\(A_{n}\) theory, L.S. category, and strong category (Q2384740)

For every space \(X\) there is a Ganea-Svarz fibration \((\Omega X)^{*(n+1)}\to B_n\Omega X \to X\), and it is clear that the functor \(B_n\Omega\) is a comonad. The author introduces the notion of a weak homotopy space over \(B_n\Omega\) and proves that if \(X\) admits such a structure, then cat(\(X\))=Cat(\(X\)) under some dimensional and connectivity restriction on \(X\).