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
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