A lemma for microlocal sheaf theory in the $\infty$-categorical setting

TL;DR

This paper generalizes Kashiwara's extension lemma from classical sheaf theory to the setting of $ig$-categories and $ig$-sheaves, enabling advanced microlocal sheaf theory in the $ig$-categorical context.

Contribution

It extends Kashiwara's extension lemma to unbounded derived categories and $ig$-sheaves, introducing $ig$-categorical microlocal sheaf theory.

Findings

Generalization of the extension lemma to unbounded derived categories.

Extension of Kashiwara's result to $ig$-categories of spaces.

Definition of micro-support for sheaves valued in stable $(ig,1)$-categories.

Abstract

Microlocal sheaf theory of \cite{KS90} makes an 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 $\mathbb{R}$ with values in the category $Sets$ of sets be constant. In a 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 a 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.