[P1] _________________________ AMAST Links 02 05

CoComplete Toposes Having No Small Set of Generators

One possible very concise definition of Grothendiek topos is as an elementary topos E such that:

1. E is locally small;
2. E has small sums;
3. E has a small set of generators.

One knows that a lex category B satisfies 1. and 2. iff the global sections functor Gamma:B -> Set has a lex left adjoint Delta.

For an elementary topos E satisfying 1. and 2. the condition 3. is equivalent to the existence of an S in E such that any X of E is a Subquotient of Delta(I) x S for some I in Set.

Now the question is:

What is the status of Locally Small CoComplete Elementary Toposes?

Locally small cocomplete elementary toposes are sufficient for interpreting set theory. Can they - maybe - be characterised via this property? Do there exist models of set theory giving rise to cocomplete topoi which don't have a small set of generators?

The background of my question is that elementary toposes as such don't provide models of IZF (how should one simulate the Goedel-Bernays-Neumann hierarchy?). It would be nice if cocomplete topoi were the precise analogon of IZF!

Grateful for any hints,

Thomas Streicher