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.
@article{urn:eudml:doc:41149,
title = {Hopfian and co-Hopfian objects.},
journal = {Publicacions Matem\`atiques},
volume = {36},
year = {1992},
pages = {293-317},
zbl = {0792.16009},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41149}
}
Varadarajan, Kalathoor. Hopfian and co-Hopfian objects.. Publicacions Matemàtiques, Tome 36 (1992) pp. 293-317. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41149/