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I split off ∞-connected (∞,1)-topos from locally ∞-connected (∞,1)-topos and added a proof that a locally ∞-connected (∞,1)-topos is ∞-connected iff the left adjoint $\Pi$ preserves the terminal object, just as in the 1-categorical case. I also added a related remark to shape of an (∞,1)-topos saying that when H is locally ∞-connected, its shape is represented by $\Pi(*)$.
I hope that these are correct, but it would be helpful if someone with a little more $\infty$-categorical confidence could make sure I’m not assuming something that doesn’t carry over from the 1-categorical world.
Thanks, Mike!
This does look correct. For the single possibly nonevident point– that every $\infty$-groupoid is the $(\infty,1)$-colimit over itself of the diagram constant on the point – I have added a link to this proposition.
Thanks.
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