A model for function spaces

Let f : X → Y be a map between simply connected spaces and f ¯ : L ( V ) → L ( W ) its Quillen model. If the differential on L ( V ) has only linear and quadratic parts, we show that there is a Lie algebra structure on Hom T V ( T V ⊗ ( Q ⊕ s V ) , L ( W ) ) , making of it a Lie model of the function space map ( X , Y ; f ) .