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from James Borger’s comment
But I am sure that there’s a rich, rich overlap between the higher-categorical world and the biring/plethory world. Just to mention one data point, in Drinfeld’s recent paper “Prismatization”, he explains how ring stacks give rise to cohomology theories. He’s particularly interested in prismatic cohomology, but crystalline, de Rham, Dolbeault are super fun baby cases. A ring stack is, sort of by definition, just Spec of a derived biring which is concentrated in two degrees. So “slightly categorified birings” = de-Rham-like cohomology theories.
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