I have the following question. Let's denote by $\Omega$ the cobar functor, by $C$ some DG-coalgebra (not co-commutative) over a field $k$. Also suppose $V=k\times\dots\times k$ is just the product of $n$ copies of the base field $k$, and $V^*$ is the linear dual of $V$, which is a coalgebra in a natural way.
Q: How can I describe the algebra $\Omega(C\otimes_k V^*)$?
The question boils down to describing $\Omega(C\oplus C)$. The first guess was that $\Omega(C\oplus C)$ is just the free product of copies $\Omega(C)$. I doubt this is true but I am struggling with proving or disproving this.
What can be said about the cobar construction of a direct sum af any two coalgebras?
Sorry if this question is silly! I might be overlooking something obvious.
Thanks a lot for your help!