# Why do we care about $(\infty,2)$-categories?

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?

• 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. Aug 9, 2021 at 19:00
• 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. Aug 10, 2021 at 11:06
• @IgorSikora I've crossposted it there, thansk for the tip Aug 12, 2021 at 15:56
• The cross-posted question on MO Aug 19, 2021 at 23:29