  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 [OT19], 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.
