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Abstract: I will give an overview of my recent results on the category of localizing motives -- the target of the universal localizing invariant (of stable infinity-categories over some base ring), commuting with filtered colimits. In particular, I will explain the most striking property of this category: it is rigid (as a presentable symmetric monoidal category) in the sense of Gaitsgory and Rozenblyum. I will also sketch some applications of rigidity and of its proof. In particular, I will explain the corepresentability of some of the classical invariants, such as TR and TC, when restricted to connective E_1 rings. The corepresenting objects are respectively the reduced motive of the affine line and the unit object of the kernel of A^1-localization.

If time permits, I will explain how rigidity of the category of motives allows to construct a refined version of (topological) Hochschild homology. The target of this refined invariant is the so-called rigidification of the category of S^1-representations (over some base ring). This has potential applications to constructing cohomology theories of algebraic varieties.