Re: Functorial injective hulls

Re: Functorial injective hulls

[Jiri Adamek]

Bill's question concerning minimal functorial injective extensions
seems very interesting. Bill's comment was:
 
>  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 ?
 
I have two different answers:
1. NO in case of Pos (and order-embeddings): there does not exist a minimal
pair (F,f) consisting of an endofunctor  F  of Pos whose values are
complete lattices and a natural transformation  f: Id -> F  whose
components are order-embeddings
2. YES in case of Set (and monomorphisms): the embedding  Id -> Id + K,
where  K  is the constant functor with value 1 , is minimal.



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Re: Functorial injective hulls

[F W Lawvere]

 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 ?

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

Re: Functorial injective hulls

[Walter Tholen]

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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Functorial injective hulls

[Peter Freyd]

     Some comments on:

>TITLE: Injective Hulls are not Natural
>AUTHORS: J. Adamek, H. Herrlich, J. Rosicky and W. Tholen
>ABSTRACT: In a category with injective hulls and a cogenerator, the 
>embeddings into injective hulls can never form a natural transformation, 
>unless all objects are injective. In particular, assigning to a field
>its algebraic closure, to a poset or boolean algebra its MacNeille
>completion, and to an R-module its injective envelope is not functorial, 
>if one wants the respective embeddings to form a natural transformation.


What is meant by saying that an object is "injective" varies a bit 
from place to place. If it means that the object represents a 
contravariant functor that carries monics into epics and if one defines
an "injective hull" of an object  A  to mean

   a monic  A --> E  where  E  is injective and such that
   A --> E --> X  monic implies  E --> X is monic 

then there are non-trivial examples of functorial injective hulls: take
any poset with a top element and view it as a category; the only
injective object is the top and the unique map from any object to the
top is easily verified to be an injective hull.

Apparently, therefore, the meaning of injective is a mutation obtained
by changing the word "monic" in the above description to something
stronger, such as "extremal monic" or "regular monic". (In Cats and
Alligators the notions of projective and injective are not dual: a
co-projective would be the mutation of injective obtained by using
extremal monics.)

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

In the days when all categories were abelian (that is, in the days
when people actually talked about injective hulls) it was also the 
case that all monic-epics were isos, and this easy proof was a pretty
standard exercise. It goes as follows. Suppose that  E  is a functor,
u  a natural transformation from the identity functor to  E  such that
u:A --> E(A)  is an injective hull for all  A. We wish to show that  u
is epic.

If  B  is injective then there must be  E(B) --> B  such that 
B --> E(B) --> B  is the identity map. The definition of injective 
hull forces  E(B) --> B  to be monic which, in turn, forces  u_B  to
be an iso. We may replace  E  with a naturally equivalent functor with
the property that  u_B  is the identity map whenever  B  is injective.

For an arbitrary  A  consider

                               u
                           A ---> E(A)
                      
                        u  |       |  E(u)
                               1
                         E(A) --> E(A)
                      
                      E(u) |       |  E(u)
                               1
                         E(A) --> E(A)


and conclude that  E(u)  is an idempotent. Using again (and for the 
last time) the definition of injective hull we have that  E(u)  is
monic. The only monic idempotent is the identity map.

                 u        x           u        y
Suppose that  A --> E(A) --> C  =  A --> E(A) --> C. Consider

              u                              u
          A ---> E(A)                    A ---> E(A)
                                    
       u  |       |  1                u  |       |  1
              1                              1
        E(A) --> E(A)                  E(A) --> E(A)
                                    
       x  |       |  E(x)             y  |       |  E(y)
              u                              u
          C ---> E(C)                    C ---> E(C)


If one considers just the outer rectangles one sees that the left hand
verticals are the same, hence so must be the right hand verticals.
But  u  is monic, thus  E(x) = E(y)  implies   x = y.