Timeline for Can the cobordism hypothesis be formulated as a statement about adjoint functors?
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| Dec 15, 2017 at 12:06 | comment | added | Jan Steinebrunner | 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? | |
| Dec 13, 2017 at 14:54 | comment | added | Noah Snyder | 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. | |
| Dec 13, 2017 at 14:22 | comment | added | Chris Schommer-Pries | 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. | |
| Dec 13, 2017 at 11:32 | history | asked | Jan Steinebrunner | CC BY-SA 3.0 |