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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMay 20th 2020
    • (edited May 20th 2020)

    added this pointer on the homotopy groups of the embedded cobordism category:

    diff, v19, current

    • CommentRowNumber2.
    • CommentAuthorJosh
    • CommentTimeMay 6th 2023
    There aren't many examples given here (or that I could find in general). Aside from the usual cobordism category of manifolds, is there anything preventing pseudomanifolds, or more generally pure simplicial complexes, from forming a cobordism category?
    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMay 6th 2023

    Some good answers to this question are given as answers to MO:q/59677.

    • CommentRowNumber4.
    • CommentAuthorperezl.alonso
    • CommentTimeAug 26th 2023

    Out of curiosity, what does the category of presheaves of a cobordism category look like? Seems like it would provide curious extensions of functorial field theories.

  1. \empty is not an initial object, as there are multiple morphisms \empty\to\empty

    Henrique Ennes

    diff, v23, current

    • CommentRowNumber6.
    • CommentAuthorDavid_Corfield
    • CommentTimeMar 28th 2024

    Does more need to be said then about this special object, 00? Maybe that M+0M + 0 is isomorphic to MM?

    • CommentRowNumber7.
    • CommentAuthorDavid_Corfield
    • CommentTimeMar 28th 2024

    Seems that what was already there was a standard definition, see e.g. this MO question. I guess the notion isn’t looking to capture categories of cobordisms.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeMar 28th 2024
    • (edited Mar 28th 2024)

    Yes, the original definition was correct: This is Stong’s notion of “cobordism categories” where the cobordisms form the objects, not the morphisms.

    I have rolled back the last edit, then rewritten the Idea-section to bring this out, and made reference to Stong more explicit.

    But I am on just on my phone in a stolen minute on a family vacation; there remains much room to expand further.

    (And is it just me or did the edit-comment-box disappear?)

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeMar 28th 2024

    Now I have an edit box, strange:

    So I have adjusted the Idea-section, moved one paragraph and added more prominent pointer to Stong 1968.

    diff, v24, current

    • CommentRowNumber10.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMar 28th 2024

    Most of the literature is about the other type of cobordism category (not Stong’s). Do we have an article about the other type?