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added this pointer on the homotopy groups of the embedded cobordism category:
Marcel Bökstedt, Anne Marie Svane, A geometric interpretation of the homotopy groups of the cobordism category, Algebr. Geom. Topol. 14 (2014) 1649-1676 (arXiv:1208.3370)
Marcel Bökstedt, Johan Dupont, Anne Marie Svane, Cobordism obstructions to independent vector fields, Q. J. Math. 66 (2015), no. 1, 13-61 (arXiv:1208.3542)
Some good answers to this question are given as answers to MO:q/59677.
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.
Does more need to be said then about this special object, $0$? Maybe that $M + 0$ is isomorphic to $M$?
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.
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?)
Most of the literature is about the other type of cobordism category (not Stong’s). Do we have an article about the other type?
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