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From: ptj@maths.cam.ac.uk
To: Steve Vickers <s.j.vickers@cs.bham.ac.uk>
Cc: "categories@mta.ca list" <categories@mta.ca>
Subject: Re: Direct image functors
Date: 08 Nov 2022 10:46:47 +0000	[thread overview]
Message-ID: <E1osxHT-0000cM-I8@rr.mta.ca> (raw)
In-Reply-To: <E1osE7Z-0004Ze-QE@rr.mta.ca>

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.
>
>[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
>


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]


  reply	other threads:[~2022-11-08 10:46 UTC|newest]

Thread overview: 5+ messages / expand[flat|nested]  mbox.gz  Atom feed  top
2022-11-07 10:56 Steve Vickers
2022-11-08 10:46 ` ptj [this message]
2022-11-08 14:10 Steve Vickers
2022-11-10 11:03 ` George Janelidze
     [not found] <1C2D2A4F-AFE4-4285-A70A-A77888CFB934@cs.bham.ac.uk>
2022-11-08 21:36 ` ptj

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