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According to the Algebraic Topology discord server, both definitions have size issues (one only has condensed infinity-groupoids that are small relative to a universe/a strong limit cardinal). Other than that, the first definition (“a (infinity,1)-sheaf of infinity-groupoids on the pro-étale (infinity,1)-site of the point”) is correct, but the other definition is incorrect: the sheaf condition is wrong, it should be profinite spaces (pro-objects in finite infinity-groupoids/finite sets) rather than pro-infinity-groupoids, and a hyperdescent condition is also required.
The lead-in paragraph does not make good sense to me:
The original motivation behind condensed infinity-groupoids is to create well-behaved categories of mathematical structures such as condensed HZ-module spectra in which one could do derived analytic geometry using category-theoretic methods without resorting to not-so-well behaved categories of topological spaces.
In one reading this says that condensed $\infty$-groupoids have be introduced to define condensed module spectra, which is a circular or empty statement. The next sentence gets closer to the point with the mentioning of analytic geometry, but it remains unclear what the poor topological spaces are being blamed for.
The next sentence is alluding to what we once optimistically called condensed cohesion, but which seems to be at most condensed local contractibility. I have added a pointer to that entry now.
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