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

Let ${\text{g}}_1$ and ${\text{g}}_2$ be two dg Lie algebras, then it is well-known that the $L_{\mathrm{\infty }}$ morphisms from ${\text{g}}_1$ to ${\text{g}}_2$ are in 1-1 correspondence to the solutions of the Maurer-Cartan equation in some dg Lie algebra $\mathbb{k}({\text{g}}_1,{\text{g}}_2)$. Then the gauge action by exponents of the zero degree component $\mathbb{k}({\text{g}}_1,{\text{g}}_2{)}^0$ on $MC\subset \mathbb{k}({\text{g}}_1,{\text{g}}_2{)}^1$ gives an explicit “homotopy relation” between two $L_{\mathrm{\infty }}$ morphisms. We prove that the quotient category by this relation (that is, the category whose objects are $L_{\mathrm{\infty }}$ algebras and morphisms are $L_{\mathrm{\infty }}$ 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 Quillen’s concept of a homotopy coincides with ours. The last result was conjectured by V.Dolgushev [D].