Structured flow categories and twisted presheaves

An orientation theory for flow categories without bubbling is determined by a functor of $\infty$-categories $\mu \colon \mathcal{C} \to U/O$. For any such functor, we construct a stable $\infty$-category $\mathcal{F}low^{\mu}$ of $\mu$-structured flow categories and bimodules. We also construct the expected functors between such $\infty$-categories, giving a tractable framework for manipulating orientations, local systems, and filtrations in exact Floer homotopy theory. Classifying spaces for certain bordism theories determined by $\mu$ appear as mapping spaces in $\mathcal{F}low^{\mu}$, and we use a Pontrjagin--Thom construction to naturally identify $\mathcal{F}low^{\mu}$ with the $\infty$-category of $\mu$-twisted presheaves on $\mathcal{C}$.