On a remark of Barr

On a remark of Barr

[categories]

Date: Tue, 11 Nov 1997 11:35:10 -0500 (EST)
From: Peter Freyd <pjf@saul.cis.upenn.edu>

Mike Barr on the question: "is every small topos fully representable
into a set-valued functor category?":

  Unfortunately, a beautiful observation of Makkai's shows that that
  is impossible. Makkai pointed out that under a full embedding that
  preserves finite limits, finite sums and epis the boolean algebra of
  complemented subobjects, which is classified by maps into 1 + 1,
  would have to be preserved. But in a functor category that lattice
  is complete and atomic (that is, completely distributive), so that
  fact, which is not true for toposes in general, becomes a necessary
  condition for the existence of an embedding.  (Is it sufficient?)

One can go further. The embedding will necessarily preserve all finite
colimits and all infinite coproducts that happen to exist. 

Extend the Makkai observation as follows. For any pair of objects  A
and  B  note that the lattice of partial maps with complemented 
domains can be constructed as the set of maps from  A  to  1+B  (give
1+B  the "flat ordering" of CS: start with  B  with the trivial 
partial ordering and adjoin a bottom). The fact that it's a complete
lattice is enough to show that arbitrary disjoint unions are 
coproducts. 

There is an argument for the case that the topos does not have a 
natural numbers object but I'll assume here that it does have such.
The standard points of the natural numbers object are complemented,
hence their union must exist; that union is a coproduct; it must
therefore be the entire natural numbers object. It is preserved by
the embedding.

In Aspects of Topoi there's a lemma entitled "one coequalizer for
all" that now suffices to show that the representation preserves
all coequalizers.