  We give complete characterisation of topologically injective (bounded below),
topologically surjective (open mapping), isometric and coisometric (quotient
mapping) multiplication operators between $L_p$ spaces defined on different
$\sigma$-finite measure spaces. We prove that all such operators invertible
from the left or from the right. As the consequence we prove that all objects
of the category of $L_p$ spaces considered as left Banach modules over algebra
of bounded measurable functions are metrically, extremelly and relatively
projective, injective and flat.
