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

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.

]]>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?)

]]>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.

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

]]>\empty is not an initial object, as there are multiple morphisms \empty\to\empty

Henrique Ennes

]]>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.

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

]]>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)