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nLab > Latest Changes: category object in an (infinity,1)-category

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    • CommentRowNumber1.
    • CommentAuthorMike Shulman
    • CommentTimeNov 28th 2012

    It was pointed out to me today that in the very special case of internal (0,1)-category objects in Set, what we are calling a “pre-category” reduces to a preordered set, while adding the “univalence/Rezk-completeness” condition to make it a “category” promotes it to a partially ordered set. I feel like surely I knew that once, but if so, I had forgotten. It provides some extra weight behind this term “precategory”, especially since some category theorists like to say merely “ordered set” to mean “partially ordered set”.

    • CommentRowNumber2.
    • CommentAuthorTobyBartels
    • CommentTimeNov 28th 2012

    Interesting!

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeNov 28th 2012
    • (edited Nov 28th 2012)

    In general, the sSetsSet-nerve of a category is complete Segal precisely if the only isomorphisms are identities (what’s the name again for such a category?).

    I have added a paragraph on this to Segal space – Examples – In Set

    (this could alternatively go to various other entries, but now I happen to have it there, and linked to from elsewhere).

    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeNov 28th 2012

    Sometimes it is called “rigid”.

  5. Sometimes it is called “gaunt”.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeNov 28th 2012
    • (edited Nov 28th 2012)

    Ah, of course. I knew that term of yours, I wrote all those notes on your article, after all. But I forgot. Thanks for reminding me. There is now an entry gaunt category, so that I shall never forget again.

    • CommentRowNumber7.
    • CommentAuthorMike Shulman
    • CommentTimeNov 29th 2012

    Right, thanks. That’s better than “rigid”.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeDec 3rd 2012
    • (edited Dec 3rd 2012)

    I have been further expanding the Definition-section at category object in an (infinity,1)-category, adding a tad more details concerning proofs of some of the statements. But didn’t really get very far yet.

    Also reorganized again slightly. I am afraid that in parts the notation is now slightly out of sync. I’ll get back to this later today.

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeDec 4th 2012

    I have edited still a bit further. This will probably be it for a while, unless I spot some urgent mistakes or omissions.

    • CommentRowNumber10.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 2nd 2021

    Added reference to

    • Louis Martini, Yoneda’s lemma for internal higher categories, (arXiv:2103.17141)

    diff, v56, current

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeNov 15th 2022

    added these references on development of ∞\infty-category theory internal to any (∞,1)-topos:

    internal (∞,1)-Yoneda lemma:

    internal (infinity,1)-limits and (infinity,1)-colimits:

    internal cocartesian fibrations and straightening functor:

    internal presentable (∞,1)-categories:

    diff, v58, current

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeApr 8th 2023

    added pointer to the recent:

    diff, v61, current

1 to 12 of 12

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