  We use sheaves of spectra to quantize a Hamiltonian $\coprod\limits_n
BO(n)$-action on $\varinjlim\limits_{N}T^*\mathbf{R}^N$ that naturally arises
from Bott periodicity. We employ the category of correspondences developed in
[GaRo] to give an enrichment of stratified Morse theory by the
$J$-homomorphism. This provides a key step in the following work [Jin] on the
proof of a claim in [JiTr]: the classifying map of the local system of brane
structures on an (immersed) exact Lagrangian submanifold $L\subset
T^*\mathbf{R}^N$ is given by the composition of the stable Gauss map
$L\rightarrow U/O$ and the delooping of the $J$-homomorphism $U/O\rightarrow
B\mathrm{Pic}(\mathbf{S})$.
  We put special emphasis on the functoriality and (symmetric) monoidal
structures of the categories involved, and as a byproduct, we produce several
concrete constructions of (commutative) algebra/module objects and (right-lax)
morphisms between them in the (symmetric) monoidal $(\infty, 2)$-category of
correspondences, generalizing the construction out of Segal objects in [GaRo],
which might be of interest by its own.

Symplectic Geometry;Algebraic Topology;K-Theory and Homology