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nLab > Latest Changes: manifold diagram

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  1.  
    • CommentRowNumber1.
    • CommentAuthorChristoph Dorn
    • CommentTimeSep 23rd 2022

    extended stub, including: - many motivational examples - a definitional sketch

    v1, current

    • CommentRowNumber2.
    • CommentAuthorChristoph Dorn
    • CommentTimeFeb 23rd 2023

    made definition explicit and self-contained, expanded article

    diff, v3, current

    • CommentRowNumber3.
    • CommentAuthorChristoph Dorn
    • CommentTimeMar 13th 2023

    Added pointer to introductory n-category café post

    diff, v4, current

    • CommentRowNumber4.
    • CommentAuthorChristoph Dorn
    • CommentTimeMar 14th 2023

    corrected terminology

    diff, v5, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeMay 11th 2023

    Given that the terminology “framed” here clashes with its ordinary use, and given that ℝ n\mathbb{R}^n equipped with such “framing” is then called a “directed space” anyways (here), why not speak of a “direction” or the like, instead of a “framing”?

    • CommentRowNumber6.
    • CommentAuthorChristoph Dorn
    • CommentTimeMay 12th 2023

    I started writing “framed-directed” instead of “framed” in a few places where the context isn’t clear, and I hope that improves the situation a bit. “Framed-directed” can then be abbreviated to “framed” if the context is clear. The following points were considered when the terminology was chosen:

    • “directed” is a highly used term. in particular, the term “directed topological space” refers to a general idea and less so to a concrete notion. there are various ways of making that term concrete though, “framed-directed” topological spaces are one of them.
    • the notion of “framed-directedness” and “ordinary framedness” are closely related (as explained to some extent at directed topological space). for instance, every stratum in a manifold diagrams is a framed manifold (in the ordinary sense). Moreover, a “framed-directed space” XX, if XX is a manifold, is in particular a “ordinarily framed (namely, parallelizable) manifold XX”. Generally, I’d argue that “framings” always refer to “choosing directions” in one way or the other.
    • While “framed manifold” has an agreed-upon meaning, “framed (stratified) space” doesn’t yet.
    • The terminology has been used with similar meaning in the context of stratified spaces in the work of Ayala-Francis-Tanaka-Rozenblyum: namely, they speak of “vari-framings”.
    • CommentRowNumber7.
    • CommentAuthorChristoph Dorn
    • CommentTimeMay 12th 2023

    highlighting combinatorialization property

    diff, v8, current

    • CommentRowNumber8.
    • CommentAuthorChristoph Dorn
    • CommentTimeJun 3rd 2023
    • added talk reference

    diff, v9, current

1 to 8 of 8

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