Comparing monomorphisms and epimorphisms in pro and pro\(^*\)-categories (Q946608)

The motivation of this paper comes from shape theory. The author studies the monomorphisms and the epimorphisms in pro and pro\(^*\)-categories. There are canonical functors from a category to the corresponding pro-category of inverse systems, and then to the corresponding pro\(^*\)-category of inverse systems with a larger class of morphisms. Characterizations of monomorphisms (resp. epimorphisms) in the pro\(^*\)-category are given provided that the base category admits sums (resp. products). It is also discussed when a monomorphism (resp. an epimorphism) of the pro-category is still a monomorphism (resp. an epimorphism) of the pro\(^*\)-category. The author answers this question affirmatively if the base category admits sums (resp. products), and even proves that there is an equivalence. The author also shows that a morphism of the base category is a monomorphism (resp. an epimorphism) if and only if it gives rise to a monomorphism (resp. an epimorphism) in the pro-category, and if and only if it gives rise to a monomorphism (resp. an epimorphism) in the pro\(^*\)-category, without additional conditions on the base category.