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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