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 topolo
gy 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 Kellers 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.
We define a notion of colimit for diagrams in a motivic category indexed by a presheaf of spaces (e.g. an etale classifying space), and we study basic properties of this construction. As a case study, we construct the motivic analogs of the classical
extended and generalized powers, which refine the categoric
We show that an old conjecture of A.A. Suslin characterizing the image of a Hurewicz map from Quillen K-theory in degree $n$ to Milnor K-theory in degree $n$ admits an interpretation in terms of unstable ${mathbb A}^1$-homotopy sheaves of the general
linear group. Using this identification, we establish Suslins conjecture in degree $5$ for infinite fields having characteristic unequal to $2$ or $3$. We do this by linking the relevant unstable ${mathbb A}^1$-homotopy sheaf of the general linear group to the stable ${mathbb A}^1$-homotopy of motivic spheres.
In fall of 2019, the Thursday Seminar at Harvard University studied motivic infinite loop space theory. As part of this, the authors gave a series of talks outlining the main theorems of the theory, together with their proofs, in the case of infinite
perfect fields. These are our extended notes on these talks.