On 7 Nov 2022, at 9:56 pm, Steve Vickers <s.j.vickers@cs.bham.ac.uk> wrote:
Do others share my discomfort with the phrase ???direct image functor??? for
Dear Steve An abstract notion of inverse image is provided by a cartesian morphism c : E --> F for a functor P : S --> T where we think of E as the inverse image of F under Pc : PE --> PF. Dually, opcartesian morphisms give direct image. Applying the Grothendieck opfibration construction to Shv : Top --> CAT, we can look at the opcartesian morphism E --> f_*E over f : X --> Y to obtain f_* of the sheaf E as a direct image in that abstract sense. Just a suggestion. All the best, Ross the right adjoint f_* in a geometric morphism f: X -> Y?
It seems to me that a direct image functor should be left adjoint of the
inverse image, not right adjoint, because in sets and functions, we have f(A) subset B iff A subset f^{-1}(B). [For admin and other information see: http://www.mta.ca/~cat-dist/ ]