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

Let g1 and g2 be two dg Lie algebras, then it is well-known that the L∞ morphisms from g1 to g2 are in 1 − 1 correspondence to the solutions of the Maurer-Cartan equation in some dg Lie algebra k(g1, g2). Then the gauge action by exponents of the zero degree component k(g1, g2) 0 on MC ⊂ k(g1, g2) 1 gives an explicit ”homotopy relation” between two L∞ morphisms. We prove that the quotient category by this relation (that is, the category whose objects are L∞ algebras and morphisms are L∞ 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].