# (∞,n)-category of triangulated cobordisms

What is an accepted definition of a (∞,n)-category of triangulated cobordisms?

Is there one that has a forgetful functor to (Rezk - Hopkins -) Lurie's smooth cobordisms? Does it shed light on how Turaev-Viro-invariants can be understood as extended TQFTs?

One naive approach would fail: We could just take the smooth cobordism category, which has as $n$-morphisms $n$-manifolds embedded into $\mathbb{R}^\infty \times \mathbb{R}^n$ (such that the projections onto the last $n$ components are Morse functions) together with $n$ tuples $(t^i_1 , \ldots t^i_{k_i}), i \in \{1\ldots n\}$ such that the projections are regular on the $t^i_j$. (I'm leaving out tons of details about the topology and the simplicial structure, but I think they aren't important here.) Now we would want to demand that the manifold be triangulated such that the preimages of the projections on the $t^i_j$ lie in the $(n-1)$-skeleton of the triangulation to assure that composition happens along triangulated $(n-1)$-manifolds.

But this is the wrong definition: If we think about e.g. 2d ETQFTs, we could assign a vector space $V$ to the interval and want a triangle with a certain orientation to be mapped to a morphism $V \otimes V \to V$, e.g. a multiplication. But this would mean that the two source edges and the target edge would share two points. Thus, there is no embedding such that the source edges lie in the preimage of a $t_0$ and the target edge lies in the preimage of a $t_1$ since there is a distance between the hyperplanes defined by the $t_i$.

I've tried to search for a good definition, but I couldn't find anything.

Note that I'm not worried about a definition for an ordinary category of triangulated cobordisms, but rather a higher categorical version that can be related to the cobordism theorem.

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I do not have any smooth cobordisms that I am aware of. – Charles Rezk Jul 25 '13 at 15:16
Sorry, I have been very superficial. I meant Lurie's smooth cobordisms that use your complete Segal spaces. – Turion Jul 25 '13 at 15:21
And do you want smooth or PL manifolds? – Theo Johnson-Freyd Jul 25 '13 at 19:12
I'm personally more interested in the PL case, but it seems most cobordism work is done in the smooth case. Also, since the smooth category and the PL category aren't equivalent for higher dimensions, probably the classification of PL ETQFTs doesn't work that smoothly? (Sorry for the pun) But a good definition for higher PL cobordisms sounds like half of the work! – Turion Jul 26 '13 at 14:43
So how about the following: Triangulated cobordisms are normal smooth cobordisms equipped with a thickening of the triangulation that is compatible with composition? (Now the only part is how to figure out which level of composition, since an n-ball can be composed in n different ways.) – Turion Jul 29 '13 at 11:14