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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeJan 5th 2011
• (edited Nov 8th 2012)

have tried to brush-up the entry locally infinity-connected (infinity,1)-topos.

Kicked out a bunch of material that we had meanwhile copied over to their dedicated entries and tried to organize the remaining material a bit better. Need to work on locally infinity-connected site

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeJan 6th 2011

have added at locally infinity-connected (infinity,1)-topos the observation that the potentially two notions of geometric fundamental $\infty$-groupoid of an object in such a beast agree

$\Pi_X(X \in \mathbf{H}/X) \simeq \Pi(X \in \mathbf{H}) \,.$
• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeNov 8th 2012

added the remark (and simple proof) here that the 1-topos underlying a (locally/globally/strongly) $\infty$-connected $\infty$-topos is itself a (locally/globally/strongly) connected topos.

• CommentRowNumber4.
• CommentAuthorDavidCarchedi
• CommentTimeJun 10th 2020

I can’t seem to find the thread for the page this has now been merged with (locally n-connected (n+1,1)-topos), but Example 3.4 is wrong (or at least its proof is): The subsite on contractible opens is not closed under finite limits, so it does not necessarily yield the same infinity-topos of sheaves unless you take hypersheaves (unless you have a proof of hypercompletion in this setting).

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeJun 11th 2020

Hi David,

thanks for the alert. Right, that is the kind of gap that you and Marc had been addressing back then. I have been trying to fix this where I could, but missed it here and likely elsewhere. I have fixed it in the entry now in a cheap way by just demanding hypercompleteness (here).