Absolute extensors and the geometry of multiplication of monads in the category of compacta

An investigation is made of the geometry of the multiplication mappings $\mu X$ for monads $\mathbf T=(t,\eta,\mu)$ whose functorial parts are (weakly) normal (in the sense of Shchepin) functors acting in the category of compacta. A characterization is obtained for a power monad as the only normal monad such that the multiplication mapping $\mu I^\tau$ is soft for some $\tau>\omega_1$. It is proved that the multiplication mappings $\mu_GX$ and $\mu_NX$ of the inclusion hyperspace monad and the monad of complete chained systems are homeomorphic to trivial Tychonoff fibrations for openly generated continua $X$ that are homogeneous with respect to character.