# Hyper-derived functors and Cartan-Eilenberg resolutions

I'm confused by the significance of Cartan-Eilenberg resolutions when constructing hyper-derived functors.

Let $$F$$ be a right-exact functor, and let $$A^\bullet$$ be a chain complex. According to this Wikipedia page, the left hyper-derived functor of $$F$$ on a chain complex $$A^\bullet$$ can be constructed by taking a Cartan-Eilenberg resolution of projective objects $$P^{\bullet,\bullet}$$ of $$A^\bullet$$, applying $$F$$ to $$P^{\bullet,\bullet}$$ and take cohomology of the total complex.

On the other hand, according to this Wikipedia page, the left hyper-derived functor of $$F$$ on $$A^\bullet$$ is given by $$H^i(F(P^\bullet))$$, where $$P^\bullet \to A^\bullet$$ is any quasi-isomorphism.

Question. What is the significance/advantage of using a Cartan-Eilenberg resolution instead of an arbitrary quasi-isomorphism?

Comments, hints, and references are welcome.

To use arbitrary quasi-isomorphisms $$P^\bullet \to A^\bullet$$, where $$P^\bullet$$ is a complex of projectives, you first need to show that such quasi-isomorphisms exist. That's what the Cartan-Eilenberg resolution accomplishes in the case when $$A^\bullet$$ is bounded above: it gives you a double complex of projectives such that its total complex is the complex $$P^\bullet$$ quasi-isomorphic to $$A^\bullet$$. Compare it to how we first show that any module over a commutative ring has a free resolution, even though we then use arbitrary projective resolutions in some computations.
The fact that a Cartan-Eilenberg resolution also induces resolutions of the homology of $$A^\bullet$$ is also sometimes useful, even though it's not needed to make sure hyperhomology is well-defined.
• @Svinto the extra properties are very, very useful. Without them you cannot conclude one of the most important properties of the Cartan-Eilenberg resolution, which is that (a) maps $f:A\to B$ of complexes always lift to maps $P\to Q$ of the Cartan-Eilenberg double complexes (b) such lifts are unique up to double-complex homotopy. Commented Dec 21, 2023 at 21:31