A conditon on maps between sheaves

A conditon on maps between sheaves

[Andrej Bauer]

Dear categorists,

I have come across a condition on maps between sheaves which I am
unable to recognize as with my feeble knowledge of sheaf theory. I
would appreciate any hints as to what this condition is about.

Succinctly but imprecisely my condition can be expressed as: the
inverse image of a sufficiently small section is again a section.

More precisely, let p : E -> B be p' : E' -> B be two etale maps over
a base space B and let f : E -> E' be a continuous map such that p = f
p'. The mystery condition on f is as follows: for every x in B there
is a neighborhood U of x, such that for every section s : U -> E' of
p' there exists a unique section t : U -> E of p for which t(U) =
f^(-1)(s(U)).

It follows from this condition that f is mono as a morphism in Sh(B)
because such an f is injective on each fiber. But I think the
condition says more than that. Am I looking at a standard notion?

With kind regards,

Andrej


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Re: A conditon on maps between sheaves

[zoran skoda]

Dear Andrej,

I do not see that the condition as you stated it implies that the map is
mono. For example, one can take an example such that for every x the U with
above property is the whole base B, hence we have a mono on global sections
over B, but this solely is very weak and does not imply we have mono
locally, hence on stalks. Maybe you wanted that, in fact, for every nei W
around x there is open U around x which is within U ?

Zoran

On Sat, Mar 12, 2011 at 2:33 AM, Andrej Bauer <andrej.bauer@andrej.com>wrote:

> Dear categorists,
>
> I have come across a condition on maps between sheaves which I am
> unable to recognize as with my feeble knowledge of sheaf theory. I
> would appreciate any hints as to what this condition is about.
>
> Succinctly but imprecisely my condition can be expressed as: the
> inverse image of a sufficiently small section is again a section.
>
> More precisely, let p : E -> B be p' : E' -> B be two etale maps over
> a base space B and let f : E -> E' be a continuous map such that p = f
> p'. The mystery condition on f is as follows: for every x in B there
> is a neighborhood U of x, such that for every section s : U -> E' of
> p' there exists a unique section t : U -> E of p for which t(U) =
> f^(-1)(s(U)).
>
> It follows from this condition that f is mono as a morphism in Sh(B)
> because such an f is injective on each fiber. But I think the
> condition says more than that. Am I looking at a standard notion?
>
> With kind regards,
>
> Andrej
>

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Re: A conditon on maps between sheaves

[Steve Vickers]

Dear Andrej,

For example: take B to be the circle and E' its Moebius double cover,
which has no global sections. Then for every x in B you can take U = B
and your condition holds vacuously for any f whatsoever.

If E = E'+E' then the codiagonal f has your property but is not mono.

Regards,

Steve.

zoran skoda wrote:
> Dear Andrej,
>
> I do not see that the condition as you stated it implies that the map is
> mono. For example, one can take an example such that for every x the U with
> above property is the whole base B, hence we have a mono on global sections
> over B, but this solely is very weak and does not imply we have mono
> locally, hence on stalks. Maybe you wanted that, in fact, for every nei W
> around x there is open U around x which is within U ?
>
> Zoran
>
> On Sat, Mar 12, 2011 at 2:33 AM, Andrej Bauer <andrej.bauer@andrej.com>wrote:
>
>> Dear categorists,
>>
>> I have come across a condition on maps between sheaves which I am
>> unable to recognize as with my feeble knowledge of sheaf theory. I
>> would appreciate any hints as to what this condition is about.
>>
>> Succinctly but imprecisely my condition can be expressed as: the
>> inverse image of a sufficiently small section is again a section.
>>
>> More precisely, let p : E -> B be p' : E' -> B be two etale maps over
>> a base space B and let f : E -> E' be a continuous map such that p = f
>> p'. The mystery condition on f is as follows: for every x in B there
>> is a neighborhood U of x, such that for every section s : U -> E' of
>> p' there exists a unique section t : U -> E of p for which t(U) =
>> f^(-1)(s(U)).
>>
>> It follows from this condition that f is mono as a morphism in Sh(B)
>> because such an f is injective on each fiber. But I think the
>> condition says more than that. Am I looking at a standard notion?
>>
>> With kind regards,
>>
>> Andrej


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

Re: A conditon on maps between sheaves

[Andrej Bauer]

Steve Vickers wrote:
> For example: take B to be the circle and E' its Moebius double cover,
> which has no global sections. Then for every x in B you can take U = B
> and your condition holds vacuously for any f whatsoever.
>
> If E = E'+E' then the codiagonal f has your property but is not mono.

I apologize for the noise, I got my conditions all wrong when I tried
to "optimize" them for the categories list. As it turns out my
condition means that I have a map of etale spaces which is bijective
on fibers (and the spaces in question are Hausdorff locally compact).
So is there a name for that other than "bijective on fibers"?

With kind regards,

Andrej


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

Re: A conditon on maps between sheaves

[Prof. Peter Johnstone]

On Sun, 27 Mar 2011, Andrej Bauer wrote:

> Steve Vickers wrote:
>> For example: take B to be the circle and E' its Moebius double cover,
>> which has no global sections. Then for every x in B you can take U = B
>> and your condition holds vacuously for any f whatsoever.
>>
>> If E = E'+E' then the codiagonal f has your property but is not mono.
>
> I apologize for the noise, I got my conditions all wrong when I tried
> to "optimize" them for the categories list. As it turns out my
> condition means that I have a map of etale spaces which is bijective
> on fibers (and the spaces in question are Hausdorff locally compact).
> So is there a name for that other than "bijective on fibers"?

Yes -- it's called an isomorphism. The fibre functors are jointly
conservative.

Peter Johnstone

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