Cauchy completion of fuzzy quasi-uniform spaces

Abstract

This paper wants to study the completion of fuzzy quasi-uniform spaces from a categorical point of view. Firstly, we introduce the concept of prorelations and describe fuzzy quasi-uniform spaces as enriched categories. Then we construct the Yoneda embedding in fuzzy quasi-uniform spaces through promodules, and prove the validness of Yoneda Lemma for right adjoint promodules. Finally, we study the Cauchy completion of fuzzy quasi-uniform spaces by the Yoneda embedding. We show that the inclusion functor from the category of  T_0 separated complete fuzzy quasi-uniform spaces to the category of fuzzy quasi-uniform spaces has a left adjoint functor. The monad related to this adjunction is just the T_0 completion monad of fuzzy quasi-uniform spaces.