\(L_{\infty }\) models of based mapping spaces (Q548524)

Higher homotopy Lie algebras, known as strongly homotopy Lie algebras, abbreviated sh-Lie algebras and denoted by \(L_{\infty}\)-algebras, have been introduced as a generalization of Lie algebras and Lie superalgebras. \(L_{\infty}\) algebras were introduced as a model for Lie algebras that satisfy the Jacobi identity up to all higher homotopies and they reflect more accurately the homotopy type of a space. Models of a mapping space \(\text{map}^*_f(X,Y)\) have been obtained in different contexts by various authors when \(X\) is a finite type complex. In the present paper, the authors study the Lie model and \(L_{\infty}\) model of the mapping space \(\text{map}^*_f(X,Y)\) which in this case is no longer of finite type. They explicitly describe and give the Lie model and the \(L_{\infty}\) model of \(\text{map}^*_f(X,Y)\) as the suspension of Lie algebras of derivations.