Dear Andr'e, of course, logos would be a natural name when understood as elementary toposes and logical morphisms. In the Elephant one finds Top, BTop and LogTop as names for elementary toposes and various different notions of morphisms. There is a problem with just geometric morphisms since they dont correspond to Grothendieck toposes - only bounded ones do. I personally find logical morphisms the basic notion and geometric morphisms as a derived notion. As I learnt from Benabou quite some time ago one should consider geometric morphisms as fibred toposes. If SS is a (base) topos then P : EE -> SS is a fibration of toposes iff P is a Grothendieck fibration whose fibers are toposes and whose reindexing functors are logical. One may characterize them as fibrations P such that P is a logical morphism of toposes. Fibrations of cocomplete toposes over SS correspond to finite limit preserving functors from SS to some topos (namely Delta(I) = \coprod_I 1_I). Fibrations of locally small cocomplete toposes correspond to finite limit preserving functors to some topos which, moreover, have a right adjoint. The bounded gm's to SS correspond to fibrations of locally small cocomplete toposes which moreover admot a small generating family. This, for me justifies the notion of bounded gm's. The fibrational point of view doesn't suggest how to compose them (see discussion pp.66-67 of my notes on fibrations). But the corresponding Delta's are obviously closed under composition. I then would add the "observation" that these functors maybe understood as "glorified" frame morphisms. This, of course, doesn't reflect history but that's not the issue here. It's about "rational reconstruction" of a notion. Thomas PS That's what I think Benabou had in mind with his remark after one of your lectures at IHES last November. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]