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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJun 23rd 2019

    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?

    diff, v219, current

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeJun 23rd 2019

    Isn’t that essentially the “locally \infty-connected” condition at infinity-connected (infinity,1)-site?

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJun 23rd 2019

    Yes, in that terminology I am asking: How strong is the condition that a cohesive \infty-topos admits any locally \infty-connected \infty-site?

  1. 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.

    • CommentRowNumber5.
    • CommentAuthorMike Shulman
    • CommentTimeJun 23rd 2019

    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.

  2. Re #5: That sounds right!

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