Savchenko A. Metrics and uniform properties of functors in topological categories

The thesis contains construction of a fuzzy metric on countable powers and bouquets of fuzzy metric spaces. This construction is consistent with the topology of countable product and bouquet spaces respectively. In the zero-dimensional case the construction of a fuzzy metric on countable powers allows us to determine an extension operator for fuzzy metrics that preserves certain operations on them. The fuzzy counterpart of the Hausdorff metric was first defined by Romaguera and Rodrigues-Lopes. In the thesis, it is proved that the hyperspace functor generates a monad on the category of fuzzy metric spaces and nonexpanding mappings. It is shown that the G-symmetric power functor admits an extension onto the Kleisli category of the hyperspace monad (i.e. the category of fuzzy metric spaces and nonexpanding compact-valued mappings). This extension is unique. We also provide a classification of the extensions of the hyperspace functor on the Kleisli category of the hyperspace monad.