The cobar construction $\Omega$ is a left adjoint, so it preserves pushouts,
$\Omega C_*(X)=\Omega C_*(X_0)\amalg_{\Omega C_*(Y)}\Omega C_*(X_1)$. Now use everywhere  the natural quasi-isomorphism $\Omega C_*(X)\simeq C_*(\Omega X)$ for simply connected spaces.
For non-simply connected spaces the result sketched in the previous paragraph is probably false. I think a counterexample can be found in the following way, although I haven't been brave enough to tackle the final step of the computation. Take $X_0=X_1=D^2$, $Y=X_0\cap X_1=S^1$ and $X=X_0\cup_YX_1=S^2$, in this case $C_*(\Omega S^2)\simeq k\langle x\rangle $ where $x$ has degree $1$ and trivial differential. The homotopy push-out $C_*(\Omega X_0)\amalg_{ C_*(\Omega Y)} C_*(\Omega X_1)$ is quasi-isomorphic to the suspension of $C_*(\Omega Y)$ in the category of augmented chain algebras, since $D^2$ is contractible. The augmented chain algebra $C_*(\Omega Y)$ is quasi-isomorphic to $k\langle y,z\rangle/(yz-1)$ concentrated in degree $0$. In order to compute its suspension, we have first to obtain a graded-free DG-resolution. This can be done by means of curved Koszul duality theory. I think that this algebraic suspension is going to be quite different to $k\langle x\rangle $.
In any case, in the non-simply connected case the natural replacement of your statement could maybe use the Baues-Tonks twisted cobar construction. 
