Direct image functors

[Steve Vickers <s.j.vickers@cs.bham.ac.uk>]

Do others share my discomfort with the phrase “direct image functor” for 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).

This is clearest in the localic case. If the frame homomorphism f^* has a left adjoint g, and moreover a Frobenius condition is satisfied, then Joyal and Tierney showed that g(U) is indeed the direct image of each open U: thus f  matches the classical characterization of an open map. (Without Frobenius, g(U) is the up-closure of the direct image.)

Moving to non-localic toposes, and their sheaves, it gets more complicated. I  wouldn’t suggest that left adjoints are always best thought of as direct images. For instance, with a locally connected f, the left adjoint of f^* gives (fibrewise) sets of connected components.

However, my question is whether the right adjoint deserves that title. In the case where Y is 1, it is well known that f_* gives the global sections. In  general f_* is more a _sections_ functor than a direct image functor.

To see why, here’s a pointwise calculation in the notation of type theory. Suppose U = Σ_{x:X} U(x) and V = Σ_{y:Y} V(y) are bundles over X and Y. (For our topos purposes, calculating sheaves, we take them both to be local homeomorphisms, ie the fibres are all discrete spaces.) Then  f^*(V) = Σ_x V(f(x)), and a map θ: f^*(V) -> U has θ_xy: V(y) -> U(x) for each x, y with f(x) = y. This is a map
   Σ_y V(y) -> Σ_y Π_{f(x) = y} U(x),
displaying f_*(U)(y) as the set of sections of U over the fibre of f over y.

(If you don’t trust these pointwise calculations, think of them as providing intuitions from the case where there are sufficient global points. But actually they are more generally valid.)

Steve.

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