Proper kernels in microlocal sheaf theory

TL;DR

This paper characterizes when convolution functors in microlocal sheaf theory preserve compact objects, linking geometric conditions on kernels with categorical properties, and introduces the notion of proper objects in stable infinity-categories.

Contribution

It provides a categorical classification of cocontinuous functors that preserve compact objects in microlocal sheaf categories, using minimal geometric input and new notions like proper objects.

Findings

Convolution functors preserve compactness iff kernels restrict to compact objects.

Sheaves are proper iff they have perfect stalks, aligning with Nadler's results.

The classification relies on the notion of proper objects in stable infinity-categories.

Abstract

Let $X$ and $Y$ be real analytic manifolds and let $\Lambda \subseteq T^*X$ and $\Sigma \subseteq T^*Y$ be closed conic subanalytic singular isotropics. Given a sheaf $K \in \mathrm{Sh}_{-\Lambda \times \Sigma}(X \times Y)$ microsupported in $-\Lambda \times \Sigma$, consider the convolution functor $(-) \ast K \colon \mathrm{Sh}_{\Lambda}(X) \rightarrow \mathrm{Sh}_{\Sigma}(Y)$ from sheaves microsupported in $\Lambda$ to sheaves microsupported in $\Sigma$. We show that the convolution functor $(-) \ast K$ preserves compact objects if and only if for each $x \in X$, the restriction $K|_{\{x\} \times Y} \in \mathrm{Sh}_\Sigma(Y)$ is a compact object. By a result of Kuo-Li, the functor sending a sheaf kernel $K$ to the convolution functor $(-) \ast K$ is an equivalence between the category $\mathrm{Sh}_{-\Lambda \times \Sigma}(X \times Y)$ of sheaves microsupported in $-\Lambda \times \Sigma$ and the category of cocontinuous functors from $\mathrm{Sh}_\Lambda(X)$ to $\mathrm{Sh}_\Sigma(Y)$. We therefore classify all cocontinuous functors that preserve compact objects between the two categories. Our approach is entirely categorical and requires minimal input from geometry: we introduce the notion of a proper object in a compactly generated stable infinity-category and study its properties under strongly continuous localizations to obtain the result. The main geometric input is the analysis of compact and proper objects of the category of $P$-constructible sheaves for a triangulation $P$ of a manifold $Z$ via the exit path category $\mathrm{Exit}(Z, P) \simeq P$. Along the way, we show that a sheaf $F \in \mathrm{Sh}_\Lambda(X)$ is proper if and only if it has perfect stalks, which is equivalent to a result of Nadler.