  For a commutative ring $\mathbf k$ with unit, we describe and study various
differential graded $\mathbf k$-modules and $ \mathbf k$-algebras which are
models for the cohomology of polyhedral products $(\underline{CX},\underline
X)^K$. Along the way, we prove that the integral cohomology $H^*((D^1, S^0)^K;
\mathbb Z)$ of the real moment-angle complex is a Tor module, the one that does
not come from a geometric setting. We also reveal that the apriori different
cup product structures in $H^*((D^1, S^0)^K;\mathbb Z)$ and in $H^*((D^n,
S^{n-1})^K; \mathbb Z)$ for $n\geq 2$ have the same origin.
  As an application, this work sets the stage for studying the based loop space
of $(\underline{CX}, \underline X)^K$ in terms of the bar construction applied
to the differential graded $\mathbb Z$-algebras $B(\mathcal C^*(\underline X;
\mathbb Z), K) $ quasi-isomorphic to the singular cochain algebra $\mathcal
C^*((\underline{CX},\underline X)^K;\mathbb Z)$.

Algebraic Topology