>Does this tell us something interesting?

I think it does. This is where higher dimensional categories and weak adjoints show up in the ordinary category theory! I attach as an explanation my message written as an answer for the discussion about this topic nine years ago. I did not get any reply on my entry those time but, in nine years there were a lot of progress. I hope it will be more understandable now.

To: categories@mta.ca

Subject: categories: Re: Functorial injective hull.

From: mbatanin@ics.mq.edu.au (Michael Batanin)

Date: Mon, 27 Mar 2000 18:24:57 +1100

Sender: cat-dist@mta.ca

This is just to give different point of view to the problem.

1. Suppose we have a category C and a full subcategory H (think of H as a

subcategory of injective objects). Suppose we have a functor E:C –> H

together with natural transformation

i: Id –> E(C)

such that i is monic and ,moreover, E is weak left adjoint to the inclusion

functor H –> C. Then I claim that under two additional conditions E is

genuine adjoint and i is unit of the adjunction. The conditions are:

a. the natural transformation i is identity on H.

b. E preseves monics.

The proof is just a repetition of P.Freyd proof. We have to prove that

the extension in the diagram

i:A –> E(A)

| /

f | /

| /

I

is unique for any morphism f and “injective” I.

By applying E to this diagram and using condition a) we see that it is

sufficient to prove that E(i) is identity. Repeating Freyd’s proof we see

that

E(i) is idempotent. Using condition b) we conclude that it is identity.

The conditions a) and b) are obviously satisfied in the case when E is

“injective hull functor” (of coarse a) is true up to iso, again see P,Freyd

proof).

As the unit of the adjunction is monic so the functor E reflects

epimorphisms. If in C we have that mono + epi implies iso we finally have

that i is iso and ,hence, the result of Adamek, Herrlish, Rosicky and

Tholen.

2. I would not dare to simply repeat P.Freyd argument if I don’t have another

proof (in a special but important case). The proof is much more

techniqual but I believe reflects another interesting side of the problem

of functoriality

of injective resolutions.

Consider the following bicategory.

The objects are categories enriched in the closed monoidal category of

(say bounded ) chain complexes.

The 1-arrows are enriched distributors.

The 2-arrows are homotopy classes of “coherent” natural transformations (i.e.

we localize the category of natural transformations with respect to the

class of morphisms which are level quaziisomorphisms).

We have to define the composition of 1-arrows. This requires some techniques

but in a few words the result is left derived functor of the composition of

enriched distributors.

The resulting bicategory is closed on the left and right. Now, for a chain

functor

K: A –> B

we can consider the right Kan extension of it along itself in the above

bicategory (codensity monad). The Kleisli category of it is called “strong

shape theory of K” Ssh_K. It is possible to prove, that:

c). If K is a full embedding and has an enriched left adjoint then Ssh_K

is just a Kleisli category of the corresponding monad on B.

d). If B is the category of chain complexes (say bounded)in an abelian

category with enough injectives and A is full subcategory of injective

chain complexes, then homology of Ssh_K(X,Y) are isomorphic to the right

derived functor of internal Hom of B. In particular, if X and Y are

concentrated in the dimension 0 the cohomology are just the correspobding

Ext’s.

Coming back to the original problem. If H\in C are abelian and satisfy the

conditions a),b) then, according to our calculations

the inclusion

K: Ch(H) —> Ch(C)

has a right adjoint and , hence,(by point c)

Ssh_K(X,Y) = Hom(X,E(Y))

for X,Y concentrated in dimension 0. Hence by d) the injective dimension of

C is 0 and we have the result again.

Another words the injective dimension is the obstruction for nonnaturality

of the injective hull.

The same sort of theory can be developed for simplicial (or Cat) enriched

situation see

Batanin.M, Categorical Strong Shape THeory, Cahiers de Topologie et Geom,

v,XXXVIII-1(1997)p. 3-66.

I think some other results of nonaturality (as , for example, the result of

Shakhmatov mentioned by AHRT on p.8) are related to these strong shape

categories.

Michael Batanin.

## Re: When Naturality Fails

The algebraic closure, the MacNeille completion, but also the Hahn-Banach extension (Joy of Cats 9.3.5), all live in a topos different from the one the orginal object lives in.

Maybe this provides a clue.