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!



Seifert--van Kampen for the loop space dga
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 $Z$. 


