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nLab > Latest Changes: Kan fibration

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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

    • CommentRowNumber7.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 23rd 2022

    Added:

    The original definition is due to Daniel M. Kan, see Definition 3.1 in

    • Daniel M. Kan, A combinatorial definition of homotopy groups, The Annals of Mathematics 67:2 (1958), 282–312. doi.

    This was extended to arbitrary simplicial objects via the Yoneda embedding in Definition 3.2 of

    • Daniel M. Kan, On c.s.s. categories, Boletín de la Sociedad Matemática Mexicana 2 (1957), 82–94. PDF.

    diff, v34, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeJul 6th 2022

    I have added some basic remarks on surjective Kan fibrations (now here) to record the fact that every π 0\pi_0-epi into a Kan complex may be resolved by a surjective Kan fibration.

    diff, v35, current

    • CommentRowNumber9.
    • CommentAuthorHurkyl
    • CommentTimeJul 6th 2022
    • (edited Jul 6th 2022)

    I think you can say something stronger: if f:X→Yf : X \to Y is a Kan fibration and π 0(X)→π 0(Y)\pi_0(X) \to \pi_0(Y) is surjective, then ff is an epimorphism.

    The idea is that the equivalence relation presenting Y 0→π 0(Y)Y_0 \to \pi_0(Y) is generated by Y 1Y_1, so any two vertices in the same connected component are connected by a zigzag of edges.

    If x∈X 0x \in X_0 and y∈Y 0y \in Y_0 is any vertex in the connected component of f(x)f(x), let Z→YZ \to Y be a zigzag from f(x)f(x) to yy. Since the cofibration Δ 0→Z\Delta^0 \to Z is a weak homotopy equivalence, it lifts against ff. In particular there is an x′x' with f(x′)=yf(x') = y, meaning ff is surjective in degree 00, and thus in all degrees by lemma 5.2 of the current version.

    Assuming I’ve not made a mistake, my instinct would be to replace proposition 5.3 with this result, or maybe even to fold it into lemma 5.2. I’m not sure if that would step on what you wanted to record, though.

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeJul 6th 2022

    Good point, that’s what the entry should really say, yes. Do you want to go ahead and make the edit?

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeJul 6th 2022

    I have made the edit now.

    diff, v36, current

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeApr 20th 2023

    cross-linked (here) the discussion of surjective Kan fibrations with the respective discussion of acyclic Kan fibrations

    diff, v38, current

    • CommentRowNumber13.
    • CommentAuthorjim stasheff
    • CommentTimeOct 21st 2023
    That geometric realization takes Kan fibrations to Serre fibrations is due to

    Dan Quillen, The geometric realization of a Kan fibration is a Serre fibration, Proc. AMS 19 (1968), 1499–1500
    which is remarkably terse. Is there a more revelatory proom somwhere?
    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeOct 22nd 2023

    It’s also proven as Thm. 10.10 in Goerss & Jardine’s Simplicial Homotopy Theory.

    diff, v39, current

    • CommentRowNumber15.
    • CommentAuthorjdc
    • CommentTimeMar 20th 2024

    The current version of Riehl-Verity, v4, does not contain the proof, but it cites v2 of itself for the proof.

    diff, v40, current

    • CommentRowNumber16.
    • CommentAuthorHurkyl
    • CommentTimeMar 20th 2024

    The quasi-fibration section had an off-by-one error in the range of indices.

    diff, v41, current

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