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 $MC\subset\Bbbk(\g_1,\g_2)^1$ 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 Quillen’s concept of a homotopy coincides with ours. The last result was conjectured by V.Dolgushev [D].

💡 Research Summary

The paper provides a concrete construction of the homotopical category of differential graded (dg) Lie algebras originally introduced by Quillen and later refined by Hinich. The authors start by recalling the well‑known correspondence between (L_\infty) morphisms (\g_1\to\g_2) and Maurer‑Cartan (MC) elements in a certain dg Lie algebra (\mathcal{B}(\g_1,\g_2)). This dg Lie algebra is built from the Chevalley‑Eilenberg type cochains on (\g_1) with values in (\g_2); its degree‑1 part parametrises all possible higher brackets of an (L_\infty) map, while the degree‑0 part (\mathcal{B}^0) acts on MC solutions by the exponential gauge action \