Recall that the *Lusternik–Schnirelmann category* (or *LS-category*) of a space is the integer $n$ such that there is an open cover by $n+1$ open sets which have nullhomotopic inclusions, and no such cover by fewer open sets. This is quite hard to compute, but for instance, spheres clearly have LS-category equal to 1.

However, sometimes one is interested not just in open covers by (relatively) contractible open sets, but by covers where all finite intersections of such opens are also (relatively) contractible. This is especially true when dealing with higher stacks, if one is interested in efficient cofibrant replacements.

Thus, one might want to define something intermediate, like level-$n$ LS-category of a space, which is one more than the minimum number of open sets one needs to cover said space, such that intersections of at most $n$ opens is (relatively) contractible. Then a finite good open cover gives an upper bound on the level-$n$ LS-category for all $n$. There is also the analogue for good open covers, namely where *all* finite intersections are (relatively) contractible.

For instance, for spheres one can take a homeomorphism with the boundary of a topological simplex, then take the open cover by puffing up a face of the simplex by some small amount (this is the analogue of covering a circle by three arcs). Thus the level-$n$ LS-category of the $k$-sphere is bounded above by $k+1$.

Has something like this been considered in the literature before?

My interest stems from considering this problem for Lie groups, in particular matrix Lie groups $O(n,\mathbb{K})$, where for certain cases we have explicit covers of the minimum size. Even something like an upper bound would be good.