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.