A note on the $(\mathrm{\infty },n)$-category of cobordisms

In this extended note we give a precise definition of fully extended topological field theories `a la Lurie. Using complete $n$-fold Segal spaces as a model, we construct an $(\mathrm{\infty },n)$-category of $n$-dimensional cobordisms, possibly with tangential structure. We endow it with a symmetric monoidal structure and show that we can recover the usual category of cobordisms $n\mathrm{Cob}$ and the cobordism bicategory $n{\mathrm{Cob}}^{ext}$ from it.