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Monomorphisms and epimorphisms in pro-categories. (English) Zbl 1140.18002

For a given category \({\mathcal C}\), the authors consider the category \(\text{Pro}\,{\mathcal C}\) of inverse systems in \({\mathcal C}\). In the first part of the paper they give characterisations of monomorphisms (if \({\mathcal C}\) has direct sums) and epimorphisms (if \({\mathcal C}\) has pushouts) in \(\text{Pro}\,{\mathcal C}\). It is shown that \(\text{Pro}\,{\mathcal C}\) need not be balanced if \({\mathcal C}\) is so. Furthermore, the authors introduce a generalisation of the Mittag-Leffler property to balanced categories with epimorphic images, which is then used to study pro-objects admitting monomorphisms into stable objects, and pro-objects admitting epimorphisms from a stable object. Finally, the categories \(\text{Pro}\,{\mathcal G}r\) and \(\text{Pro}\,{\mathcal H}_0\) are studied, where \({\mathcal G}r\) denotes the category of groups and \({\mathcal H}_0\) denotes the category of pointed connected CW-complexes.

MSC:

18A20 Epimorphisms, monomorphisms, special classes of morphisms, null morphisms
18D35 Structured objects in a category (MSC2010)
54C56 Shape theory in general topology
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