Are geometric categories balanced?

[Steve Vickers <s.j.vickers@cs.bham.ac.uk>]

Am I right in believing that geometric categories (Elephant A 1.4.18)
need not be balanced? (I don't know a counterexample - the geometric
categories that I can think of are all toposes.)

The reason I ask is this. One is (or certainly I am) used to thinking
of geometric logic as the logic of Grothendieck toposes. Grothendieck
toposes are balanced, and consequently a sound reasoning principle in
their internal logic is that functions are equivalent to total,
single-valued relations. One therefore thinks of this as a principle
of geometric reasoning. However, I suspect it doesn't follow just
from the pure logic - the connectives and inference rules -  of
geometric logic. This would be verified if there are unbalanced
geometric categories, since the pure logic is interpretable in
arbitrary geometric categories. I would take this as indicating that
we want geometric logic to be more than just what the pure logic says
it is.

The Elephant gives two different definitions of geometric theory: by
the pure logic in D 1.1.6, and by a more general notion of geometric
construct in B 4.2.7. It asserts their equivalence, but I think this
must be with respect to a semantics already presumed to be in
Grothendieck toposes.

Steve Vickers.