In these lecture notes by Jacob Lurie, he identifies the homotopy type of $\text{Bun}_G$ with that of a certain classifying space $B\mathcal{P}_{sm}$ when the group scheme $G$ is over $\mathbb{C}$. Does something analogous hold (e.g. for the $\ell$-adic homotopy type) when the base field is positive characteristic, i.e. $\overline{\mathbb{F}}_p$?


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