  In this paper we investigate the functoriality properties of map-graded
Hochschild complexes. We show that the category MAP of map-graded categories is
naturally a stack over the category of small categories endowed with a certain
Grothendieck topology of 3-covers. For a related topology of infinity-covers on
the cartesian morphisms in MAP, we prove that taking map-graded Hochschild
complexes defines a sheaf. From the functoriality related to "injections"
between map-graded categories, we obtain Hochschild complexes "with support".
We revisit Keller's arrow category argument from this perspective, and
introduce and investigate a general Grothendieck construction which encompasses
both the map-graded categories associated to presheaves of algebras and certain
generalized arrow categories, which together constitute a pair of complementary
tools for deconstructing Hochschild complexes.
