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I forget if I ever knew the following:
What is there to the assumption that a given cohesive $\infty$-topos admits an $\infty$-site of definition all whose objects have (under Yoneda embedding) contractible shape?
Is this automatic? Is it a weak extra assumption? A strong extra assumption?
Isn’t that essentially the “locally $\infty$-connected” condition at infinity-connected (infinity,1)-site?
Yes, in that terminology I am asking: How strong is the condition that a cohesive $\infty$-topos admits any locally $\infty$-connected $\infty$-site?
My intuition such as it is is that it is quite strong. For example in algebraic geometry, affines will typically not be contractible. Even in the pro-étale topos of Scholze and Bhatt I expect that the affine line is not contractible for instance.
Re #4: …and for that reason, those toposes are not, I believe, cohesive. (-:
Re #3: C3.6.3 of the Elephant implies that any cohesive 1-topos has a locally 0-connected site, and by Prop. 1.3 of remarks on punctual local connectedness it can be taken to have finite products as well. I don’t have time to look up the proofs right now, but I would expect that they generalize at least partially to the $\infty$-case.
Re #5: That sounds right!
Thanks for adding something, David. Just a quick note that it is not really correct to say that the base topos is sheaves on profinite sets, as your notation would suggest. I was going to correct it, but was not hesistant to do so, as the way I thought to do so might change things a bit from what you had in mind.
Yeah, could be a fibred topos or similar, rather than a map of toposes.
Re #8, I was just copying Urs from back here. What are you saying it should be?
Another case of relative cohesion we have is over $Sh_\infty\left(Sch_{\mathbb{Z}}\right)$ at differential algebraic K-theory. I’ll add that.
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