I will present joint work in progress with Chenchang Zhu in which we study the relationship between the integration of Lie $n$-algebras/algebroids and differentiation of Lie $n$-groups/groupoids, in analogy with Lie's Second Theorem. A crucial first step (arXiv:1609.01394) involves constructing a user--friendly homotopy theory for Lie $n$-groupoids. This is a subtle problem, due to the fact that the category of manifolds lacks limits. I will describe how results of Behrend and Getzler can be generalized to develop a homotopy theory for Lie $n$-groups/groupoids that is compatible with the homotopy theory of Lie $n$-algebras/algebroids. In particular, we show that Henriques' integration functor sends quasi-isomorphisms between Lie $n$-algebras to weak equivalences between their corresponding Lie $n$-groups.