Please use this identifier to cite or link to this item: http://hdl.handle.net/10077/4338

 Title: Categorical aspects of the theory of quasi-uniform spaces Authors: Brümmer, G.C.L. Keywords: functorial quasi-uniformityprereflectionBicompletionepireflectionwell-monotone quasi-uniformitysobrification Issue Date: 1999 Publisher: Università degli Studi di Trieste. Dipartimento di Scienze Matematiche Citation: G.C.L., "Brümmer Categorical aspects of the theory of quasi-uniform spaces", in: Rendiconti dell’Istituto di Matematica dell’Università di Trieste. An International Journal of Mathematics, 30 (1999) suppl., pp. 45-74. Series/Report no.: Rendiconti dell’Istituto di Matematica dell’Università di Trieste. An International Journal of Mathematics30 (1999) suppl. 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. URI: http://hdl.handle.net/10077/4338 ISSN: 0049-4704 MS Classification: 54B3054E1518A40 Appears in Collections: Rendiconti dell'Istituto di matematica dell'Università di Trieste: an International Journal of Mathematics vol.30 (1999) s.

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