Dear Colleagues,
Let A be a full isomorphism-closed coreflective subcategory of a category C, G : C -> A be a coreflection. Let (E, M) be a factorization system for C-morphisms, such that class G(E) contains class of all A-isomorphisms and is contained in class of all A-retractions. Is any of the following statements correct: 1. If functor G preserves M, then it preserves E. 2. If any M-morphism is mono, then an M-morphism belongs to Mor(A) provided that its codomain belongs to Ob(A).
John Kennison provided me with excellent counter-example to both statements. They are found in settings where M is "large" enough. Specifically, (1) is refuted as follows: Let C = Sets x {0,1} where {0,1} is the ordered category with 0 < 1. Then E = All isos of C plus all maps (f,1):(S,1) to (T,1) with f onto. And M = All maps (f,1):(S,1) to (T,1) with f one-to-one plus all maps from (S,0) And A = Sets x {0} (2) is refuted by taking an order with cardinality > 2, a minimum element, and a maximum element, for C, {min C, max C} for A, and (Iso, Mor) for factorization system on C. John suggested the simplest example, 3-chain 0 < 1 < 2. Thanks, Serge.