Epimorphisms and maximal covers in categories of compact spaces

B. Banaschewski

Canada

McMaster University

Department of Mathematics and Statistics

A. W. Hager

United States

Wesleyan University

Department of Mathematics and Computer Science

Submitted: 2013-07-26

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Accepted: 2013-07-26

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DOI: https://doi.org/10.4995/agt.2013.1616
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Keywords:

Epimorphism, Cover, Projective, Essential extension, Compact, Strongly rigid

Supporting agencies:

This research was not funded

Abstract:

The category C is "projective complete"if each object has a projective cover (which is then a maximal cover). This property inherits from C to an epireflective full subcategory R provided the epimorphisms in R are also epi in C. When this condition fails, there still may be some maximal covers in R. The main point of this paper is illustration of this in compact Hausdorff spaces with a class of examples, each providing quite strange epimorphisms and maximal covers. These examples are then dualized to a category of algebras providing likewise strange monics and maximal essential extensions.
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References:

J. Adamek, H. Herrlich and G. Strecker, Abstract and Concrete Categories, Dover 2009.

B. Banaschewski, Projective covers in categories of topological spaces and topological algebras, pp. 63-91 in General Topology and its Relations to Modern Analysis and Algebra, Academia 1971.

B. Banaschewski and A. Hager, Essential completeness of archimedean â„“-groups with weak unit, to appear.

H. Cook, Continua which admit only the identity mapping onto non-degenerate subcontinua, Fund. Math. 60 (1966) 214-249. https://doi.org/10.4064/fm-60-3-241-249

R. Engelking, General Topology, Heldermann 1989.

A. Gleason, Projective topological spaces, Ill. J. Math. 2 (1958), 482-489. https://doi.org/10.1215/ijm/1255454110

L. Gillman and M. Jerison, Rings of Continuous Functions, Springer-Verlag 1976.

A. Hager, Minimal covers of topological spaces, Ann. NY Acad. Sci. 552 (1989), 44-59. https://doi.org/10.1111/j.1749-6632.1989.tb22385.x

A. Hager and J. Martinez, Singular archimedean lattice-ordered groups, Alg. Univ. 40 (1998), 119-147. https://doi.org/10.1007/s000120050086

A. Hager and L. Robertson, Representing and ringifying a Riesz space, Symp. Math. XXI (1977), 411-431.

H. Herrlich and G. Strecker, Category Theory, Allyn and Bacon 1973.

J. Isbell, A closed non-reflective subcategory of compact spaces, Manuscript c. 1971.

J. Kennison, Reflective functors in general topology and elsewhere, Trans. Amer. Math. Soc. 118 (1965), 303-315. https://doi.org/10.1090/S0002-9947-1965-0174611-9

J. Porter and R. G. Woods, Extensions and Absolutes of Hausdorff Spaces, Springer-Verlag 1988. https://doi.org/10.1007/978-1-4612-3712-9

V. Trnková, Non-constant continuous mappings of metric or compact Hausdorff spaces, Comm. Math. Univ. Carol. 13 (1972), 283-295.

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