  In this article, we develop a general technique for gluing subcategories of
$\infty$-categories. We obtain categorical equivalences between simplicial sets
associated to certain multisimplicial sets. Such equivalences can be used to
construct functors in different contexts. One of our results generalizes
Deligne's gluing theory developed in the construction of the extraordinary
pushforward operation in \'etale cohomology of schemes. Our results are applied
in subsequent articles to construct Grothendieck's six operations in \'etale
cohomology of Artin stacks.
