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/ ]
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/ ]
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.
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Peter and Steve, If anyone protests against the direct image functor being a right adjoint, I would recall the following simple well-known story: 1. If f : X-->Y is a map of sets, then the inverse image (=pullback) functor f* : Sets/Y --> Sets/X has both left and right adjoint. Let me call them Lf and Rf, respectively. 2. If we replace Sets with an arbitrary category C with pullbacks, then f* and Lf are still there, but Rf disappears, unless C is locally cartesian closed. In particular, there is no Rf (in general) when C = Top is the category of topological spaces. 3. But if I am thinking towards topos theory, I might prefer to consider not f* : Top/Y --> Top/X, but f* : Shv(Y) --> Shv(X), where Shv(?), the category of sheaves (of sets) over "?", is equivalent to the full subcategory of Top/? with objects all local homeomorphisms with codomain "?", and, under this equivalence, the 'new' f* is the restriction of the old one. And then we have f* and Rf but not Lf (in general). Therefore, I might prefer to have name "direct image functor" for the right adjoint! Peter, might this be what you thought of as "geometric aspects"? (Surely, a geometer, considering, say, a manifold X, would be more interested in Shv(X) than in Top/X.) With apologies for trivialities- George -------------------------------------------------- From: "Steve Vickers" <s.j.vickers@cs.bham.ac.uk> Sent: Tuesday, November 8, 2022 4:10 PM To: <ptj@maths.cam.ac.uk> Cc: <categories@mta.ca> Subject: categories: Re: Direct image functors
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.
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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/ ]
Dear Pedro, That’s an interesting point about “direct image” going back to sheaf theory. Bredon’s old book reminded me that the most natural definitions of f^* and f_* use quite different representations of sheaves. With local homeomorphisms, f^* is composition of the map f with the stalk map, while with pasting presheaves f_* is composition of the presheaf with f^{-1}. I guess a sheaf theorist might see both of those as images, and then “inverse” or “direct” just indicates variance with respect to the map f. My own focus on the local homeomorphisms, whatever its merits when investigating geometricity and arithmetic universes, leads me to disregard the pasting presheaves. All the best, Steve.
On 8 Nov 2022, at 10:20, Pedro Resende <pedro.m.a.resende@tecnico.ulisboa.pt> wrote: Hi Steve,
The terminology is at odds with the localic case, but it seems to be standard in sheaf theory, independently of category theory and toposes: https://en.wikipedia.org/wiki/Direct_image_functor Then a question is what a left adjoint to an inverse image functor ought to be called in general…
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (4)
-
George Janelidze -
ptj＠maths.cam.ac.uk -
Ross Street -
Steve Vickers