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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeFeb 25th 2015

    added the pointers to the combinatorial proofs of the fiberwise detection of acyclicity of Kan fibrations, currently discussed on the AlgTop list, to the nLab here.

    • CommentRowNumber2.
    • CommentAuthorTim_Porter
    • CommentTimeMay 11th 2020
    • (edited May 11th 2020)

    Added Kan complex as a related concept.

    diff, v29, current

    • CommentRowNumber3.
    • CommentAuthorzskoda
    • CommentTimeFeb 14th 2021

    A structured/constructive analogue suited to purposes like homotopy type theory with equivalent homotopy theory is the notion of effective Kan fibration from

    • Benno van den Berg, Eric Faber, Effective Kan fibrations in simplicial sets, arXiv:2009.12670
    • Benno van den Berg, Effective Kan fibrations in simplicial sets, Bohemian Logical & Philosophical Café, Feb 2021, yt

    diff, v30, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeFeb 14th 2021

    I haven’t yet found the actual definition of “effective Kan fibration” in the article, but from the discussion around it I gather it involves equipping a Kan fibration with choices of horn fillers. That concept (Kan fibrations equipped with choices of fillers) has been discussed before, under the name algebraic Kan fibrations.

    • CommentRowNumber5.
    • CommentAuthorRichard Williamson
    • CommentTimeFeb 14th 2021
    • (edited Feb 14th 2021)

    I haven’t looked at the definition either, but I imagine it goes further than just requiring choices of horn fillers. To get things to work well constructively, one typically also needs to be able to relate the fillers in some ways, for example some kind of compatibilities under composition or at least identities. For example, in my thesis, which is constructive, a notion of ’normally cloven fibration’ is important (with respect to a cylinder/co-cylinder/interval), where there is an extra compatibility condition, not just a choice of lifts.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeApr 16th 2021

    added the example of the empty bundle (here)

    diff, v31, current

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