L.S. category of the total space in a fibration and k-monomorphisms

Let Fι↪Ef↠B be a fibration, we focus on the relation between the homotopy invariants cat E, cat F and cat B. The general formula due to Hardie says that cat E⩽(cat ι+1)·(cat f+1)−1; if the fibration is trivial, this upper bound can be replaced by cat F+cat B. In this paper, we introduce a notion of k-triviality and construct upper bound for cat E in terms of k, cat ι and cat f.