TITLE: Ambidexterity and Quantization
QUESTION [6 upvotes]: Here the nlab says about Hopkins-Lurie's ambidexterity paper:

"The discussion in the article is apparently motivated as part of what it takes to make precise the discussion of quantization in sections 3 and 8 of [Freed-Hopkins-Lurie-Teleman]"

I don't understand this statement, and would appreciate if someone could shed more detail on it. I only found the following links that seem to be related: http://www3.nd.edu/~cmnd/graduate/abstracts/Lurie.pdf and $\S 1.2$ of http://arxiv.org/pdf/1409.0837.pdf. The latter has some commentary that is related to the nlab's statement, but I would appreciate a more detailed explanation. My question is therefore as follows:

Can anyone explain the nlab's statement above?

REPLY [3 votes]: Charles Rezk is right in quoting that part of [FHLT], which is probably the main motivation to ambidexterity; another one might come from Lurie's program to categorify Fourier Theory (however, it would be better to ask Lurie himself).
The obstacles in Lurie's abstract are probably related to the construction of the TQFT suggested in [FHLT] and the role played into it by the Nakayama isomorphism, as spelled out in [Morton, Two-Vector Spaces and Groupoids] where the factor $|G|^{-1}$ shows up.
In particular, the Nakayama isomorphism appears when you consider the quantization functor $Sum$ from $Fam_{1}(Vect)$ to $Vect$ (or its higher analogues). Here the source category is the one defined by Haugseng, in the case when $C=Vect$. In general the functor goes from $Fam_{n}(C)$ to $C$, and it seemed that an isomorphism between right and left Kan extensions along maps of spaces is needed when defining it on morphisms.  
As DamienC said, I've been working on this in my PhD dissertation. In dimension 1 it turns out that only cocompleteness and dualizability need to be imposed on our category $C$ (completeness comes for free). 
Under these hypothesis one can construct a canonical map going from the right to the left Kan extension which is NOT (a priori) needed to be an iso, and build up the quantization functor $Sum$. 
The interesting fact, is that once you have your quantization functor, you can show that the above canonical map is indeed an isomorphism. Ambidexterity is therefore a consequence, not an ingredient one needs. 
Ideally, the same procedure will work also for $(\infty,n)$-categories, where now the hypotheses will be cocompleteness and full dualizability, and I'm currently working on this.
I've uploaded the thesis on dropbox at this address 
https://www.dropbox.com/s/hci6utz2x0g0qmm/Trova-Quantization.pdf?dl=0
Edit: an incorrect guess on the possibility of the result to hold also in positive characteristic has been removed thanks to the remark by Yonatan Harpaz here below.