Subject: categories: Re: Functorial injective hulls

From: "Walter Tholen" <[email-masked]>
Date: Wed, 22 Mar 2000 18:10:57 -0500

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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From: F W Lawvere <[email-masked]>
Date: Thu, 23 Mar 2000 14:50:57 -0500 (EST)

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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