I will explain recent new results about the category of localizing motives -- the target of the universal localizing invariant of stable k-linear infinity-categories (over some base k), commuting with filtered colimits. In particular, I will explain the most striking property of this category: it is rigid as a large symmetric monoidal category (in the sense of Gaitsgory and Rozenblyum).
I will also explain how to compute morphisms in this category, obtaining an effective description of the algebraic version of K-homology and more generaly of Kasparov's KK-theory. As a special case, we will deduce the corepresentability of TR (by the reduced motive of the affine line) and of the topological cyclic homology (by the unit object of the kernel of A1-localization), when restricted to the motives of connective E1-rings. Another special case is the comparison theorem of two approaches to K-theory of formal schemes: the classical continuous K-theory is equivalent to the K-theory of the category of nuclear modules, which was defined by Clausen and Scholze.
If time permits, I will explain an application to the p-adic analogue of the lattice conjecture. Namely, we construct a symmetric monoidal functor from smooth and proper dg categories over Cp to perfect modules over the p-completion of KU, with a natural map from the K(1)-local K-theory (this map is conjecturally an equivalence, but this seems to be out of reach).
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