Homological triviality of the category of $L_p$

The paper presents a complete description of topologically injective, topologically surjective, isometric and coisometric multiplication operators by a function acting between $L_p$ spaces of $\sigma$-finite measure spaces. It is proved that all such operators are invertible from the right and left. As a corollary, it is proved that in the category consisting of $L_p$-spaces with all $p\in[1,+\infty]$ considered as left Banach modules over the algebra of bounded measurable functions, all objects are metrically and topologically projective, injective, and flat.