  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.
