Spaces of nonexpanding maps: categorical properties (Q1605601)

Every monad in a category generates the category of algebras (the Eilenberg-Moore category) and the Kleisli category. General problems of lifting functors to the categories of algebras and extensions of functors to the Kleisli categories were considered by many authors. Examples of solution to these problems for monads generated by functors close to being normal in the category of compact Hausdorff spaces are collected in the monograph by \textit{A. Teleiko} and \textit{M. Zarichnyi} [``Categorical topology of compact Hausdorff spaces'', Math. Stud. Monograph Series. 5. Lviv: VNTL Publishers. 263 p. (1999; Zbl 1032.54004)]. \textit{V. S. Levyts'ka} [Visn. L'viv. Univ., Ser. Mekh.-Mat. 51, 22-26 (1998; Zbl 1125.54305)] deals with the problem of extension of contravariant functors onto the Kleisli categories of a monad. A criterion of existence of such extension was proposed and applied to the case of the contravariant functor of continuous functions in the pointwise convergence topology acting in the category of Tykhonov spaces and continuous maps. The aim of this paper is to extend the results to metric categories. The authors deal with the category of compact metric spaces and nonexpanding maps.