Let \(f:X\to Y\) be a map between nilpotent CW-complexes of finite type and \(\text{map}^*_f(X,Y)\) the path-component of the space of based maps from \(X\) to \(Y\) containing the map \(f\). Let \(C\) be a finite type graded differential coalgebra model of \(X\) and let \(M\) be a Lie algebra model of \(Y\). Let \({\mathcal L}(C)\) denote the Quillen functor on \(C\) and let \(\gamma^{\prime}:{\mathcal L}(C) \to M\) be a model for \(f\). Then it is known that the graded vector space \(s^{-1}Der_{\gamma^{\prime}}({\mathcal L}(C),M)\) of derivations becomes a Lie model of \(\text{map}^*_f(X,Y)\). Similarly, let \({\mathcal Q}\) denote the Quillen minimal model of \(X\) and let \(\gamma :{\mathcal Q}\to M\) be any Lie model of \(f\). Then it is also known that there is a natural \(L_{\infty}\)-structure on \(s^{-1}Der_{\gamma}({\mathcal Q},M)\) encoding the higher Whitehead products on the rational homotopy groups of \(\text{map}^*_f(X,Y)\).
As the first step, the authors prove that the two \(L_{\infty}\)-algebra \(s^{-1}Der_{\gamma^{\prime}}({\mathcal L}(C),M)\) and \(s^{-1}Der_{\gamma}({\mathcal Q},M)\) are quasi-isomorphic.
Next, they consider the Lawrence-Sullivan completed Lie model of the interval \({\mathcal I}\) and they prove that \(s^{-1}Der ({\mathcal I},M)\) and \(s^{-1}Der({\mathcal L}(C),M)\) are quasi-isomorphic as \(L_{\infty}\)-algebras for any differential graded Lie algebra \(M\). In particular, they show that \(s^{-1}Der ({\mathcal I},M)\) is an \(L_{\infty}\)-model of the based path space \(PY=\text{map}^*(I,Y)\) for any Lie model \(M\) of a nilpotent complex \(Y\).
Finally, let \(\chi :L\to M\) be a differential Lie algebra homomorphism and \(C_{\chi}\) be its suspended mapping cone. Then they prove that \(C_{\chi}\) and \(L\times_Ms^{-1}Der ({\mathcal I},M)\) are strongly isomorphic as \(L_{\infty}\) algebras, and they obtain an explicit and close relation between the Lawrence-Sullivan model for the interval and the Fiorenza-Manetti mapping cone.