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From: Peter Freyd <pjf@saul.cis.upenn.edu>
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Subject: More comments on Functorial injective hulls
Date: Thu, 23 Mar 2000 14:12:31 -0500 (EST)
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   Some comments on:

>Theorem. Let  H  be a class of morphisms in a category  C  such that
>1. all H-injective objects form a cogenerating class, and
>2. the class of all H-essential morphisms which are epimorphic
>	is precisely the class of isomorphisms of  C .
>Then C cannot have natural H-injective hulls (i.e. they cannot
>form an endofunctor together with a natural transformation from Id)
>unless every object in  C  is H-injective.


Walter wrote "We are able to compensate for the loss of mono through
condition 1". Wouldn't it be simpler just to say that condition 1 
implies that everything in  H  is a monomorphisms?
                                              x        y
(Let  A --> B  be an  H-morphism  and let  X --> A, X --> A  be such
         x                 y
that  X --> A --> B  =  X --> A --> B. If  x  were  different from
y  then there would be   A --> E, E  an  H-injective object, so that
   x                                  y
X --> A --> E  were different from X --> A --> E.  But there would
have to be  B --> E  such that  A --> E  =  A --> B --> E  and 
    x                                            y
 X --> A --> B  would have be different from  X --> A --> B.)


So H-morphism is a strengthening of monic and that put's us back to
the situation I outlined:

  If the strengthening of monic is such that it becomes an iso
  whenever epic then there's an easy proof of the impossibility of
  functoriality, with or without a cogenerator.