$$L_{\infty }$$ rational homotopy of mapping spaces

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In this paper we describe explicit \(L_\infty \) algebras modeling the rational homotopy type of any component of the spaces \(\text{ map}(X,Y)\) and \(\text{ map}^*(X,Y)\) 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 \(L_\infty \) algebras model the universal covers of the corresponding mapping spaces.