Homotopy theory provides much motivation for studying $(\infty,1)$-categories in their relations to homotopical algebra, derived geometry, stable homotopy stuffs, cohomology, physics, and so on. As for $2$-categories, one doesn't even have to motivate them since they're all over the place.

However, I'm having a hard time motivating myself to study $(\infty,2)$-categories. I've been learning a bunch of facts about them: how the Duskin nerve can be regarded as an embedding from bicategories to the complicial sets model, how the Lack-Paoli nerve can be regarded as an embedding to a "simplicially enriched model", but in the end I can't see why we would want to deal with $(\infty,2)$-categories in the first place.

All I've seen so far is their use in low dimensional TQFT, and as a way to comprise the $(\infty,2)$-category of $(\infty,1)$-categories, both in the context of specific models as well as in $\infty$-cosmological contexts.

So (do we care, and if so) why do we care about $(\infty,2)$-categories?

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    $\begingroup$ I mean, if you think 2-categories are everywhere, then you should probably think $(\infty,2)$-categories are everywhere for the same reasons. Any general situation that produces a 2-category when done to sets or categories produces an $(\infty,2)$-category when done to spaces or $\infty$-categories. $\endgroup$ Aug 9 '21 at 19:00
  • 1
    $\begingroup$ As this is quite a nice research-level question, I would suggest to move it to MathOverflow - there might be people working in that stuff who will be able to answer this question. $\endgroup$ Aug 10 '21 at 11:06
  • $\begingroup$ @IgorSikora I've crossposted it there, thansk for the tip $\endgroup$ Aug 12 '21 at 15:56
  • $\begingroup$ The cross-posted question on MO $\endgroup$
    – Lee Mosher
    Aug 19 '21 at 23:29

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