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?