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