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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.
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
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 ***************************************************************** 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
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. xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx alternative e-mail address (in case reply key does not work): J.Adamek@tu-bs.de xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
participants (4)
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F W Lawvere -
Jiri Adamek -
Peter Freyd -
Walter Tholen