A∞-spaces and L–S category in the category of fibrewise spaces

We give the projective spaces for an A ∞ -space in Fib B B the topological category of cell-wise trivial fibre spaces over polyhedra and obtain a relationship between a pointed L–S category and an A ∞ -space in Fib B B . As an application, we calculate the fibrewise pointed L–S category of Klein bottle.