Re: Functorial injective hulls

[F W Lawvere <wlawvere@ACSU.Buffalo.EDU>]

 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 ?

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F. William Lawvere			
Mathematics Dept. SUNY Buffalo, Buffalo, NY 14214, USA
716-829-2144  ext. 117		   
HOMEPAGE:  http://www.acsu.buffalo.edu/~wlawvere
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On Wed, 22 Mar 2000, Walter Tholen wrote:

> Peter,
> 
> Jirka Adamek had prepared a draft response to your earlier remark that a poset
> with top element should disprove the assertion in the Abstract of our paper
> (with Herrlich and Rosicky) which he had circulated. His response is attached
> below, slightly edited by me - hence I take full responsibility for its
> contents.
> 
> Our proof of the Theorem adds only one twist to the proof you have just
> circulated: monomorphisms get substituted by an absolutely ARBITRARY class H of
> morphisms;  H-injective then indeed means that the contravariant hom sends H to
> epis; and H-essential is as you described as well (: an h in H such that g.h is
> in H only if g is in H). We are able to compensate for the loss of mono through
> condition 1, while condition 2 obviously replaces your (epi&mono is iso). For
> full details, please consult the paper.
> 
> Best wishes,
> Walter.
> 
> 
> =============================================================================
> Dear Peter,
> The precise result we prove in our paper is the following:
> 
> 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.
> 
> The abstract we have given in our posting was meant to be an abbreviation of
> this precise statement. While condition 1 holds true for the set H of all
> (mono)morphisms in a poset with top element, condition 2 fails.
> 
> Best regards,
> J.A., H.H., J.R., W.T.
> 
> 
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> alternative e-mail address (in case reply key does not work):
> J.Adamek@tu-bs.de
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> 
> 
> 
>