Isomorphisms in pro-categories.



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Dydak, J. and Romero Ruiz del Portal, Francisco (2004) Isomorphisms in pro-categories. Journal of Pure and Applied Algebra, 190 (1-3). pp. 85-120. ISSN 0022-4049

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A morphism of a category which is simultaneously an epimorphism and a monomorphism is called a bimorphism. In (Dydak and Ruiz del Portal (Monomorphisms and epimorphisms
in pro-categories, preprint)) we gave characterizations of monomorphisms (resp. epimorphisms)in arbitrary pro-categories, pro-C, where C has direct sums (resp. weak push-outs). In this paper, we introduce the notions of strong monomorphism and strong epimorphism. Part of their
signi5cance is that they are preserved by functors. These notions and their characterizations lead us to important classical properties and problems in shape and pro-homotopy. For instance,strong epimorphisms allow us to give a categorical point of view of uniform movability and
to introduce a new kind of movability, the sequential movability. Strong monomorphisms are connected to a problemof K. Borsuk regarding a descending chain of retracts of ANRs. If f : X → Y is a bimorphism in the pointed shape category of topological spaces, we prove that
f is a weak isomorphism and f is an isomorphism provided Y is sequentially movable and X or Y is the suspension of a topological space. If f : X → Y is a bimorphism in the pro-category pro-H0 (consisting of inverse systems in H0, the homotopy category of pointed connected CW
complexes) we show that f is an isomorphism provided Y is sequentially movable.

Item Type:Article
Uncontrolled Keywords:Bimorphism; Pro-category; Balanced category
Subjects:Sciences > Mathematics > Topology
ID Code:18169
Deposited On:07 Feb 2013 11:39
Last Modified:12 Dec 2018 15:13

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