Kan simplicial manifolds, also known as "Lie infinity-groupoids", are simplicial Banach manifolds which satisfy conditions similar to the horn lling conditions for Kan simplicial sets. Group-like Lie infinity-groupoids (a.k.a "Lie infinity-groups") have been used to construct geometric models for the higher stages of the Whitehead tower of the orthogonal group. With this goal in mind, Andre Henriques developed a smooth analog of Sullivan's realization functor from rational homotopy theory which produces a Lie infinity-group from certain commutative dg-algebras (i.e. L_infinity-algebras).
In this talk, I will present a homotopy theory for both these commutative dg-algebras and for Lie infinity-groups, and discuss some examples that demonstrate the compatibility between the two. Conceptually, this work can be interpreted either as a C^\infty-analog of classical results of Bouseld and Gugenheim in rational homotopy theory, or as a homotopy-theoretic analog of classical theorems from
Lie theory. This is based on joint work with A. Ozbek (UNR grad student) and C. Zhu (Gottingen).