Relative Sullivan algebra

A relative Sullivan algebra is an inclusion $(A,d)\to (A\otimes \Lambda W,D)$ with a well-ordered homogeneous basis of $W$ such that the differential of a generator lies in the algebra on $A$ and earlier generators. It need not be minimal: a differential can have linear terms from earlier generators. Such extensions supply cofibrant replacements for commutative differential graded algebras. A relative acyclic extension of a sphere model represents its path-space fibration and computes a homotopy-fibre model after base change.