Derived, coderived, and contraderived categories of locally presentable abelian categories

For a locally presentable abelian category B with projective generator, we construct the derived and contraderived model structures on of complexes, proving in particular existence enough homotopy complexes objects. We also show that D(B) is generated, as triangulated coproducts, by generator B. Grothendieck A, injective coderived complexes. Assuming Vopěnka's principle, prove D(A) products, cogenerator A. More generally, define notion an exact object size function any such κ-directed colimits chains admissible monomorphisms has Hom sets. In particular,