On Mon, 25 Sep 2006, Steve Vickers wrote:
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.)
Of course -- (cocomplete) quasitoposes are geometric categories as well, and they needn't be balanced. Incidentally, Gordon Monro wrote a couple of papers about the interpretation of logic in quasitoposes, which appeared in JPAA 42 (1986).
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.
I've never really found a satisfactory conceptual explanation of why these two definitions come out equivalent. Undoubtedly it's connected with the fact that, in the recursive definition, one is thinking in terms of interpretations in geometric categories, but is there more to it than that? Peter Johnstone