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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
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
$\Pi_X(X \in \mathbf{H}/X) \simeq \Pi(X \in \mathbf{H}) \,.$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.
I canâ€™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).
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