Proper kernels in microlocal sheaf theory

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.