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nLab > Latest Changes: equivalence of 2-categories

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeNov 16th 2015

    I have received an email asking for clarification at the (old) entry equivalence of 2-categories, as to the meaning of “essentially full”. I have briefly added a parenthetical “i.e. essentially surjective on hom-categories”. But the entry deserves to be expanded a bit more, maybe somebody feels inspired to do so?

  2. Tweaked things slightly to emphasise that the notion of biequivalence is relevant both when one is working with weak 2-functors and when one is working with strict ones.

    diff, v14, current

    • CommentRowNumber3.
    • CommentAuthorAlexanderCampbell
    • CommentTimeSep 16th 2020
    • (edited Sep 16th 2020)

    It’s not true in general that a strict 2-functor between strict 2-categories is a biequivalence (i.e. is biessentially surjective on objects and an equivalence on hom-categories) iff it is part of an “equivalence of 2-categories” as currently defined on this page: the ’inverse’ of the 2-functor might only be a pseudofunctor, not a strict 2-functor. See Example 3.1 in Steve Lack’s paper A Quillen model structure for 2-categories.

  4. Feel free to edit the page, it could do with some re-structuring and additions; I was writing quickly earlier just to try to improve things a little. I will adjust the offending paragraph now.

  5. Now done.

    diff, v15, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeSep 28th 2021

    added the classical recognition theorem for 2-equivalences assuming AC, copied over from the same material that I just wrote into 2-functor (see the other thread)

    As a reference I have added pointer to Thm. 7.4.1 in

    but maybe this prompts somobody else to add pointer to original references, if any.

    diff, v16, current

1 to 6 of 6

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