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Let $G$ be a finite group. It can be think of as a $1$-category with one object and $|G|$ many morphisms. If $A$ happens to be abelian, then one can think of it to an $n$-category. Conversly, this doesn't work if $G$ isn't abelian, as the Eckmann-Hilton argument shows.

As an analogy, a monoidal category can be thought as a $2$-category with one object. If it's braided, then it can be furthermore thought of as a $3$-category. In this case, the monoidal structures and the braided structures (up to suitable equivalence) can be classified by some cohomology classes.

In general, can we (and how) to classify ways to "lift" an $n$-category to an $(n+1)$-category?

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    $\begingroup$ Deloopings of an $n$-category to a 1-object $(n+1)$-category should precisely correspond to monoidal structures. In general, delooping an $n$-category $k$ times to a $(k-1)$-connected $(n+k)$-category should correspond to putting an $E_k$-monoidal structure on it. $\endgroup$ Feb 18 at 22:43
  • $\begingroup$ @AchimKrause could you provide a pointer? The tool I have for monoidal categories are called Davydov cohomology.. but I don't see how it generalizes to higher and more general categories. $\endgroup$
    – Student
    Feb 18 at 22:44
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    $\begingroup$ I only know that this is worked out for $(n, 0)$-categories, i.e. $n$-groupoids. In that context it is equivalent to the "recognition principle" for loop spaces. That's a classic result from homotopy theory, but for example it's developed in Lurie's Higher Algebra. I don't even know a good formalism for monoidal $(n, n)$-categories to make the full statement in. $\endgroup$ Feb 18 at 22:51
  • $\begingroup$ Can one parametrize the "$E_n$-monoidal" structures and obstructions by some cohomology? $\endgroup$
    – Student
    Feb 19 at 1:13
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    $\begingroup$ As I understand it Davydov-Yetter cohomology is about deformations of a given tensor category structure and not about existence, which is a very hard problem in general c.f. mathoverflow.net/a/285907/184. $\endgroup$ Feb 19 at 14:13

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