The $E_2$ term of the Adams spectral sequence may be identified with certain derived functors, and this also holds for a number other sequences. Our goal is to show how higher terms such sequences are determined by truncations relative defined in simplicial functors called mapping algebras

We exhibit a surprising connection between the following two concepts: Brown representability which arises in stable homotopy theory, and flat covers which arise in module theory. It is shown that Brown representability holds for a compactly generated triangulated category if and only if for every additive functor from the category of compact objects into the category of abelian groups a flat c...