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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.
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