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    • CommentRowNumber1.
    • CommentAuthorMike Shulman
    • CommentTimeOct 19th 2010

    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 Π\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 Π(*)\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.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeOct 20th 2010
    • (edited Oct 21st 2010)

    Thanks, Mike!

    This does look correct. For the single possibly nonevident point– that every \infty-groupoid is the (,1)(\infty,1)-colimit over itself of the diagram constant on the point – I have added a link to this proposition.

    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeOct 21st 2010


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