Comparing monomorphisms and epimorphisms in pro and pro*-categories

Given a category pair (C,E)), where D is dense in C, the abstract coarse shape category Sh*((C,D)) was recently founded. It is realized via the category pro*-D defined on the class of all inverse systems in D. In this paper monomorphisms and epimorphisms in the category pro*-C are considered, for various categories C. The characterizations of epimorphisms (monomorphisms) in the category pro*-C are given, provided C admits products (sums). Since, one may consider the category pro-C as a subcategory of pro*-C. we discuss in which cases an epimorphism (monomorphism) in pro-C is an epimorphism (monomorphism) in pro*-C as well. We answered this question affirmatively for a category C admitting products (sums). It is shown by examples that the answer is generally negative, i.e. there exists a certain category C and an epimorphism (monomorphism) in pro-C which is not an epimorphism (monomorphism) in pro*-C. (C) 2008 Elsevier B.V. All rights reserved.