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  1. Creating the page, linked to from isofibration currently. Not yet finished, but contains so far the definition and some remarks on expressing it as a lifting condition. In a later edit, I will discuss the second condition, and remark on viewing Lack fibrations as ’Hurewicz fibrations’.

    v1, current

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeJul 4th 2020

    The lifting against the free-standing equivalence is not unique. Also fixed a bit of the wording around the error: no need for scare quotes since it actually was an error, and as you said it didn’t change the definition of fibration but the definition of the free-standing equivalence.

    diff, v3, current

  2. Thank you, no idea why I added unique there; writing on automatic without thinking! What I meant by the scare quotes was that one can make a definition as Lack did originally and maybe it makes sense for some things, so is not really ’wrong’ in an absolute sense; but I don’t mind removing them.

  3. I had included ’unique’ in my edit to isofibration yesterday too, removed now.

  4. Added an equivalent characterisation of the second condition in the definition of a Lack fibration.

    More to come.

    diff, v4, current

    • CommentRowNumber6.
    • CommentAuthorvarkor
    • CommentTimeDec 4th 2023

    Renamed the page to “equifibration”, which is the natural naming scheme following “isofibration” and has been used in the literature. Lack’s terminology “equiv-fibration” is still better than “Lack fibration”, but is a little unnatural.

    diff, v5, current

    • CommentRowNumber7.
    • CommentAuthorMike Shulman
    • CommentTimeMar 8th 2024

    I don’t like “equifibration” – it sounds to me like something related to an equifibered natural transformation. I suggest we go back to “equiv-fibration”.

    • CommentRowNumber8.
    • CommentAuthorvarkor
    • CommentTimeMar 9th 2024

    For what it’s worth, the previous page name was “Lack fibration” rather than “equiv-fibration”. I think “equiv-fibration” is better than “Lack fibration”, but it sounds a little unnatural to me. (I also think “equifibred natural transformation” is not a good name, and that the alternatives are better, e.g. “cartesian natural transformation”.) But I wouldn’t oppose changing the page name to “equiv-fibration” if people find that clearer.

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeMar 9th 2024

    If we stay with “equifibration” here it seems to need at least a warning that this clashes with established terminology.

    How about “2-isofibration”, instead?

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeMar 9th 2024

    Independent of the naming, I have adjusted wording and hyperlinking of the lead-in paragraph.

    diff, v6, current

    • CommentRowNumber11.
    • CommentAuthorvarkor
    • CommentTimeMar 9th 2024

    If we stay with “equifibration” here it seems to need at least a warning that this clashes with established terminology.

    Mike pointed out the term is similar to “equifibred natural transformation”, but I don’t think that it clashes with any existing terminology. (And, to clarify, “equifibration” is, itself, established terminology: it wasn’t introduced in this article.)

    How about “2-isofibration”, instead?

    I think it’s not entirely clear whether “2-isofibration” should refer to this concept, or something stricter, e.g. lifting 2-isomorphisms.

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeMar 9th 2024

    Who says “equifibration”? I see it (so far only) in Campbell 2020, Def. 4.1 (have added links to the entry). Are there other authors?

    The term “equifibered” is quite well-established for constructions making Cartesian squares (that’s how I always knew this term, but Google search confirms its prevalence).

    If “2-isofibration” is ambiguous, it is still better than a clash, I’d think. You could disambiguate by saying “weak 2-isofibration”.

    (Incidentally, the type theorist crowd has long ago discarded the category theorist’s convention that “equivalence” is specifically the categorified form of “isomorphism”.)

    diff, v7, current

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeMar 10th 2024

    Added a sentence warning about the clash of Campbell’s terminology.

    And made some minor adjustments to wording and hyperlinking, throughout

    diff, v8, current

    • CommentRowNumber14.
    • CommentAuthorMike Shulman
    • CommentTimeMar 11th 2024

    I also think that “equifibred” meaning pullback squares is significantly more common than “equifibration” meaning equivalence-lifting. So although it’s not exactly a “clash” per se, it’s potentially confusing. I could live with “weak 2-isofibration” although I would prefer “equiv-fibration”.

    Who are these type theorists who have discarded “equivalence”?

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeMar 12th 2024

    Who are these type theorists who have discarded “equivalence”?

    They call themselves “homotopy type theorists”, a strange bunch. Like to call equivalences in \infty-categories “isomorphisms” and then go further and say that “isomorphic homotopy types are equal”.

    • CommentRowNumber16.
    • CommentAuthorMike Shulman
    • CommentTimeMar 13th 2024

    I mean, can you point to a specific reference that says that? Maybe sometimes people say “isomorphic types are equal” in informal surveys, but the HoTT Book for instance uses “equivalence” for sameness of types and reserves “isomorphism” for equivalences between 0-types, and in my experience this is the most common terminology.