$\newcommand{\CH}{\mathrm{CH}} \newcommand{\F}{\mathscr{F}} $ Let $X$ be a smooth projective variety over a field $k$. Let $\F$ be a local system on $X$, i.e. a locally constant sheaf (for Zariski topology).
How can we define the Chow groups of $X$ with coefficients in $\F$ ?
Such a definition should satisfy the following conditions:
1) If $\F$ is the constant sheaf $\Bbb Z_X$ on $X$ (resp. $\Bbb Q_X$), then we should find that $\CH^j(X, \F) = \CH^j(X)$ is the usual Chow group (resp. $\CH^j(X) \otimes_{\Bbb Z} \Bbb Q$ is the Chow group taking equivalence classes of divisors with rational coefficients).
2) If $H^{\bullet}$ is a Weil cohomology theory, we should get (functorial?) cycle class maps $$\CH^j(X, \F) \to H^{2j}(X, \F).$$
I did not find any reference about such constructions. Thank you for your advice.