[Here](http://ncatlab.org/nlab/show/Ambidexterity+in+K(n)-Local+Stable+Homotopy+Theory) the nlab says about Hopkins-Lurie's ambidexterity paper:



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> "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]"
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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:



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> Can anyone explain the nlab's statement above?
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Ambidexterity and Quantization
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.


