Re: A conditon on maps between sheaves

[zoran skoda <zskoda@gmail.com>]

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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