$L_{i}nfty$ rational homotopy of mapping spaces

In this paper we describe explicit $L_{\mathrm{\infty }}$ algebras modeling the rational homotopy type of any component of the spaces $\text{map}(X,Y)$ and ${\text{map}}^{\ast }(X,Y)$ of free and pointed maps between the finite nilpotent CW-complex $X$ and the finite type nilpotent CW-complex $Y$. When $X$ is of finite type, non necessarily finite, we also show that the algebraic covers of these $L_{\mathrm{\infty }}$ algebras model the universal covers of the corresponding mapping spaces.