  How does one measure the failure of Hochschild homology to commute with
colimits?
  Here I relate this question to a major open problem about dynamics in contact
manifolds -- the assertion that Reeb orbits exist and are detected by
symplectic homology. More precisely, I show that for polarizably Weinstein
fillable contact manifolds, said property is equivalent to the failure of
Hochschild homology to commute with certain colimits of representation
categories of tree quivers.
  So as to be intelligible to algebraists, I try to include or black-box as
much of the geometric background as possible.
