Categorical aspects of the theory of quasi-uniform spaces

Abstract

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