An explicit structure for the components of mapping spaces

In this paper, we describe an explicit L ∞ -algebra structure on the differential graded vector space of Lie γ-derivations between two differential graded Lie algebras where the source is minimal free. If γ is a Lie model for a based map between a finite simply connected complex and a nilpotent complex of finite type over Q , then the above L ∞ -algebra is a model for the corresponding component of the space of based maps.