We prove that, for nice classes of infinite-dimensional smooth groups
, natural
constructions in smooth topology and symplectic topology yield homotopically coherent group
actions of
.
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
are
naturally equivalent to the (continuous) homotopy groups, and a second showing
that a particular strict category localizes to yield the homotopy type of
.
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 Oh and Tanaka (2019), 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.