rectification; Abhyankar-Sathaye embedding problem; variable model categories; deformation theory; deformation problem; differential graded algebras; differential graded Lie algebra; Maurer-Cartan equations; Maurer-Cartan modulus gauge action; Maurer-Cartan elements; Tate-Quillen resolution; Palamodov resolvent; Tate-Quillen resolution; cofibrant replacement; projective model structure 