From mboxrd@z Thu Jan  1 00:00:00 1970
X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/1456
Path: news.gmane.org!not-for-mail
From: "Walter Tholen" <tholen@pascal.math.yorku.ca>
Newsgroups: gmane.science.mathematics.categories
Subject: Re: Functorial injective hulls
Date: Wed, 22 Mar 2000 18:10:57 -0500
Message-ID: <1000322181057.ZM23382@pascal.math.yorku.ca>
NNTP-Posting-Host: main.gmane.org
Mime-Version: 1.0
Content-Type: text/plain; charset=us-ascii
X-Trace: ger.gmane.org 1241017844 31335 80.91.229.2 (29 Apr 2009 15:10:44 GMT)
X-Complaints-To: usenet@ger.gmane.org
NNTP-Posting-Date: Wed, 29 Apr 2009 15:10:44 +0000 (UTC)
To: categories@mta.ca
Original-X-From: rrosebru@mta.ca Thu Mar 23 13:57:15 2000 -0400
Original-Received: (from Majordom@localhost)
	by mailserv.mta.ca (8.9.3/8.9.3) id NAA21202
	for categories-list; Thu, 23 Mar 2000 13:54:38 -0400 (AST)
X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f
X-Mailer: Z-Mail (4.0.1 13Jan97)
Original-Sender: cat-dist@mta.ca
Precedence: bulk
Original-Lines: 48
Xref: news.gmane.org gmane.science.mathematics.categories:1456
Archived-At: <http://permalink.gmane.org/gmane.science.mathematics.categories/1456>

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.


xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
alternative e-mail address (in case reply key does not work):
J.Adamek@tu-bs.de
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx