  We prove that dg manifolds of finite positive amplitude, i.e. bundles of
positively graded curved $L_\infty[1]$-algebras, form a category of fibrant
objects. As a main step in the proof, we obtain a factorization theorem using
path spaces. First we construct an infinite-dimensional factorization of a
diagonal morphism using actual path spaces motivated by the AKSZ construction.
Then we cut down to finite dimensions using the Fiorenza-Manetti method. The
main ingredient in our method is the homotopy transfer theorem for curved
$L_\infty[1]$-algebras. As an application, we study the derived intersections
of manifolds.
