Ambidexterity and Quantization 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?

 A: I hope someone more knowledgable about this replies.  But it seems to me the nlab comment is pointing to the remarks after equation (3.9) in [FHLT]:

To guarantee that this formula describes a well-defined functor
    from $Fam_n(C)$ to $C$, we need to make certain assumptions on $C$: namely,
    that it is additive in a strong sense which guarantees that the colimit 
    [of $\chi\colon X\to C$] exists and coincides with the limit [of $\chi$].

Their basic examples of $C$ in [FHLT] which are "additive in a strong sense" seem to be  complex vector spaces (for 1-dimensional field theories), and higher categorical analogues of complex vector spaces (for higher dimensional field theories).
The Ambidexterity paper is about constructing new examples of $(\infty,1)$-categories, with the property that all functors $\chi\colon X\to C$ are "additive in a strong sense".  Namely, $(\infty,1)$-categories of $K(h)$-local spectra (for $h$ depending on $X$).
Presumably, for [FHLT] you would really like a target $(\infty,n)$-category, not just $(\infty,1)$: Hokpins-Lurie only produces new $(\infty,1)$ examples.
A: To my understanding the situation is roughly like this. Let $\mathcal{C}$ be an $\infty$-category admiting small limits and colimits and let $f: X \to Y$ be a map of spaces whose homotopy fiber is $n$-truncated for some $n$. One can then define what it means for $f$ to be $\mathcal{C}$-ambidextrious. When $f$ is $\mathcal{C}$-ambidexterious we may construct a natural equivalence between the left Kan extension $f_!: LocSys(X,\mathcal{C}) \longrightarrow LocSys(Y,\mathcal{C})$ and the right Kan extension $f_*: LocSys(X,\mathcal{C}) \to LocSys(Y,\mathcal{C})$. Now let $A \in \mathcal{C}$ be an object and for a space $X$ let $A_X$ denote the constant local system on $X$ with value $A$. If $f: X \longrightarrow Y$ is $\mathcal{C}$-ambidextrous then we get a natural map $f_*A_X \simeq f_!A_X \simeq f_!f^*A_Y \longrightarrow A_Y$ and hence a map $\lim_X A_X \longrightarrow \lim_Y A_Y$. Such maps are sometimes called transfer maps, or "wrong-way maps". Now consider the subcategory of $Fam^{am}_1 \subseteq Fam_1$ containing all objects but only the morphisms
$$ X \longleftarrow Z \longrightarrow Y $$
such that the map $Z \longrightarrow Y$ is $\mathcal{C}$-ambidextrous. Given an object $A \in \mathcal{C}$ we may use the transfer maps above to construct a functor
$$ Fam^{am}_1 \longrightarrow \mathcal{C} $$
sending $X$ to $\lim_X A_X$. Now suppose that $\mathcal{C}$ is symmetric monoidal and that the functor $X \longrightarrow \lim_X A_X$ is monoidal (i.e., we have a Kunneth formula). If $X$ is a space such that the constant map $X \longrightarrow \ast$ and the diagonal map $X \longrightarrow X \times X$ are $\mathcal{C}$ ambidextrous then $X$ will be a dualizable object of $Fam^{am}_1$ and as a result $\lim_X A_x$ will be a dualizable object of $\mathcal{C}$, yielding a 1-dimensional topological field theory $Bord_1 \longrightarrow \mathcal{C}$ sending the point to $\lim_X A_X$. The main theorem of Lurie and Hopkins' paper on ambidexterity is that if $\mathcal{C} = Sp_{K(n)}$ is the $\infty$-category of $K_n$-local spectra then every map $f: X \longrightarrow Y$ between $\pi$-finite spaces is $Sp_{K(n)}$-ambidextrious (where $\pi$-finite here means having finitely many homotopy groups, all of which are finite). Taking $A = L_{K(n)}(S)$ to be the $K(n)$-localization of the sphere spectrum we get that the functor $X \mapsto \lim_XA_X$ is monoidal (at least when restricted to $\pi$-finite spaces). In particular, the cohomology spectrum of a $\pi$-finite space with coefficients in $L_{K(n)}(S)$ is a dualizable spectrum.
We may now take this construction one step up. Let $Fam^{am}_2 \subseteq Fam_2$ be the sub $(\infty,2)$-category which contains all objects and all $1$-morphisms, but only those $2$-morphisms for which in the highest level span $Z \longleftarrow P \longrightarrow W$ the map $P \longrightarrow W$ is $Sp_{K(n)}$-ambidextrous (this is a particular case of the construction described in Remark 4.2.5 of Lurie and Hopkins's paper on ambidexterity). As above we may use the transfer maps in order to construct a map $Fam^{am}_2 \longrightarrow St^{K(n)}_\infty$ (where $St^{K(n)}_\infty$ is a suitable $(\infty,2)$-category of stable $Sp_{K(n)}$-module $\infty$-categories) which sends an object $X$ to the $\infty$-category $LocSys(X, Sp_{K(n)})$. It can be shown that this functor is monoidal (with respect to a suitable tensor product on $St^{K(n)}_\infty$) and by the Lurie-Hopkins theorem every $\pi$-finite space is fully dualizable in $Fam^{am}_2$. It follows that if $X$ is $\pi$-finite then $LocSys(X, Sp_{K(n)})$ is fully dualizable in $St^{K(n)}_\infty$, yielding a 2-dimensional topological field theory $Bord_2 \longrightarrow St^{K(n)}_\infty$ sending the point to $LocSys(X,Sp_{K(n)})$. Presumably one can take this construction further and obtain in this way topological field theories in every dimension. For example, for every $\pi$-finite space $X$ there should be a $3$-dimensional topological field theory with values in the $(\infty,3)$-cateogry of stable $St^{K(n)}_\infty$-module $(\infty,2)$-categories sending the point to the $(\infty,2)$-category of local systems of $Sp_{K(n)}$-module stable $\infty$-cateogries.
The case discussed in the Lurie-Hopkins-Freed-Teleman paper in the section entitled "finite path integrals" roughly corresponds to the above when one replaces the spectrum K_n with the field of complex numbers.
A: 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.
