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categories: Functorial injective hulls



     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.