Re: About the cartesian closedness of the category of all small diagrams

For a fibration of ccc's over a ccc one knows that the total category is again ccc and this structure is preserved (could be in Bart Jacob's book).

This is wrong! Let EE be the free topos (with nno). Then the fibration Fam(EE) over Set is a fibered ccc but if it were a cartesian closed functor between toposes then EE would have small sums which it hasn't. It is not clear at all which fibrations of ccc's over ccc's have the property that they are ccc-preserving functors between ccc's. It seems as if such fibrations have to be internally complete. That's what I definitely overlooked. Sorry for that, Thomas [For admin and other information see: http://www.mta.ca/~cat-dist/ ]