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nLab > Latest Changes: fundamental theorem of covering spaces

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  1.  
    • CommentRowNumber1.
    • CommentAuthorBartek
    • CommentTimeDec 21st 2021

    Added the proof of the fundamental theorem of covering spaces in HoTT

    diff, v13, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeDec 21st 2021
    • (edited Dec 21st 2021)

    For a more faithful type-theoretic reflection of traditional covering-space theory one ought to understand it as a topic in cohesive homotopy type theory:

    What deserves to be called the fundamental groupoid of a geometric homotopy type XX is not really ‖X‖ 1\Vert X \Vert_1, but ‖eshX‖ 1\Vert \esh X \Vert_1, where esh\esh is the shape modality (see at shape via cohesive path ∞-groupoid). With this, the fundamental theorem of covering space theory is a reflection of the defining adjunction esh⊣♭\esh \dashv \flat with the flat modality, which is such that ♭Sets\flat Sets is the actual classifier for covering spaces over geometric homotopy types.

    I have appended (here) a paragraph along these lines.

    diff, v14, current

    • CommentRowNumber3.
    • CommentAuthorHurkyl
    • CommentTimeDec 21st 2021

    Added another paragraph to the idea section describing this as related to the equivalence ∞Grpd /X≃∞Grpd X\infty Grpd_{/X} \simeq \infty Grpd^X.

    diff, v15, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeDec 21st 2021

    As before, since “XX” denotes a topological space in the preceding sentences, I have changed it to eshX\esh X and added brief commentary.

    diff, v16, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeDec 26th 2021

    added pointer to

    which explicitly states and proves the cohesive HoTT-version of the theorem.

    diff, v17, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJan 18th 2023

    added pointer to

    and added cross-linking with the entry on transport

    diff, v19, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeFeb 14th 2023

    added a graphics (here) illustrating path lifting in a covering space

    diff, v21, current

    • CommentRowNumber8.
    • CommentAuthorDmitri Pavlov
    • CommentTimeNov 17th 2023

    Added:

    A detailed treatment is available in

    diff, v24, current

1 to 8 of 8

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