There is no monad based on Hartman-Mycielski functor

TL;DR

This paper proves that the Hartman-Mycielski functor, when extended to a monad structure, cannot exist, clarifying limitations in the functor's categorical properties.

Contribution

It demonstrates the non-existence of a monad based on Radul's Hartman-Mycielski functor, a significant insight into the functor's categorical structure.

Findings

No monad structure exists for the Hartman-Mycielski functor

Clarifies limitations of the functor's categorical properties

Provides theoretical insight into functorial compactifications

Abstract

We show that there is no monad based on the normal functor $H$ introduced earlier by Radul which is a certain functorial compactification of the Hartman-Mycielski construction $HM$.