Cartan–Eilenberg complexes and resolutions

We study Cartan–Eilenberg projective, injective and flat complexes and show how they can be used to get Cartan–Eilenberg resolutions. We argue that every complex has a Cartan–Eilenberg injective envelope. Then we show that a complex is a Cartan–Eilenberg flat complex if and only if it is the direct limit of finitely generated Cartan–Eilenberg projective complexes.