Algebraic models of non-connected spaces and homotopy theory of   $L_\infty$ algebras
  We develop a homotopy theory of $L_\infty$ algebras based on the Lawrence-Sullivan construction, a complete differential graded Lie algebra which, as we show, satisfies the necessary properties to become the right cylinder in this category. As a result, we obtain a general procedure to algebraically model the rational homotopy type of non-connected spaces. 
Urtzi Buijs and Aniceto Murillo
math.AT