On Wed, 17 Apr 2002, Christopher Townsend wrote:
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.
Yes, this is correct. It's most easily proved "from the other end", by defining v to be the classifying map of f^*(True), and then showing that its transpose is left adjoint to the classifying map of f_*(True) (and so preserves joins). I don't know any reference where this is written out in general, but a special case of it (for hyperconnected morphisms) appears in my paper "Factorization theorems I" in CTGD 22 (1981). Peter Johnstone 20-Apr-2002 15:56:55 -0300,4123;000000000000-00000000