  We study the cohomology ring of the complement $\mathcal{M}(\mathcal{A})$ of
a manifold arrangement $\mathcal{A}$ in a smooth manifold $M$ without boundary.
We first give the concept of monoidal cosheaf on a locally geometric poset
$\mathfrak{L}$, and then define the generalized Orlik--Solomon algebra
$A^*(\mathfrak{L}, \mathcal{C})$ over a commutative ring with unit, which is
built by the classical Orlik--Solomon algebra and a monoidal cosheaf
$\mathcal{C}$ as coefficients. Furthermore, we construct a monoidal cosheaf
$\hat{\mathcal{C}}(\mathcal{A})$ associated with $\mathcal{A}$, so that the
generalized Orlik--Solomon algebra $A^*(\mathfrak{L},
\hat{\mathcal{C}}(\mathcal{A}))$ becomes a double complex with suitable
multiplication structure and the associated total complex
$Tot(A^*(\mathfrak{L}, \hat{\mathcal{C}}(\mathcal{A})))$ is a differential
algebra. Our main result is that $H^*(Tot(A^*(\mathfrak{L},
\hat{\mathcal{C}}(\mathcal{A}))))$ is isomorphic to
$H^*(\mathcal{M}(\mathcal{A}))$ as algebras. Our argument is of topological
with the use of a spectral sequence induced by a geometric filtration
associated with $\mathcal{A}$. In particular, we also discuss the mixed Hodge
complex structure on our model if $M$ and all elements in $\mathcal{A}$ are
complex smooth varieties, and show that it induces the canonical mixed Hodge
structure of $\mathcal{M}(\mathcal{A})$. As an application, we calculate the
cohomology of chromatic configuration spaces, which agrees with many known
results in some special cases. In addition, some explicit formulas with respect
to Poincar\'e polynomial and chromatic polynomial are also given.
