Re: Functorial injective hulls

[Jiri Adamek <adamek@iti.cs.tu-bs.de>]

Bill's question concerning minimal functorial injective extensions
seems very interesting. Bill's comment was:
 
>  But by contrast, functorial injective resolutions do exist, usually
> by some sort of double-dualisation monad. What if the "hull" or minimality
> requirement is imposed on the process qua functor instead of at each
> object? Do such functors exist ?
 
I have two different answers:
1. NO in case of Pos (and order-embeddings): there does not exist a minimal
pair (F,f) consisting of an endofunctor  F  of Pos whose values are
complete lattices and a natural transformation  f: Id -> F  whose
components are order-embeddings
2. YES in case of Set (and monomorphisms): the embedding  Id -> Id + K,
where  K  is the constant functor with value 1 , is minimal.



xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
alternative e-mail address (in case reply key does not work):
J.Adamek@tu-bs.de
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx