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This might be a bit of a soft question, and I apologize in advance for this. Here it is:

What is the relationship between the "homotopical" categorification (e.g. we consider every category as an $(\infty,1)$-category, or every sheaf as a discrete $\infty$-sheaf) with the "representation-theoretic" categorification (e.g. we replace polynomials with complexes, add quantum parameters etc.)?

Ideally (for me), the answer would look like some survey-type paper with an explanation of the second phenomenon in terms of the first one and several examples. But if there is some kind of folk wisdom regarding this, I would also be delighted.

The nlab entry on this, although clearly written with something like this in mind, looks like a stub, and I can't extract the general picture from it.

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    $\begingroup$ One motivation for Khovanov-type categorification is to get nontrivial 3+1 TQFTs. The idea is that 2+1 TQFTs (more specifically, 3-2-1) are essentially the same as modular tensor categories (MTCs). This is the Reshetikhin-Turaev construction. If you try to go one dimension up you get invertible 3+1 TQFTs, which are not so interesting. However, the idea is that replacing an MTC with a 2-category gives enough room to get a non-invertible 3+1 TQFT by repeating Reshetikhin-Turaev. $\endgroup$
    – Calvin McPhail-Snyder
    Commented Apr 17, 2023 at 15:28
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    $\begingroup$ This is still a pretty imprecise explanation (hence why it's a comment not an answer) but maybe it's more in the direction you're looking for? $\endgroup$
    – Calvin McPhail-Snyder
    Commented Apr 17, 2023 at 15:28
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    $\begingroup$ The short version is that to pass to higher-dimensional TQFT you need more and more algebraic data in your input: from Froenius algebras for 1+1 dimensions to MTCs (special 2-categories with a single object) for 2+1 to more general types of 2-category for 3+1. $\endgroup$
    – Calvin McPhail-Snyder
    Commented Apr 17, 2023 at 15:30
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    $\begingroup$ @CalvinMcPhail-Snyder Yes, this seems like a great starting point! Are there some introductory references for this? $\endgroup$
    – Grisha Taroyan
    Commented Apr 17, 2023 at 18:00
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    $\begingroup$ Unfortunately I don't know any particular survey that addresses this: I've learned it as folk wisdom. I think it's sometimes called the "ladder construction". $\endgroup$
    – Calvin McPhail-Snyder
    Commented Apr 19, 2023 at 20:45

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