Direct Reflections

Authors: Brümmer, G.C.L.1; Giuli, E.2; Holgate, D.B.3

Source: Applied Categorical Structures, Volume 8, Number 3, September 2000 , pp. 545-558(14)

Publisher: Springer

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Abstract:

A pointed endofunctor (and in particular a reflector) (R, r) in a category X is direct iff for each morphism f : X → Y the pullback of R f against r_Y exists and the unique fill-in morphism u from X to the pullback is such that R u is an isomorphism. (This is close to the concept of a simple reflector introduced by Cassidy, Hébert and Kelly in 1985.) We give sufficient conditions for directness, and for directness to imply reflectivity. We also relate directness to perfect morphisms, and we give several examples and counterexamples in general topology.

Keywords: Samuel compactification; direct reflection; perfect morphism; topological bicompletion

Document Type: Regular Paper

Affiliations: 1: Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch 7701, South Africa, e-mail: gclb@maths.uct.ac.za 2: Dipartimento di Matematica Pura ed Applicata, Università degli Studi di L'Aquila, via Vetoio, loc. Coppito, 67100 L'Aquila, Italia, e-mail: Giuli@aquila.infn.it 3: Departement Wiskunde, Universiteit van Stellenbosch, Stellenbosch 7600, South Africa, e-mail: DH2@land.sun.ac.za

Publication date: September 1, 2000

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