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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!
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