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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 24th 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!

    • CommentRowNumber7.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 7th 2020

    Since there’s a certain interest at the moment, I noted earlier in the article the possibility of a relative notion of cohesion, and gave an instance in Remark 2.2.

    diff, v220, current

    • CommentRowNumber8.
    • CommentAuthorRichard Williamson
    • CommentTimeApr 8th 2020
    • (edited Apr 8th 2020)

    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.

    • CommentRowNumber9.
    • CommentAuthorDavidRoberts
    • CommentTimeApr 8th 2020

    Yeah, could be a fibred topos or similar, rather than a map of toposes.

    • CommentRowNumber10.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 8th 2020

    Re #8, I was just copying Urs from back here. What are you saying it should be?

    • CommentRowNumber11.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 8th 2020
    • (edited Apr 8th 2020)

    Another case of relative cohesion we have is over Sh (Sch )Sh_\infty\left(Sch_{\mathbb{Z}}\right) at differential algebraic K-theory. I’ll add that.

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeJan 2nd 2023

    just noticed/remembered that “cohesive homotopy theory” has been redirecting directly to cohesive (infinity,1)-topos, all along. In order to offer better user experience, I am removing the redirect and will given cohesive homotopy theory a small entry of its own, which points back to cohesive (infinity,1)-topos as well as to cohesive homotopy type theory.

    diff, v229, current

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeJul 31st 2023
    • (edited Jul 31st 2023)

    added pointer (here and elsewhere) to today’s

    diff, v230, current