  We establish an upper bound for the cochain type level of the total space of
a pull-back fibration. It explains to us why the numerical invariant for a
principal bundle over the sphere are less than or equal to two. Moreover
computational examples of the levels of path spaces and Borel constructions,
including biquotient spaces and Davis-Januszkiewicz spaces, are presented. We
also show that the chain type level of the homotopy fibre of a map is greater
than the E-category in the sense of Kahl, which is an algebraic approximation
of the Lusternik-Schnirelmann category of the map. The inequality fits between
the grade and the projective dimension of the cohomology of the homotopy fibre.
