Let $X$ be a path-connected nilpotent space (meaning $\pi_1(X)$ is nilpotent and acts nilpotently on the higher homotopy groups). Let $\rho\colon\thinspace\pi_1(X)\to \mathrm{Gl}(V)$ be a representation, where $V$ is a vector space over a field of characteristic zero, let's say the rationals. Then $\rho$ defines a local coefficient system on $X$, and we have the cohomology with local coefficients $H^\ast(X;\rho)$.

It is well known that Sullivan's theory of minimal models works well for nilpotent spaces, and one can (more or less) easily obtain a minimal, nilpotent cdga $M_X$ with $H(M_X)\cong H^*(X;\mathbb{Q})$, whose isomorphism type determines the rational homotopy type of $X$. (Here I am assuming some finiteness hypotheses, such as are satisfied if $X$ is a manifold or finite CW complex.)

My question is, is there a (more or less) easy way to obtain a minimal (in some sense) dg-module $M_{X,\rho}$ over $M_X$ which has $H(M_{X,\rho})\cong H^\ast(X;\rho)$ as modules over $H^\ast(X;\mathbb{Q})$?

Of course I would like to see a reference with an example worked out in detail, or an explanation of why such a construction is infeasible.

  So far I have looked at

Gómez-Tato, Antonio Théorie de Sullivan pour la cohomologie à coefficients locaux.[Sullivan's theory for cohomology with local coefficients] Trans. Amer. Math. Soc. 330 (1992), no. 1, 235–305.

and parts of Sullivan's original paper

Sullivan, Dennis Infinitesimal computations in topology. Inst. Hautes Études Sci. Publ. Math. No. 47 (1977), 269–331.

Let $X$ be a path-connected nilpotent space (meaning $\pi_1(X)$ is nilpotent and acts nilpotently on the higher homotopy groups). Let $\rho\colon\thinspace\pi_1(X)\to \mathrm{Gl}(V)$ be a representation, where $V$ is a rational vector space. Then $\rho$ defines a local coefficient system on $X$, and we have the cohomology with local coefficients $H^\ast(X;\rho)$.

It is well known that Sullivan's theory of minimal models works well for nilpotent spaces, and one can (more or less) easily obtain a minimal, nilpotent cdga $M_X$ with $H(M_X)\cong H^*(X;\mathbb{Q})$, whose isomorphism type determines the rational homotopy type of $X$. (Here I am assuming some finiteness hypotheses, such as are satisfied if $X$ is a manifold or finite CW complex.)

My question is, is there a (more or less) easy way to obtain a minimal (in some sense) dg-module $M_{X,\rho}$ over $M_X$ which has $H(M_{X,\rho})\cong H^\ast(X;\rho)$ as modules over $H^\ast(X;\mathbb{Q})$?

Ideally I would like to be able to calculate $H^1(X;\rho)$ when $X$ is a nilmanifold (so $\pi_1(X)$ is nilpotent with trivial higher homotopy groups) and calculate cup products $a_1\cup a_2\in H^2(X;\rho_1\otimes\rho_2)$. So far I have looked at

Gómez-Tato, Antonio Théorie de Sullivan pour la cohomologie à coefficients locaux.[Sullivan's theory for cohomology with local coefficients] Trans. Amer. Math. Soc. 330 (1992), no. 1, 235–305.

and parts of Sullivan's original paper

Sullivan, Dennis Infinitesimal computations in topology. Inst. Hautes Études Sci. Publ. Math. No. 47 (1977), 269–331.

Let $X$ be a path-connected nilpotent space (meaning $\pi_1(X)$ is nilpotent and acts nilpotently on the higher homotopy groups). Let $\rho\colon\thinspace\pi_1(X)\to \mathrm{Gl}(V)$ be a representation, where $V$ is a vector space over a field of characteristic zero, let's say the rationals. Then $\rho$ defines a local coefficient system on $X$, and we have the cohomology with local coefficients $H^*(X;\rho)$.

It is well known that Sullivan's theory of minimal models works well for nilpotent spaces, and one can (more or less) easily obtain a minimal, nilpotent cdga $M_X$ with $H(M_X)\cong H^*(X;\mathbb{Q})$, whose isomorphism type determines the rational homotopy type of $X$. (Here I am assuming some finiteness hypotheses, such as are satisfied if $X$ is a manifold or finite CW complex.)

My question is, is there a (more or less) easy way to obtain a minimal (in some sense) cdga $M_{X,\rho}$ which has $H(M_{X,\rho})\cong H^*(X;\rho)$?

Of course I would like to see a reference with an example worked out in detail, or an explanation of why such a construction is infeasible.

So far I have looked at

Gómez-Tato, Antonio Théorie de Sullivan pour la cohomologie à coefficients locaux.[Sullivan's theory for cohomology with local coefficients] Trans. Amer. Math. Soc. 330 (1992), no. 1, 235–305.

and parts of Sullivan's original paper

Sullivan, Dennis Infinitesimal computations in topology. Inst. Hautes Études Sci. Publ. Math. No. 47 (1977), 269–331.

Let $X$ be a path-connected nilpotent space (meaning $\pi_1(X)$ is nilpotent and acts nilpotently on the higher homotopy groups). Let $\rho\colon\thinspace\pi_1(X)\to \mathrm{Gl}(V)$ be a representation, where $V$ is a vector space over a field of characteristic zero, let's say the rationals. Then $\rho$ defines a local coefficient system on $X$, and we have the cohomology with local coefficients $H^\ast(X;\rho)$.

It is well known that Sullivan's theory of minimal models works well for nilpotent spaces, and one can (more or less) easily obtain a minimal, nilpotent cdga $M_X$ with $H(M_X)\cong H^*(X;\mathbb{Q})$, whose isomorphism type determines the rational homotopy type of $X$. (Here I am assuming some finiteness hypotheses, such as are satisfied if $X$ is a manifold or finite CW complex.)

My question is, is there a (more or less) easy way to obtain a minimal (in some sense) dg-module $M_{X,\rho}$ over $M_X$ which has $H(M_{X,\rho})\cong H^\ast(X;\rho)$ as modules over $H^\ast(X;\mathbb{Q})$?

Of course I would like to see a reference with an example worked out in detail, or an explanation of why such a construction is infeasible.

So far I have looked at

Gómez-Tato, Antonio Théorie de Sullivan pour la cohomologie à coefficients locaux.[Sullivan's theory for cohomology with local coefficients] Trans. Amer. Math. Soc. 330 (1992), no. 1, 235–305.

and parts of Sullivan's original paper

Sullivan, Dennis Infinitesimal computations in topology. Inst. Hautes Études Sci. Publ. Math. No. 47 (1977), 269–331.