Topologies and sheaves on causal manifolds

TL;DR

This paper explores the relationship between topologies and sheaves on causal manifolds, establishing an equivalence of categories under certain conditions, thus generalizing previous results in the field.

Contribution

It introduces a new equivalence of categories between sheaves micro-supported by a cone and sheaves on a $ ext{gamma}$-topology on causal manifolds, extending prior work to more general settings.

Findings

The notions of $ ext{gamma}$-open sets and $ ext{lambda}$-open sets coincide.

An equivalence of derived categories is established under the existence of a future time function.

Generalizes a classical result to causal manifolds with variable cones.

Abstract

A causal manifold $(M,\gamma)$ is a manifold $M$ endowed with a closed proper cone $\gamma$ in the tangent bundle $TM$ such that the projection $TM\to M$ is surjective when restricted to the interior of $\gamma$. Let $\lambda$ be the antipodal of the polar cone of $\gamma$. An open set $U$ of $M$ is called $\gamma$-open if its Whitney normal cone contains the interior of $\gamma$. Similarly, $U$ is called $\lambda$-open if the micro-support of the constant sheaf on $U$ is contained in $\lambda$. We begin by proving that the two notions coincide. Next, we prove that if $(M,\gamma)$ 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 $\lambda$ and the derived category of sheaves on the manifold $M$ endowed with the $\gamma$-topology. This generalizes a result of~\cite{KS90} which dealt with the case of a constant cone in a vector space.