I have the following question about geometric morphisms, surely well known, for which I would appreciate any references... If f:E->E' is a geometric morphism (with direct image f_* and inverse image f^*) then f^* preserves monics, and so there is a mapping from the homset E'(A',Omega') to E(f^*(A'),Omega) which takes the classifying map of a monic i':B'>->A' to the classifying map of f^*(i'):f^*(B')>->f^*(A'). I beleive that this mapping is given by taking h' to v o f^*(h') (i.e. f^*(h') followed by v) where v:f^*(Omega')->Omega is the adjoint transpose of the map Omega'(!):Omega'->f_*(Omega), the unique SUP-lattice hom. that sends True' (of Omega') to True of f_*(Omega) (f_*(Omega) is known to be a frame). The question is equivalent to proving that the pullback of True along v is f^*(True') (i.e. that v classifies f^*(True)). Is there an explicit (or any) reference to this anywhere in the literature? Thank you, Christopher Townsend NB Omega' and Omega are, of course, the subobject classifiers in E' and E respectively. 18-Apr-2002 09:04:13 -0300,5719;000000000000-00000000