Algebraic models of non-connected spaces and homotopy theory of L∞L∞ algebras

We develop a homotopy theory of L∞L∞ 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.