nLab
nCob

As a category, $n\mathrm{Cob}$ has $(n-1)$-dimensional compact oriented manifolds as objects and $n$-dimensional oriented cobordisms between these as morphisms. It plays a fundamental role in TQFT.

For some purposes, we want $n\mathrm{Cob}$ to be an n-category, but this is less well understood.

category: category