  We develop sheaf-theoretic methods to deal with non-smooth objects in
symplectic geometry. We show the completeness of a derived category of sheaves
with respect to the interleaving distance and construct a sheaf quantization of
a Hamiltonian homeomorphism. We also develop Lusternik--Schnirelmann theory in
the microlocal theory of sheaves. With these new sheaf-theoretic methods, we
prove an Arnold-type theorem for the image of a compact exact Lagrangian
submanifold under a Hamiltonian homeomorphism.
