An explicit model for the homotopy theory of finite-type Lie $n$–algebras

Abstract

Lie $n$–algebras are the $L_{\infty }$ analogs of chain Lie algebras from rational homotopy theory. Henriques showed that finite-type Lie $n$–algebras can be integrated to produce certain simplicial Banach manifolds, known as Lie $\infty$–groups, via a smooth analog of Sullivan’s realization functor. We provide an explicit proof that the category of finite-type Lie $n$–algebras and (weak) $L_{\infty }$–morphisms admits the structure of a category of fibrant objects (CFO) for a homotopy theory. Roughly speaking, this CFO structure can be thought of as the transfer of the classical projective CFO structure on nonnegatively graded chain complexes via the tangent functor. In particular, the weak equivalences are precisely the $L_{\infty }$–quasi-isomorphisms. Along the way, we give explicit constructions for pullbacks and factorizations of $L_{\infty }$–morphisms between finite-type Lie $n$–algebras. We also analyze Postnikov towers and Maurer–Cartan/deformation functors associated to such Lie $n$–algebras. The main application of this work is our joint paper with C Zhu (1127–1219), which characterizes the compatibility of Henriques’ integration functor with the homotopy theory of Lie $n$–algebras and that of Lie $\infty$–groups.