Dydak, J.Romero Ruiz del Portal, Francisco2023-06-202023-06-202007Z. Cerin, Homotopy properties of locally compact spaces at infinity—calmness and smoothness, Pacific J. Math. 79 (1978) 69–91. J. Dydak, The Whitehead and the Smale theorems in shape theory, Dissertationes Math. 156 (1979) 1–51. J. Dydak, Epimorphism and monomorphism in homotopy, Proc. Amer. Math. Soc. 116 (1992) 1171–1173. J. Dydak, F.R. Ruiz del Portal, Bimorphisms in pro-homotopy and proper homotopy, Fund. Math. 160 (1999) 269–286. J. Dydak, F.R. Ruiz del Portal, Isomorphisms in pro-categories, J. Pure Appl. Algebra 190 (2004) 85–120. J. Dydak, J. Segal, Shape Theory: An Introduction, Lecture Notes in Math., vol. 688, Springer-Verlag, 1978. E. Dyer, J. Roitberg, Homotopy-epimorphism, homotopy-monomorphism and homotopy-equivalences, Topology Appl. 46 (1992) 119–124. S. Ghorbal, Epimorphisms and monomorphisms in homotopy theory, PhD Thesis, Universitè Catholique de Louvain, 1996 (in French). T.W. Hungerford, Algebra, Springer-Verlag, New York, 1974. S. Mardesic, J. Segal, Shape Theory, North-Holland Publ. Co., Amsterdam, 1982. B. Mitchell, Theory of Categories, Academic Press, New York and London, 1965. M.A. Morón, F.R. Ruiz del Portal, On weak shape equivalences, Topology Appl. 92 (1999) 225–236. G. Mukherjee, Equivariant homotopy epimorphisms, homotopy monomorphisms and homotopy equivalences, Bull. Belg.Math.Soc. 2 (1995) 447–461.0166-864110.1016/j.topol.2006.06.009https://hdl.handle.net/20.500.14352/50343A morphism of a category which is simultaneously an epimorphism and a monomorphism is called a bimorphism. We give haracterizations of monomorphisms (respectively, epimorphisms) in pro-category pro-C, provided C has direct sums (respectively,pushouts). Let E(C) (respectively, M(C)) be the subcategory of C whose morphisms are epimorphisms (respectively, monomorphisms) of C. We give conditions in some categories C for an object X of pro-C to be isomorphic to an object of pro-E(C) (respectively,pro-M(C)). A related class of objects of pro-C consists of X such that there is an epimorphism X→P ∈ Ob(C) (respectively, a monomorphism P Ob(C) →X). Characterizing those objects involves conditions analogous (respectively, dual) to the Mittag–Leffler property. One should expect that the object belonging to both classes ought to be stable. It is so in the case of pro-groups. The natural environment to discuss those questions are balanced categories with epimorphic images. The last part of the paper deals with that question in pro-homotopy.engjournal articlehttp://www.sciencedirect.com/science/journal/01668641http://www.sciencedirect.comrestricted access515.1MonomorphismEpimorphismPro-categoriesBalanced categoriesCategories with epimorphic imagesTopología1210 Topología