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Lately there has been a lot of progress on the foundations of $(\infty,2)$-categories (for example, all currently-known models for them were shown to be equivalent and finally we have a construction of the Gray tensor product).

While I'm aware they are used in Gaitsgory–Rozenblyum, I don't know of any other applications (out of ignorance, not because there aren't other ones)...

  • Q$_1$: What are some current applications of $(\infty,2)$-categories (say in homotopy theory or DAG)?
  • Q$_2$: What are some expected future applications of them, say as the theory gets in a better overall shape and starts to be more widely-adopted?
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    $\begingroup$ There are a few papers about $(\infty,1)$-categories that uses $(\infty,2)$-categorical arguments. For example arxiv.org/pdf/1712.06469.pdf $\endgroup$ Oct 1, 2020 at 16:42
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    $\begingroup$ I don't agree that all currently-known models for $(\infty,2)$-categories have been shown to be equivalent. E.g. the comical sets model of Campion--Kapulkin--Maehara has not yet been shown to be equivalent to any other model, not to mention the non-homotopical models defined by Batanin and many others. $\endgroup$ Oct 1, 2020 at 19:36

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Topological field theory (TFT) is a major client of higher-dimensional category theory. For $(\infty, 2)$-categories specifically, this specializes to two-dimensional TFT. One significant research area in this field is taking physics ideas, making them mathematically rigorous, and then using the resulting mathematical objects to prove interesting theorems internal to mathematics.

For example, string theorists originally predicted a form of duality now known as “mirror symmetry,” coming out of a more general physics phenomenon called T-duality. The context is that given $X$ a Calabi-Yau $3$-fold, physicists studied a two-dimensional supersymmetric quantum field theory called "the sigma model with target $X$" and found that it admits two twists, called the A and B models, which are topological field theories (in the physicists' sense). Mirror symmetry predicts that given $X$, there is another CY3 called the mirror of $X$ and denoted $X^\vee$, such that the A model with target $X$ is equivalent to the B model with target $X^\vee$, and vice versa.

Now, because these are topological field theories, it should be possible to make sense of them mathematically: the A and B model should admit descriptions as symmetric monoidal functors from an $(\infty, 2)$-category of bordisms in dimension 2 to some target $(\infty, 2)$-category. This has amounted to a lot of hard work by many teams of researchers, but large parts of this story have been translated into mathematics, and the higher-categorical language is important here!

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    $\begingroup$ Shameless request: Could you provide some references along this line? $\endgroup$
    – Student
    Nov 25, 2021 at 10:16

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