Some categorical properties of the functors $O_\tau$ and $O_R$ of weakly additive functionals

In the paper, the spaces of weakly additive $\tau$-smooth and Radon functionals are investigated. It is proved that the functors of weakly additive $\tau$-smooth and Radon functionals weakly preserve the density of Tychonoff spaces, and the functor of weakly additive $\tau$-smooth functionals forms a monad in the category of Tychonoff spaces and their continuous mappings. Examples and remarks are given showing that these functors fail to satisfy certain Shchepin normality conditions. Problems having positive solutions for normal functors are presented.