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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Adámek, J., Herrlich, H. and Strecker, G. E.: Abstract and Concrete Categories, Wiley, New York, 1990.
Borubaev, A. A.: Absolutes of uniform spaces, Russian Math. Surveys 43 (1988), 233–234.
Brümmer, G. C. L.: Completions of functorial topological structures, in W. Gähler et al. (eds.), Recent Developments of General Topology and its Applications, Proc. Conf. Berlin, 1992, Akademie Verlag, Berlin, 1992, pp. 60–71.
Brümmer, G. C. L.: Categorical aspects of the theory of quasi-uniform spaces, Rend. Ist. Mat. Univ. Trieste, to appear.
Brümmer, G. C. L. and Giuli, E.: A categorical concept of completion of objects, Comment. Math. Univ. Carol. 33 (1992), 131–147.
Brümmer, G. C. L. and Hager, A.W.: Completion-true functorial uniformities, Seminarberichte Fachbereich Math. Inf. FernUniv., Hagen 19 (1984), 95–104.
Clementino, M. M., Giuli, E. and Tholen, W.: Topology in a category: Compactness, Portugaliae Math. 53 (1996), 397–433.
Cassidy, C., Hébert, M. and Kelly, G. M.: Reflective subcategories, localizations and factorization systems, J. Austral. Math. Soc. Ser. A 38 (1985), 287–329.
Hager, A. W.: Perfect maps and epi-reflective hulls, Canad. J. Math. 27 (1975), 11–24.
Herrlich, H.: A generalization of perfect maps, in General Topology and its Relations to Modern Analysis and Algebra III, Symp. Prague, 1971, Academia, Prague, 1972, pp. 187–191.
Herrlich, H.: Perfect subcategories and factorisations, in Topics in Topology, Conf. Keszthely, Hungary, 1972, Colloq. Math. Soc. János Bolyai 8, 1974, pp. 387–403.
Henriksen, M. and Isbell, J. R.: Some properties of compactifications, Duke Math. J. 25 (1958), 83–106.
Holgate, D.: The pullback closure, perfect morphisms and completions, Ph.D. Thesis, University of Cape Town, 1995.
Holgate, D.: The pullback closure operator and generalisations of perfectness, Appl. Cat. Structures 4 (1996), 107–120.
Holgate, D.: Linking the closure and orthogonality properties of perfect morphisms in a category, Comment. Math. Univ. Carolin. 39 (1998), 587–607.
Nummela, E. C.: The completion of a topological group, Bull. Austral. Math. Soc. 21 (1980), 407–417.
Tholen, W.: Prereflections and reflections, Comm. Algebra 14 (1986), 717–740.
Weck-Schwarz, S.: T0–objects and separated objects in topological categories, Quaestiones Math. 14 (1991), 315–325.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Brümmer, G.C.L., Giuli, E. & Holgate, D.B. Direct Reflections. Applied Categorical Structures 8, 545–558 (2000). https://doi.org/10.1023/A:1008756203227
Issue Date:
DOI: https://doi.org/10.1023/A:1008756203227