A Hamiltonian $\coprod _{n}BO(n)$-action, stratified Morse theory and the $J$-homomorphism

We use sheaves of spectra to quantize a Hamiltonian $\coprod\limits_n BO(n)$-actionon$\varinjlim\limits_{N}T^$\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}$^N$isgivenbythecompositionofthestableGaussmap$L\rightarrow U/O$andthedeloopingofthe$J$-homomorphism$U/O\rightarrow B$\mathrm{P}\mathrm{i}\mathrm{c}$($\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 $(\mathrm{\infty },2)$-category of correspondences, generalizing the construction out of Segal objects in [GaRo], which might be of interest by its own.