Equalisers of power

Equalisers of power

[Paul Taylor]

Michael Barr asked,

> Suppose C is a complete category and E is an object.

By which I understand that C has all finite limits and powers of E,
although I usually write Sigma instead of E.

> Form the full subcategory of C whose objects are equalizers of two
> arrows between powers of E.

This full subcategory consists of the objects that I call "sober".

> Is that category closed in C under equalizers?

Yes.

Write  $X  for the exponential  E^X.
Then $ is a self-adjoint contravariant endofunctor of the category C,
and the covariant functor $$ is part of a monad.

For any object X there is a diagram with equal composites
                       eta $$ X
         eta X        --------->
     X  ------>  $$ X             $$$$ X
                      --------->
                       $$ eta X
and X is by definition "sober" if this is an equaliser.
Any sober object in this sense belongs to Mike's subcategory.

Any  $Y  is sober, because  eta $Y  is split by  $ eta Y
(see the chapter on Beck's triplability theorem in "Toposes,
Triples and Theories", for example, for details).

Now if  Y and Z are sober and  X >--->  Y ====>  Z  is an equaliser,
we can form a  3x3  square of objects,  whose rows and columns (with
one a priori exception)  are equalisers,   and then  check that the
last is an equaliser too,  ie X is sober.

In other words,  Mike's subcategory is closed under equalisers and
consists of the sober objects.     []

I don't remember the details of the papers in question, but
investigations of this kind go back to Lambek & Rattray c1975.

Then in Synthetic Domain Theory (SDT) c1990,  Rosolini, Phoa, Hyland
and I looked at various properties that select "predomains" as special
objects of a topos.   One of these was Hyland's notion of "repleteness",
which Rosolini, Fiore and Makkai showed to be slightly weaker than
sobriety,  when you chararacterise these things in particular
concrete categories.  Conceptually, though, they amount to the same
thing.  Streicher also looked looked at sobriety in a concrete setting.

I would say that it is a mistake to see sobriety as a property of
objects in a category that has been given in advance.  It should
really be seen as a property of the category:  that the functor $
reflects invertibility.   Alternatively, we may see it is an axiom
in a richer logic,  where it has the concrete interpretation that
- N is sober iff it admits definition by description and
- R is sober iff it is Dedekind complete.
See S 14 of "The Dedekind Reals in ASD"  by Bauer and me for details
www.PaulTaylor.EU/ASD/dedras

The idea of Abstract Stone Duality came out of my earlier involvement
in SDT,  and was also motivated by Pare's theorem that $ (now the
contravariant powerset functor) is monadic in any elementary topos.
Mathematically, the most important consequence of this hypothesis
is  the Heine--Borel theorem,  that [0,1] subset R is compact in the
"finite open subcover" sense.    I gave a summary of this in my
posting to "categories" on 18 August 2007.

The monadic hypothesis led to an account of (computably based)
locally compact spaces,  from which it is very difficult to escape.

Stepping back from monadicity, and almost going back to Mike Barr's
question,  the Heine--Borel theorem is a consequence of having "the
right" relationship between  $  (powers of the Sierpinski space) and
equalisers.   Within the category of "topological spaces"  as found
in the textbooks,  or that of locales,  this relationship is called
either the "subspace topology" or "injectivity of Sierpinski".

However, I have a counterexample (which I am not willing to spell
out in ASCII) to show that this cannot extend verbatim to cartesian
closed categories of spaces.    I shall present this,  the weaker
property that I think should generalise it,  my syntactic proof
that this property is consistent (I have no concrete model)  and
some of its applications,  at CT08 in Calais next month (assuming,
of course, that the programme committee accepts my submission).

Paul Taylor