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In the introduction to the paper Higher traces, noncommutative motives, and the categorified Chern character, Hoyois, Scherotzke and Sibilla write the following.

An important insight emerging from topology is that climbing up the chromatic ladder is related to studying invariants of spaces that are higher-categorical in nature.

The only example that I am aware of is the theory of categorified vector bundles, and its relation with chromatic-level-$2$ cohomology theories, including elliptic cohomology, and the algebraic K-theory spectrum of complex K-theory. See for instance this nLab page and the references therein.

Question. Are there other examples of this phenomenon, and is there a conceptual reason why categorification should be related to chromatic redshifting?

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The paper https://arxiv.org/abs/1312.5699 by Torlief Veen involves arbitrarily large chromatic redshifts via higher topological Hochschild homology. I know of no other examples like that.

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