Subject: categories: Re: Functorial injective hulls

From: Jiri Adamek <[email-masked]>
Date: Thu, 30 Mar 2000 10:56:04 +0200 (MET DST)

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.

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