Any news about equivalences of periodic triangulated or $\infty$-categories? There is a very old question (October 2009) Equivalence of derived categories which is not Fourier-Mukai which has been bumped by improving links to the literature in one of the answers and attracted my attention.
That question is about bounded derived categories, and that answer about extensions of Orlov's description of equivalences via Fourier-Mukai transforms, but I thought - nowadays I see lots of activity around various kinds of periodic derived categories, so maybe there is some recent progress about description of their equivalences? In particular, what might play the rôle of Fourier-Mukai transforms in this context?
Here I must confess that I should probably first place a question about what precisely is a periodic derived category. There is another (less old, March 2016) question N-periodic derived categories related to this that is still unanswered, so I put a bounty there.
For purposes of this question, you may assume that I have in mind either of the following (ordered by my increasing ignorance):

*

*derived category of (unbounded but) $n$-periodic complexes;

*orbit category, as described by Keller in "On triangulated orbit categories" (2005, Documenta Vol. 10  pp. 551-581, Corrigendum)

*one of the categories exhibiting $v_n$-periodicities in the stable homotopy context, as appearing in the work of Bousfield, Franke, Patchkoria, ...

*possibly also some of the categories related to matrix factorizations but I don't even know whether they are triangulated

What is known about equivalences between such categories?
Another old (May 2013) unanswered question that seems to be relevant: How do I find abelian subcategories of periodic triangulated categories?
Also probably relevant question: 2-limits of triangulated categories. The answer by Dan Petersen to the latter suggests that it could be more natural to place my question in the context of $\infty$-categories, so please imagine a relevant entry added to the list above (I am not even competent enough to formulate it properly in that context).
 A: $\newcommand{\QCoh}{\mathrm{QCoh}} \newcommand{\Mod}{\mathrm{Mod}} \newcommand{\LMod}{\mathrm{LMod}} \newcommand{\spec}{\mathrm{Spec}} \newcommand{\Fun}{\mathrm{Fun}} \newcommand{\Z}{\mathbf{Z}} \newcommand{\cC}{\mathcal{C}} \newcommand{\Pic}{\mathrm{Pic}} \newcommand{\Eoo}{\mathbf{E}_\infty}$Let $k$ be a commutative ring. One can form a simplicial $k$-algebra $k[t_n^{\pm 1}]$, where $t_n$ lives in homological degree $n$. If $n$ is odd, it won't be commutative if $k$ isn't of characteristic $2$, since otherwise the Koszul sign rule kicks in to imply that $2t_n^2 = 0$. If $n$ is even, it is a simplicial commutative $k$-algebra, and it is a simplicial commutative $k$-algebra if $2=0$ in $k$.
(There are variants of all of this over general $\Eoo$-rings $k$, but this story is more subtle and encounters interesting topological phenomena; since this question doesn't have the algebraic-topology tag, I'll not talk about this.) Let $\LMod_{k[t_n^{\pm 1}]}$ denote the $\infty$-category of left modules over $k[t_n^{\pm 1}]$ in simplicial commutative $k$-algebras. The category $\LMod_{k[t_n^{\pm 1}]}$ won't have a monoidal structure for odd $n$ (but it does acquire one if $n$ is even). The unit $k \to k[t_n^{\pm 1}]$ gives a functor $\Mod_k \to \LMod_{k[t_n^{\pm 1}]}$.
Now let $X$ be a $k$-scheme, and let $\QCoh(X)$ denote the $k$-linear $\infty$-category obtained by viewing $X$ as a simplicial $k$-scheme (so its homotopy category is the unbounded quasicoherent derived category). Then one can define the $n$-periodified module category $\QCoh(X)^{[n]}$ to be $\QCoh(X) \otimes_{\Mod_k} \LMod_{k[t_n^{\pm 1}]}$ (see Lurie's comment in one of the linked questions). It will be convenient to not worry about the case of odd $n$, so let me assume $n$ is even. Then $\QCoh(X)^{[n]}$ can be rewritten as $\QCoh(X \times_{\spec(k)} \spec(k[t_n^{\pm 1}]))$. (An alternative definition that one might have given is that $\QCoh(X)^{[n]}$ is the fixed points of the $n$-fold iterate of the shift/suspension functor on $\QCoh(X)$; this action can be thought of as a symmetric monoidal functor $\Z \to \QCoh(X)$ which sends $1$ to $\mathcal{O}_X[n]$. But it is not quite formal that this agrees with the definition given above, even in the case when $X = \spec(k)$. And again, things become much more subtle/interesting in the setting of structured ring spectra.)
The question is now: suppose $X$ and $Y$ are $k$-schemes; what is $\Fun_{\Mod_{k[t_n^{\pm 1}]}}(\QCoh(X)^{[n]}, \QCoh(Y)^{[n]})$? (This denotes the $\infty$-category of $k[t_n^{\pm 1}]$-linear colimit-preserving functors.) Suppose that $\QCoh(X)$ is self-dual as a $k$-linear $\infty$-category (see Ben-Zvi + Francis + Nadler, where they call $X$ "perfect" if this condition holds; also Section 9.4.3 of SAG). Then $\QCoh(X)^{[n]}$ is self-dual as a $\Mod_{k[t_n^{\pm 1}]}$-module $\infty$-category, so we have
$$\Fun_{\Mod_{k[t_n^{\pm 1}]}}(\QCoh(X)^{[n]}, \QCoh(Y)^{[n]}) \simeq \QCoh(X)^{[n]} \otimes_{\Mod_{k[t_n^{\pm 1}]}} \QCoh(Y)^{[n]} \\
\simeq \QCoh((X\times_{\spec(k)} Y) \times_{\spec(k)} \spec(k[t_n^{\pm 1}])) \simeq \QCoh(X\times_{\spec(k)} Y)^{[n]}.$$
These are just $n$-periodic quasicoherent sheaves on $X\times_{\spec(k)} Y$.
Of course, you didn't need to start life with a scheme over $k$: if $\cC$ is a self-dual $k[t_n^{\pm 1}]$-linear $\infty$-category, and $\cC'$ is any other $k[t_n^{\pm 1}]$-linear $\infty$-category, then
$$\Fun_{\Mod_{k[t_n^{\pm 1}]}}(\cC, \cC') \simeq \cC \otimes_{\Mod_{k[t_n^{\pm 1}]}} \cC'.$$
The identifications are, as you expect, via Fourier-Mukai transforms.
This latter equivalence lets you work with matrix factorization categories, for example. In this case, $n=-2$, and $t_{-2}$ is understood as the generator of $\mathrm{H}^2(\mathbf{C}P^\infty; k)$. Namely, $k[t_{-2}]$ is to be understood the homotopy fixed points for the trivial $S^1$-action on $k$, and $k[t_{-2}^{\pm 1}]$ is the Tate construction for this $S^1$-action. (Technical warning: it's not true that $k[t_{-2}]$ can be identified with $C^\ast(\mathbf{C}P^\infty; k)$ if $k$ isn't of characteristic zero.) Both the $2$-periodic and $4$-periodic stories (as well as the $1$-periodic story when $k$ is of characteristic $2$) can be understood via $S^1$- and $\mathrm{SU}(2)$-actions (resp. $\Z/2$-actions in the $1$-periodic case) in this way.
