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.