engleski

Comparing monomorphisms, epimorphisms and bimorphisms in pro and pro*-categories

Given a category pair (C, D), where D is dense in C, the abstract coarse shape category Sh*(C, D)∗ was recently founded. In the most interesting case C=HTop (the homotopy category of topological spaces) and D=HPol (the homotopy category of spaces having the homotopy type of polyhedra), one obtains the coarse shape category of topological spaces Sh*≡ Sh*(HTop, HPol). Its isomorphisms induce the classification of topological spaces strictly coarser then the shape type classification and the shape category Sh can be considred as the subcategory of the Sh*. The category Sh*(C, D) is realized via the category pro*-D defined on the class of all inverse systems in D in the same way as Sh(C, D) is defined via the category pro-D. In this talk monomorphisms, epimorphisms and bimorphisms in the category pro*-D are considered, for various categories D. Since, one may consider the category pro-D as the subcategory of pro*-D, we discuss in which cases an epimorphism (monomorphism) in pro-D is an epimorphism (monomorphism) in pro*-D as well. We answered this question affirmatively for a categoriy D admitting products (sums). However, it is showed by examples that the answer is generally negative. Since the study of coarse shape isomorphisms reduces to the study of isomorphisms in the appropriate category pro*-D, in this talk, we also consider isomorphisms and bimorphisms in a category pro*-D, for various categories D. We discuss in which cases pro*-D is a balanced category (category in which every bimorphism is an isomorphism). We are interested in the question whether the fact that one of the categories: D, pro-D and pro*-D is balanced implies that the other two categories are balanced.

Category; Pro-category; Shape; Monomorphism; Epimorphism; Topological space; Polyhedron

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