* A conditon on maps between sheaves @ 2011-03-12 1:33 Andrej Bauer 2011-03-13 16:25 ` zoran skoda 0 siblings, 1 reply; 5+ messages in thread From: Andrej Bauer @ 2011-03-12 1:33 UTC (permalink / raw) To: categories list 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/ ] ^ permalink raw reply [flat|nested] 5+ messages in thread
* Re: A conditon on maps between sheaves 2011-03-12 1:33 A conditon on maps between sheaves Andrej Bauer @ 2011-03-13 16:25 ` zoran skoda 2011-03-14 14:27 ` Steve Vickers 0 siblings, 1 reply; 5+ messages in thread From: zoran skoda @ 2011-03-13 16:25 UTC (permalink / raw) To: Andrej Bauer; +Cc: categories list 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/ ] ^ permalink raw reply [flat|nested] 5+ messages in thread
* Re: A conditon on maps between sheaves 2011-03-13 16:25 ` zoran skoda @ 2011-03-14 14:27 ` Steve Vickers 2011-03-27 18:27 ` Andrej Bauer 0 siblings, 1 reply; 5+ messages in thread From: Steve Vickers @ 2011-03-14 14:27 UTC (permalink / raw) To: Andrej Bauer; +Cc: zoran skoda, categories list 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/ ] ^ permalink raw reply [flat|nested] 5+ messages in thread
* Re: A conditon on maps between sheaves 2011-03-14 14:27 ` Steve Vickers @ 2011-03-27 18:27 ` Andrej Bauer 2011-03-28 9:40 ` Prof. Peter Johnstone 0 siblings, 1 reply; 5+ messages in thread From: Andrej Bauer @ 2011-03-27 18:27 UTC (permalink / raw) To: Steve Vickers; +Cc: zoran skoda, categories list 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/ ] ^ permalink raw reply [flat|nested] 5+ messages in thread
* Re: A conditon on maps between sheaves 2011-03-27 18:27 ` Andrej Bauer @ 2011-03-28 9:40 ` Prof. Peter Johnstone 0 siblings, 0 replies; 5+ messages in thread From: Prof. Peter Johnstone @ 2011-03-28 9:40 UTC (permalink / raw) To: Andrej Bauer; +Cc: Steve Vickers, zoran skoda, categories list 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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 5+ messages in thread
end of thread, other threads:[~2011-03-28 9:40 UTC | newest] Thread overview: 5+ messages (download: mbox.gz / follow: Atom feed) -- links below jump to the message on this page -- 2011-03-12 1:33 A conditon on maps between sheaves Andrej Bauer 2011-03-13 16:25 ` zoran skoda 2011-03-14 14:27 ` Steve Vickers 2011-03-27 18:27 ` Andrej Bauer 2011-03-28 9:40 ` Prof. Peter Johnstone
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