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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMar 9th 2012
    • (edited Nov 26th 2012)

    I have added to all Segal space-related entries, as well as to the Example section at category object in an (infinity,1)-category statements like

    That list can be further expanded. But I have to quit now.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeNov 26th 2012
    • (edited Nov 26th 2012)

    prompted by Mike’s remark here I added to Segal space a section Examples – in 1Grpd with a remark on how a Segal space in 1GrpdGrpd1Grpd \hookrightarrow \infty Grpd induces a 2-category equipped with proarrows.

    It’s not very polished and still sort of incomplete. But I need to quit now.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeNov 26th 2012
    • (edited Nov 26th 2012)

    I have now also spelled out the converse construction in Examples – From a category, spelling out how for 𝒞\mathcal{C} a category and p:𝒦𝒞p \colon \mathcal{K} \to \mathcal{C} a functor out of a groupoid, the “nn-fold comma catgeory” construction X np / nX_n \coloneqq p^{/^n} yields a Segal space X X_\bullet, which is complete if pp is the core inclusion.

    So far the writeup is a bit rambling, I am just writing stuff out as I go along. Eventually when I have a more leisure I should go an polish this. There should be, I suppose, a statement and proof that

    • a 2-coskeletal Segal space X X_\bullet in 1Grpd1Grpd is precisely the comma-Cech nerve of a functor X 0𝒞X_0 \to \mathcal{C} and is complete Segal precisely if this is the core inclusion of 𝒞\mathcal{C}, and all 2-coskeletal Segal spaces in 1Grpd1Grpd arise this way.

    (Here X X_\bullet is “kk-coskeletal” if X nX Δ nX_n \to X^{\partial \Delta^n} is a 1-monomorphism for all nk+1n \geq k+1).

    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeNov 27th 2012

    This is very nice, thanks!

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