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

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. In this paper, 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 non-negatively 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 arXiv:1609.01394 with C. Zhu which characterizes the compatibility of Henriques' integration functor with the homotopy theory of Lie $n$-algebras and that of Lie $\infty$-groups.