There is no monad based on Hartman-Mycielski functor

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 $HM$.

💡 Research Summary

The paper investigates whether the normal functor H, introduced by Radul as a functorial compactification of the Hartman‑Mycielski construction HM, can be equipped with a monad structure. After recalling the classical Hartman‑Mycielski construction—where for any topological space X one forms HM(X) by taking finite convex combinations of points and endowing the resulting set with a suitable topology—the author presents Radul’s functor H. For each space X, H(X) is the compactification of HM(X) obtained by completing the space of finite convex combinations under the uniform topology; for a continuous map f:X→Y, H(f) acts by sending each convex combination in H(X) to the corresponding combination of images in H(Y). Radul proved that H is a normal functor: it preserves embeddings, carries homeomorphisms to homeomorphisms, and respects limits of inverse systems.

A monad on a category consists of a functor M together with two natural transformations: the unit η:Id⇒M and the multiplication μ:M²⇒M, satisfying the usual unit and associativity axioms. The author first examines the possibility of defining a unit for H. A natural candidate is the map that sends each point x∈X to the degenerate convex combination concentrated at x; this map is continuous, natural, and indeed satisfies the unit axioms when paired with any prospective multiplication. Hence the existence of a suitable η poses no obstacle.

The main difficulty lies in constructing a multiplication μ. An element of H(H(X)) can be viewed as a finite convex combination of elements of H(X), each of which is itself a finite convex combination of points of X. To obtain an element of H(X) one would need to “flatten’’ this two‑level combination into a single level while preserving continuity and the uniform structure that defines H. The paper shows that such a flattening cannot be performed by a continuous natural transformation.

The core of the argument is a counterexample built on the discrete two‑point space X={a,b}. In H(X) each element corresponds to a weight t∈