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 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 ∞-category of spaces and apply it to generalize the extension lemma to ∞-sheaves, the ∞-categorical version of sheaves.
Finally, we define the micro-support of sheaves with values in a stable (∞,1)-category.