Re: Question on E-coreflective subcategories

Re: Question on E-coreflective subcategories

[Serge P. Kovalyov]

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.

Question on E-coreflective subcategories

[Serge P. Kovalyov]

Dear Category Theory gurus,

In my reserach I have encountered the following problem.
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).

Examples known to me satisfy both statements, but I fail to prove any.

Thanks,
Serge.