Summary: We prove that, for nice classes of infinite-dimensional smooth groups \(G\), natural constructions in smooth topology and symplectic topology yield homotopically coherent group actions of \(G\). This yields a bridge between infinite-dimensional smooth groups and homotopy theory.
The result relies on two computations: one showing that the diffeological homotopy groups of the Milnor classifying space \(BG\) are naturally equivalent to the (continuous) homotopy groups, and a second showing that a particular strict category localizes to yield the homotopy type of \(BG\).
We then prove a result in symplectic geometry: these methods are applicable to the group of Liouville automorphisms of a Liouville sector. The present work is written with an eye toward Y.-G. Oh and H. L. Tanaka [“Continuous and coherent actions on wrapped Fukaya categories”, Preprint, arXiv:1911.00349], where our constructions show that higher homotopy groups of symplectic automorphism groups map to Fukaya-categorical invariants, and where we prove a conjecture of Teleman from the 2014 ICM in the Liouville and monotone settings.