Rigidity of the category of localizing motives

In this paper we study the category of localizing motives ${\mathrm{Mot}}^{\mathrm{loc}}$ – the target of the universal finitary localizing invariant of idempotent-complete stable categories as defined by Blumberg-Gepner-Tabuada. We prove that this (presentable stable) category is rigid symmetric monoidal in the sense of Gaitsgory and Rozenblyum. In particular, it is dualizable. More precisely, we prove a more general version of this result for the category ${\mathrm{Mot}}_{\mathcal{E}}^{\mathrm{loc}}$ – the target of the universal finitary localizing invariant of dualizable modules over a rigid symmetric monoidal category $\mathcal{E}\mathrm{.}$ We obtain general results on morphisms and internal $\mathrm{Hom}$ in the categories ${\mathrm{Mot}}_{\mathcal{E}}^{\mathrm{loc}}$ of localizing motives. As an application we compute the morphisms in multiple non-trivial examples. In particular, we prove the corepresentability statements for $\mathrm{TR}$ (topological restriction) and $\mathrm{TC}$ (topological cyclic homology) when restricted to connective ${\mathbb{E}}_1$-rings. As a corollary, for a connective ${\mathbb{E}}_{\mathrm{\infty }}$-ring $R$ we obtain a $\mathrm{TR}(R)$-module structure on the nil $K$-theory spectrum $NK(R)\mathrm{.}$ We also apply the rigidity theorem to define refined versions of negative cyclic homology and periodic cyclic homology. This was announced previously in \cite{E24b}, and certain very interesting examples were computed by Meyer and Wagner in \cite{MW24}. Here we do several computations in characteristic $0,$ in particular showing that in seemingly innocuous situations the answer can be given by an interesting algebra of overconvergent functions.