  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.
