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I would like to formulate the cobordism hypothesis for general tangential structure as a statement about adjoint $(\infty,1)$-functors.

For a space $Y$ with an action of $O(n)$ let $X=Y\times_{O(n)} EO_n$ and let $\zeta$ be the pullback of the universal bundle under the map $X \to BO_n$. Then I will write $\mathit{Bord}_n^{\ Y}$ for what is usually called $\mathit{Bord}_n^{(X,\zeta)}$. I believe that using the notation from Lurie's paper we then have $\widetilde{X} \simeq Y$.

Therefore Theorem 2.4.18 in the same paper states that for $\mathcal{C}$ a symmetric monoidal $(\infty,n)$-category with duals $$ \operatorname{Fun}^\otimes(\mathit{Bord}_n^{\ Y}, \mathcal{C}) \simeq \operatorname{Map}_{O(n)}(Y,\operatorname{Fun}^\otimes(\mathit{Bord}_n^{fr}, \mathcal{C})). $$ Here I used that by the cobordism hypothesis for framed manifolds we have that $$ \operatorname{Fun}^\otimes(\mathit{Bord}_n^{\ \!\operatorname{fr}}, \mathcal{C}) \xrightarrow{\ \ \sim\ \ } \mathcal{C}^{\sim}. $$ And the $O(n)$ action is the one coming from $\mathit{Bord}_n^{fr}$.

My question is now if the cobordism hypothesis implies (or is equivalent to) the statement that there is an adjunction of $(\infty,1)$-functors $$ \mathit{Bord}_n^{(\underline{\quad})}: O(n)\text{-}\mathit{Spaces} \rightleftarrows \mathit{Cat}_{(\infty,n)}^\otimes : \operatorname{Fun}^\otimes(\mathit{Bord}_n^{\ \!\operatorname{fr}}, \mathcal{C}). $$

If I'm not mistaken this should follow from the above, but I am unsure as to how many technical details one has to check to actually prove this. Has this way of stating it maybe already appeared somewhere so I could cite it?

Also, the main reason why I would like to have this statement is because I would like to know that $\mathit{Bord}_n^{(\underline{\quad})}$ preserves homotopy colimits. Is there a way to see that more immediately, not invoking the (almighty) cobordism hypothesis?

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    $\begingroup$ The only way I know of to prove that $Bord_n^{(-)}$ preserves homotopy colimits uses the cobordism hypothesis. You don't need the full version, you can prove it more or less directly using induction and Thm 3.1.8 in Lurie's paper (the "inductive" version of the cob. hypo.). It is interesting to note that this fact is only true in the fully-local case, where your bordism category is extended all the way down to points. The corresponding statement for the partially extend bordism higher category is actually false. $\endgroup$ – Chris Schommer-Pries Dec 13 '17 at 14:22
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    $\begingroup$ Note that for the failure in the non-local case that CSP mentions, you don’t need anything obscure or large, it already fails for unoriented 2-dimensional TQFTs. These are commutative Frobenius algebras with a special element x corresponding to the crosscap which satisfies an identity $x^3=x h$ where h is the handle element, and that’s not the same as being a Z/2Z homotopy fixed point. $\endgroup$ – Noah Snyder Dec 13 '17 at 14:54
  • $\begingroup$ Thank you both for the very helpful comments. What you are saying sounds to me as if the first part of my question (cobordism hypothesis implies the above adjunction) is clear to you. In that case do you know any reference stating this? I haven't found any, but I don't really understand why. Isn't formulating some statement as an adjunction considered 'nicer' than just saying there is an equivalence; or does the adjoint functor statement simply not add anything interesting, so no-one cares about writing it down? $\endgroup$ – J. Steinebrunner Dec 15 '17 at 12:06

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