Topologies and sheaves on causal manifolds
Pierre Schapira
TL;DR
The paper extends KS90's equivalence between micro-supported and topological-derived categories from linear spaces to causal manifolds by developing $γ$- and $λ$-topologies on $M$ and introducing a future time function. It proves that these topologies coincide and, under a future time function, establishes an equivalence $\mathrm{D}^{\mathrm{b}}_{{γ}^{\circ a}}({\bf k}_M) \simeq \mathrm{D}^{\mathrm{b}}({\bf k}_{M_γ})$ with quasi-inverse $\varphi_γ^{-1}$, generalizing the KS framework to causal manifolds. The work builds on preorders $\preceq_{γ}$ and $\preceq_{λ}$ to analyze causal propagation and shows how the interplay between microlocal data and topology yields global equivalences, even beyond vector spaces. This provides a robust microlocal-sheaf-theoretic bridge for causal geometry and has potential implications for hyperbolic D-modules and spacetime analysis.
Abstract
A causal manifold $(M,γ)$ is a manifold $M$ endowed with a closed proper cone $γ$ in the tangent bundle $TM$ such that the projection $TM\to M$ is surjective when restricted to the interior of $γ$. Let $λ$ be the antipodal of the polar cone of $γ$. An open set $U$ of $M$ is called $γ$-open if its Whitney normal cone contains the interior of $γ$. Similarly, $U$ is called $λ$-open if the micro-support of the constant sheaf on $U$ is contained in $λ$. We begin by proving that the two notions coincide. Next, we prove that if $(M,γ)$ admits a ``future time function'' the functor of direct images establishes an equivalence of triangulated categories between the derived category of sheaves on $M$ micro-supported by $λ$ and the derived category of sheaves on the manifold $M$ endowed with the $γ$-topology. This generalizes a result of~\cite{KS90} which dealt with the case of a constant cone in a vector space.