Re: Are geometric categories balanced?

Re: Are geometric categories balanced?

[Prof. Peter Johnstone]

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

Are geometric categories balanced?

[Steve Vickers]

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.