From: Peter Freyd <pjf@saul.cis.upenn.edu>
To: categories@mta.ca
Subject: More comments on Functorial injective hulls
Date: Thu, 23 Mar 2000 14:12:31 -0500 (EST) [thread overview]
Message-ID: <200003231912.OAA11052@saul.cis.upenn.edu> (raw)
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.
next reply other threads:[~2000-03-23 19:12 UTC|newest]
Thread overview: 2+ messages / expand[flat|nested] mbox.gz Atom feed top
2000-03-23 19:12 Peter Freyd [this message]
[not found] ` <pjf@saul.cis.upenn.edu>
2000-03-24 19:06 ` Walter Tholen
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