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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMar 14th 2012
    • (edited Mar 14th 2012)

    added to complete Segal space a discussion of what an ordinary category looks like when regarded as a complete Segal space.

    (This is meant to be pedagogical, therefore the recollection of all the basics at the beginning.)

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeDec 4th 2012

    added to the References at complete Segal space pointers to Bergner’s groupoidal version.

    • CommentRowNumber3.
    • CommentAuthoradeelkh
    • CommentTimeNov 14th 2013

    Added a remark that they are also called Rezk categories (at least by Joyal).

    • CommentRowNumber4.
    • CommentAuthoradeelkh
    • CommentTimeNov 14th 2013
    • (edited Nov 14th 2013)

    Created the stub

    and added links from Segal category and complete Segal space.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeNov 14th 2013

    Thanks! Could you please cross-link that with Segal condition? There the Segal maps appear all over the place.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeNov 14th 2013
    • (edited Nov 14th 2013)

    I did briefly cross-link now (also slightly edited your intro sentence at Segal map, please check).

    There might be more cross-linking suitable to internal category in an (infinity,1)-category and maybe also at nerve.

    • CommentRowNumber7.
    • CommentAuthorMike Shulman
    • CommentTimeNov 14th 2013

    I also like calling them “Rezk categories”. It has the double advantage of crediting Charles and indicating in the terminology that they are a kind of category.

    • CommentRowNumber8.
    • CommentAuthoradeelkh
    • CommentTimeNov 14th 2013

    Ah sorry, I hadn’t seen Segal condition.

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeNov 14th 2013

    @adeelkh: No problem, it’s good to have an entry Segal map as you created. But let’s make sure it’s cross-linked properly

    @Mike: yes, we should further push that (change of) convention on the nLab, maybe it gets to stick

    • CommentRowNumber10.
    • CommentAuthorZhen Lin
    • CommentTimeNov 14th 2013

    What do we call Segal spaces, in that case?

    • CommentRowNumber11.
    • CommentAuthorMike Shulman
    • CommentTimeNov 15th 2013

    Rezk precategories.

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeApr 7th 2014
    • (edited Apr 7th 2014)

    The question keeps circulating whether the Rezk-complete Segal space canonically obtained from a relative category is equivalent as an \infty-category to the classical simplicial localization.

    The nnLab entry complete Segal space had a link to the relevant MO discussion hidden somewhere. But since I was asked about this again now I have made the statement more explicit in a new section

    The nnLab would be the ideal place to record not just the statement but also the proof in a stable way (adapting from what Chris Schommer-Pries and Denis-Charles Cisinki once said on MO, and maybe from published material which has appared since??). But I don’t have time to edit further now.

  1. Noted the nerve formula still works for quasi-categories as well.


    diff, v45, current

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeJun 7th 2022

    cross-linked with the new entry Rezk completion

    diff, v46, current

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeMay 30th 2023

    added publication data for

    and hyperlinked the references to this item

    diff, v48, current

  2. added more reference discussing the relation with the simplicial localization.

    Kensuke Arakawa

    diff, v51, current