Categorical aspects of the theory of quasi-uniform spaces

Abstract

This is a survey for the working topologist of several categorical aspects of the bicompletion of functorial quasiuniformities. We consider functors $F\::\:\mathbf{Top_{o}}\longrightarrow\mathbf{QU_{o}}$ from the $T_{o}$-topological spaces to the $T_{o}$-quasi-uniform spaces which endow the $T_{o}$-spaces with compatible quasi-unformities. Regarding the bicompletion as a functor $K:\mathbf{QU_{o}}\longrightarrow\mathbf{QU_{o}}$, we ask when the composite R=TKF is an epireflection in $\mathbf{Top_{o}}$ and when the equality KF=FR holds. Thereby we obtain analogues of important classical results from the theory of uniform spaces. We also present some new results concerning weaker versions of the above questions, e.g. when the pointed endofunctor given by TKF can be augmented to a monad. We prove that every epireflective subcategory of $\mathbf{Top_{o}}$ between the subcategory of sober spaces and the subcategory of topologically becomplete spaces can be obtained from a reflexion of the type TKF. We give full proofs of all new results and of some less known result whose proofs in the literature are in some way inaccessible. The exposition is intended for readers with little knowledge of category theory.