------------- Does anyone have any advice about this following proposed terminology? I think I want to use the term "weak topos" to mean a cartesian closed category with all finite limits and colimits, stable surjections, and a weak power object. By a "weak power object" I mean an object W with a monic M>--->W such that every monic in the category is a pullback of M in at least one way. The reasons for this are that categories of assemblies (including assemblies for extensional or modified realizability) have these properties and: 1) Except for the part about colimits this is just what you need to show the effective reflection is a topos. 2) Using the presence of coequalizers this gives a usable generalization of geometric morphisms. If a functor between weak toposes in this sense has a left exact left adjoint then that adjoint is regular (pres. finite limits and surjections) and the adjunction lifts to a geometric morphism between the toposes. And in fact these arise among categories of assemblies (as is in effect remarked by Pitts and more recently Oosten). Thanks, Colin ==============================================================================