Re: Direct image functors

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


Dear Peter,

I agree the term isn’t likely to change (to “sections functor” or anything else) at this stage. I was partly trying to find out how widely the issue was recognised, and partly trying to sharpen my discussion of it.

> I don't think anyone is likely to be deceived into thinking that it's a direct
> image in the set-theoretic sense.

I’m not so sure. I’ve seen how when people start looking more closely at the points of a topos, and the part they play in topological analogies, that there is a risk of confusion. I have known a student, learning  about the action of a geometric morphism on points, who wondered if it’s somehow closely related to the direct image functor.

By the way, I looked at the Elephant to see what you said there, and I saw “we shall see later that, in a sense, f_* ‘embodies the geometric aspects’ of the morphism f”. What did you have in mind for the “we shall see later”?

Best wishes,

Steve.


> On 8 Nov 2022, at 10:50, ptj@maths.cam.ac.uk wrote:
> Dear Steve,
> 
> You are of course right that `direct image functor' was an unfortunate
> name to choose for the right adjoint part of a geometric morphism.
> But the term has been around for sixty years now, and it's very well
> understood; so I think it's too late to change it. I don't think anyone is  likely to be deceived into thinking that it's a direct
> image in the set-theoretic sense.
> 
> Best regards,
> Peter
> 
> On Nov 8 2022, Steve Vickers wrote:
> 
>> 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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