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

[ this was nonsense ]

• 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.

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeJun 18th 2022

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTime6 days ago
• (edited 6 days ago)

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).

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTime5 days ago

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

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTime3 days ago
• (edited 3 days ago)

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