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nLab > Latest Changes: discrete fibration

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    • CommentRowNumber1.
    • CommentAuthorzskoda
    • CommentTimeOct 7th 2010
    • (edited Oct 7th 2010)

    In discrete fibration I added a new section on the Street’s definition of a discrete fibration from AA to BB, that is the version for spans of internal categories. I do not really understand this added definition, so if somebody has comments or further clarifications…

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeOct 7th 2010

    The extra condition just means that the actions of A and B on the fibers of C commute with each other up to isomorphism.

    • CommentRowNumber3.
    • CommentAuthorzskoda
    • CommentTimeOct 7th 2010
    Oh, right, this makes full sense now. Thanks.
    • CommentRowNumber4.
    • CommentAuthorDavidCarchedi
    • CommentTimeApr 9th 2013

    I changed the commutative diagram for discrete fibration. The previous one was for a discrete opfibration. (I changed d 0d_0 do d 1d_1).

    • CommentRowNumber5.
    • CommentAuthorDavidCarchedi
    • CommentTimeApr 26th 2013

    I went and undid that, since it should be d 0d_0. Sorry- was mixing up d 0d_0 and d 1d_1 in my head, thinking d 1d_1 was “target”.

    • CommentRowNumber6.
    • CommentAuthorTim_Porter
    • CommentTimeApr 27th 2013

    David. been there done that, perhaps someone should produce a T-shirt! ;-)

    • CommentRowNumber7.
    • CommentAuthorTim_Porter
    • CommentTimeJul 21st 2020

    The name was duplicated as a heading. I changed this to ‘Contents’ as in other entries.

    diff, v16, current

  8. Using “maps to” instead of “assigns to”. The ambiguity of the latter may have confused some reader in revision 14.

    Anonymous

    diff, v20, current

    • CommentRowNumber9.
    • CommentAuthorvarkor
    • CommentTimeSep 22nd 2022

    Mention the relationship between representable presheaves and slice categories.

    diff, v25, current

    • CommentRowNumber10.
    • CommentAuthorBryceClarke
    • CommentTimeMay 1st 2023

    Added publication data for the reference Categorical notions of fibration.

    I think this page could probably benefit from many more references; if I remember, I will return to add some more.

    diff, v26, current

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeMay 1st 2023

    Thanks for adding references!

    I am copying every reference also to all its author-pages, now for example here.

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeMay 7th 2023

    added pointer to:

    diff, v28, current

    • CommentRowNumber13.
    • CommentAuthorBryceClarke
    • CommentTimeMay 8th 2023

    Added property that that discrete opfibrations are the right class of the comprehensive factorisation system.

    diff, v29, current

    • CommentRowNumber14.
    • CommentAuthorvarkor
    • CommentTimeDec 29th 2023

    Added a reference to unique factorization liftings.

    diff, v30, current

    • CommentRowNumber15.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMar 27th 2024

    Added:

    Model structure for discrete fibrations

    \begin{theorem} (Moser–Sarazola, Theorem 2.18.) Suppose CC is a category. The slice category Cat/CCat/C admits a combinatorial model structure with the following properties. * Cofibrations are functors that are injective on objects. * Trivial fibrations are isomorphisms. * Fibrant objects are discrete fibrations P→CP\to C. * Weak equivalences are given by morphisms whose fibrant replacement is a trivial fibration, i.e., an isomorphism. * Fibrant replacement is induced by the weak factorization system cofibrantly generated by the morphisms 1:[0]→[1]1\colon[0]\to[1], [1]⊔ [0][1][1]→[1][1]\sqcup_{[0]}[1][1]\to[1] mapping to CC in an arbitrary way. \end{theorem}

    diff, v32, current

    • CommentRowNumber16.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMar 27th 2024

    Added:

    \begin{theorem} (Moser–Sarazola, Theorem 3.9.) Suppose CC is a category. There is a Quillen equivalence

    Cat/C⇄Fun(C op,Set),Cat/C \rightleftarrows Fun(C^{op},Set),

    where Cat/CCat/C is equipped with a model structure for discrete fibrations, Fun(C op,Set)Fun(C^{op},Set) is equipped with the projective model structure (weak equivalences are isomorphisms; cofibrations and fibrations are all maps), the right adjoint Fun(C op,Set)→Cat/CFun(C^{op},Set)\to Cat/C is given by the category of elements construction, and the left adjoint adds formal strict base changes to a fibration in order to a get a strict presheaf of sets. \end{theorem}

    diff, v32, current

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