Hopfian and Co-Hopfian objects

• Autores: K. Varadarajan
• Localización: Publicacions matematiques, ISSN 0214-1493, Vol. 36, Nº 1, 1992, págs. 293-317
• Idioma: inglés
• Títulos paralelos:
  • Objetos hopfianos y co-hopfianos
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• Resumen

  • The aim of the present paper is to study Hopfian and Co-Hopfian objects in categories like the category of rings, the module categories A-mod and mod-A for any ring A. Using Stone's representation theorem any Boolean ring can be regarded as the ring A of clopen subsets of compact Hausdorff totally disconnected space X. It turns out that the Boolean ring A will be Hopfian (resp. co-Hopfian) if and only if the space X is co-Hopfian (resp. Hopfian) in the category Top. For any compact Hausdorff space X let CR(X) (resp. CC(X)) denote the R (resp. C)-algebra of real (resp. complex) valued continuous functions on X. Using Gelfand's representation theorem we will prove that CR(X) (CC(X)) is Hopfian (respectively co-Hopfian) as an R(C) algebra if and only if X is co-Hopfian (respectively Hopfian) as an object of Top. We also study Hopfian and co-Hopfian compact topological manifolds.