Let $X$ be an Hausdorff space, which is locally compact and locally connected. Let $R$ be a commutative and unitary ring and let $D(X)$ be the derived category of unbounded complexes of sheaves of $R$-modules.

Question: is $D(X)$ a stable homotopy category, and is it unital algebraic in the sense of Hovey, Palmieri and Strickland (def 1.1.4 in their AMS memoir "Axiomatic Stable Homotopy Theory")?

Edit: A. Neeman proved that if $M$ is a non-compact manifold of $dim\geq 1$, and $R=\mathbb{Z}$, the only compact object of the category $D(M)$ is the zero object (see his paper "On the derived category of sheaves on a manifold"). Thus in that case $D(M)$ cannot be an algebraic stable homotopy category.