An explicit construction of the Quillen homotopical category of dg Lie algebras

Let $g_1$ and $g_2$ be two dg Lie algebras, then it is well-known that the $L_infty$ morphisms from $g_1$ to $g_2$ are in 1-1 correspondence to the solutions of the Maurer-Cartan equation in some dg Lie algebra $Bbbk(g_1,g_2)$. Then the gauge action by exponents of the zero degree component $Bbbk(g_1,g_2)^0$ on $MCsubsetBbbk(g_1,g_2)^1$ gives an explicit homotopy relation between two $L_infty$ morphisms. We prove that the quotient category by this relation (that is, the category whose objects are $L_infty$ algebras and morphisms are $L_infty$ morphisms modulo the gauge relation) is well-defined, and is a localization of the category of dg Lie algebras and dg Lie maps by quasi-isomorphisms. As localization is unique up to an equivalence, it is equivalent to the Quillen-Hinich homotopical category of dg Lie algebras [Q1,2], [H1,2]. Moreover, we prove that the Quillens concept of a homotopy coincides with ours. The last result was conjectured by V.Dolgushev [D].