A Lemma for Microlocal Sheaf Theory in the $\infty$-Categorical Setting

Abstract

The microlocal sheaf theory of Kashiwara and Schapira (Sheaves on manifolds, Grundlehren der mathematischen Wissenschaften 292, Springer, Berlin, 1990) makes essential use of an extension lemma for sheaves due to Kashiwara, and this lemma is based on a criterion of the same author giving conditions in order that a functor defined in ${\mathbb{R}}^{\mathrm{op}}$ with values in the category Sets of sets be constant. In the first part of this paper, using classical tools, we show how to generalize the extension lemma to the case of the unbounded derived category. In the second part, we extend Kashiwara's result on constant functors by replacing the category Sets with the $\infty$-category of spaces and apply it to generalize the extension lemma to $\infty$-sheaves, the $\infty$-categorical version of sheaves. Finally, we define the micro-support of sheaves with values in a stable $(\infty ,1)$-category.