The n-Category Café

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.

Bas Spitters hits the nail on the head. For a topos E one can talk of an E-geometric theory T, whose models live in E-toposes. The yoga of algebraic theories, the existence of free functors left adjoint to forgetful functors induced by maps of theories etc, lifts I believe to geometric theories, but stepping up one degree in the categorical hierarchy. The classic example is Monique Hakim’s thesis. This says that given a commutative ring A in a topos E, the free local ring on A lives in the E-topos Spec(A). Of course you have to formulate the E-geometric theory of local rings over A, describe the map of geometric theories from rings to it, etc. This stuff has been around a while. Has nobody’s student been tasked with writing it all down prettily yet?

I don’t understand what Bas and Gavin are getting at. If I have a field in the topos $Set$, then surely its algebraic closure is also a field in $Set$. Are you saying that rather than asking for “the free algebraically closed field containing $k$” one should ask instead for “the free topos containing an algebraically closed field containing $k$”?

Let us look at a simple case - the algebraic closure of the field of real numbers. One way of looking at the field of complex numbers is as a field in the topos of sets-with-involution, complex conjugation being the involution in this case. In Topos Theoretic Methods in Geometry, Aarhus University Various Pub. Series No. 30, May 1979, I showed with little pictures how you adjoin freely square roots of -1 to any field in a topos. What you get does not live in the same topos.
If you take a larger view of all this, what you are doing is using 2-pullbacks in the 2-category of toposes, geometric morphisms and natural maps.

>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.

Thanks to everyone for their rich comments. An overall sense of their direction still eludes me, however. And this is without including Todd Trimble’s bicategorical approach to MacNeille completion.