[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?

Are there natural examples?

  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