Relative Sullivan algebras

In §12 we showed how to model commutative cochain algebras (C,d) by Sullivan algebras, (ΛV, d). Now any cochain algebra comes equipped with the specific morphism k → (C, d), 1→ 1. In this section we introduce relative Sullivan algebras (B ⊗ ΛV,d) and use them to model general morphisms (B.d) → (C,d) of commutative cochain algebras. We shall then prove existence and uniqueness theorems in this setting. This will, in particular, eliminate the hypothesis H 1 = 0 that was frequently supposed in §12.