Monomorphisms and epimorphisms in pro-categories (Q886316)

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.