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There is a class of big ∞-toposes that come with a good supply of intrinsic notions of differential geometry and differential cohomology: called cohesive ∞-toposes (after Lawvere's cohesive toposes).

One way to get hold of cohesive ∞-toposes is to construct them over suitable sites of definition, such as ∞-cohesive sites. For instance, by using variants of the site of smooth manifolds, one obtains this way ∞-toposes for things like smooth geometry, synthetic differential geometry, supergeometry and the like. An account is here. One key technical point is that smooth manifolds are, of course, locally contractible, and that the large site of all of them has a small dense subsite of contractible spaces, which is an ∞-cohesive site.

I would like to construct cohesive ∞-toposes for further kinds of geometry. Currently I am focusing on analytic geometry in the Berkovich style (as referenced for instance here). Because Berkovich has the nice result that every k-analytic space locally embeddable into a smooth space is locally contractible, with the contractible patches being directed colimits of analytic domains.

This seems to suggest that we can faithfully embed this analytic geometry (and its higher analogs) in the ∞-sheaf ∞-topos over a site that consists, maybe, of contractible ind-objects of k-analytic spaces, or something similar.

But I am just learning some basics of Berkovich theory, and this is where my question starts: can anyone help me see if there is an ∞-cohesive site (or some variant, for instance we can use hypercovers instead of the covers mentioned there) suitable for (Berkovich style) analytic geometry?

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