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    • CommentRowNumber1.
    • CommentAuthorzskoda
    • CommentTimeJul 19th 2011
    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJul 25th 2021
    • (edited Jun 8th 2022)

    [ this was nonsense ]

    diff, v4, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJun 8th 2022

    I have replaced the previously somewhat nonsensical Idea-sentence with something more sensible (but still vague). But mainly I have added a bunch of original and of review references.

    diff, v6, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJun 19th 2022

    added these pointers:

    diff, v8, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJun 21st 2022
    • (edited Jun 21st 2022)

    Following my own scattered remarks in another thread (here) I have now written a section – here – which means to motivate, explain and spell out what I think is a neat abstract homotopy-theoretic construction of the Gauss-Manin connection on bundles of twisted generalized cohomology groups over fibers of a locally trivial topological fiber bundle.

    This is essentially a category-theoretic elaboration on a couple of informal sentences on Gauss-Manin connections given in Etingof, Frenkel & Kirillov 1998, §7.5, all of which is rather different in style from the algebro-geometric discussion in most of the literature. While the special case here is possibly uninteresting for the algebro-geometer (?) it is rather remarkable in itself in that it yields, in particular, non-trivial solutions to the Knizhnik-Zamolodchikov equation (as recalled in that EFK98, §7.5).

    This is somewhat remarkable once you observe that the evident HoTT-formulation of the model-category theory proof spelled out here is essentially a one-line tautology (the homotopy pullbacks become just substition, the fiberwise 0-truncation requires no further comment, and the Beck-Chevalley argument is nothing but the syntactical compatibility of substitution with dependent product).

    diff, v14, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJun 22nd 2022

    for completeness, I have added a remark (here) recalling why that right base change of the fiberwise mapping space really is the space of sections expected in the definition of twisted cohomology

    diff, v15, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeJun 24th 2022
    • (edited Jun 24th 2022)

    we are making a pdf-writeup of this observation, I have added the pointer:

    diff, v21, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeFeb 1st 2024

    In the first paragraph of “For fiber bundles” (here) I have added links to 1-page pdf-s reproducing the exact pages in Voisin and Etingof et al., respectively, where they give their definition of the GM-connection.

    diff, v26, current