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My apologies for coming back to this old question, but I want to address a point that I think is not really addressed so far. Namely, for $n=1$, the universal cover of a (reasonable) topological space …

As suggested by Dustin Clausen in his answer, polynomial functors induce maps on $K$-theory. In the setting of stable $\infty$-categories, you proved this in your joint work with Barwick, Glasman, and …

That's a good question! I think Barwick and Haine have thought much more about this, and maybe they already know the answer? What I say below is definitely known to them. Also beware that I've written …

Good question! Actually, it seems unlikely that perfectoid methods per se play a key role in homotopy theory. The reason is that perfectoid things are "infinitely ramified", but there are theorems to …

For a long time (and still today), I very much shared the confusion of the OP. I think Jacob Lurie gives a very clear take on the standard perspective, but Mike Shulman does have a very valid contrast …

This is an interesting question. First, I think the [PS] reference does not give the "correct" Betti stack. In my notes on 6 functors, I define a different stack $X_B$ such that $D_{\mathrm{qc}}(X_B)$ …