Proper kernels in microlocal sheaf theory
Yuxuan Hu
TL;DR
The paper addresses when the microlocal convolution with a kernel $K$ preserves compact objects between categories of sheaves with prescribed microsupports, proving a sharp criterion: $(-)\ast K$ preserves compactness iff for every $x\in X$, the slice $K|_{\{x\}\times Y}$ is compact in $\mathrm{Sh}_{\Sigma}(Y)$. It achieves this through a categorical framework: (i) Kuo–Li duality identifies cocontinuous functors with kernels; (ii) the notion of proper objects in compactly generated stable $\infty$-categories, together with strongly cocontinuous localizations, reduces the problem to $P$-constructible sheaves via exodromy (with $\mathrm{Exit}(X,P)\simeq P$); and (iii) a clean characterization in functor categories shows that, on $P$, proper objects are precisely those taking finite spectra values on each point. As a consequence, the subcategory of kernels implementing compact-preserving functors is equivalent to exact functors between the compact objects, and a practical sufficient condition—kernels with perfect stalks and compact slice supports—ensures preservation of compact objects, recovering conjectures in the literature and linking microlocal sheaf theory to wrapped Fukaya categories. The approach provides a conceptual, geometry-light proof with broad applicability to categorical microlocal analysis and its Floer-theoretic counterparts.
Abstract
Let $X$ and $Y$ be real analytic manifolds and let $Λ\subseteq T^*X$ and $Σ\subseteq T^*Y$ be closed conic subanalytic singular isotropics. Given a sheaf $K \in \mathrm{Sh}_{-Λ\times Σ}(X \times Y)$ microsupported in $-Λ\times Σ$, consider the convolution functor $(-) \ast K \colon \mathrm{Sh}_Λ(X) \rightarrow \mathrm{Sh}_Σ(Y)$ from sheaves microsupported in $Λ$ to sheaves microsupported in $Σ$. We show that the convolution functor $(-) \ast K$ preserves compact objects if and only if for each $x \in X$, the restriction $K|_{\{x\} \times Y} \in \mathrm{Sh}_Σ(Y)$ 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 $\mathrm{Sh}_{-Λ\times Σ}(X \times Y)$ of sheaves microsupported in $-Λ\times Σ$ and the category of cocontinuous functors from $\mathrm{Sh}_Λ(X)$ to $\mathrm{Sh}_Σ(Y)$. 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 $F \in \mathrm{Sh}_Λ(X)$ is proper if and only if it has perfect stalks, which is equivalent to a result of Nadler.