Trivial fibrations of the multiplication maps for monads generated by the functors of order-preserving and positively homogeneous functionals

In this paper we further investigate the geometry of monads of order-preserving functionals and of positively homogeneous functionals. We prove that for any compactum X with $w(X) = \tau$ the map $\mu_F X$, where $F\in{O,OH}$, is homeomorphic to trivial $I^\tau$-fibration if and only if $X$ is openly generated $\chi$-homogeneous compactum.

💡 Research Summary

The paper investigates the topological structure of monads generated by two specific functors on the category of compact Hausdorff spaces: the order‑preserving functional functor (O) and the positively homogeneous functional functor (OH). For a compact space (X) with weight (w(X)=\tau), each functor yields a monad ((F, \eta_F, \mu_F)) where (\eta_F: X \to FX) is the unit and (\mu_F: F^2X \to FX) is the multiplication (or “product”) map. The central question is under what conditions the multiplication map (\mu_F X) is topologically a trivial fibration with fiber the Tychonoff cube (I^\tau =