Re: Sheaves and the empty set
- From: Jannick Asmus <jannick.news@xxxxxx>
- Date: Fri, 18 May 2007 14:15:42 +0200
On 18.05.2007 10:53, Jose Capco wrote:
Dear NG,
We know very well that if F is a sheaf of rings/groups, then
F(emptyset)=the zero ring/groups..
Now we know also that sheaves can be generally defined in any complete
category C.
a sheaf C-objects over a topological space X is a contravariant
functor F: X --> C
such that for any open covering U_i, i in I of U the canonical map
F(U) --> \prod_i F(U_i)
is the equalizer of the parallel canonical map
\prod_i F(U_i) ==> \prod_i, \prod_j F(U_i \cap U_j)
(see http://groups.google.com/group/sci.math/browse_thread/thread/af9946a715ae673a/
)
The claim is that F(emptyset) is the terminal object of the category
(i.e. an object c of Obj(C) such that for all
for all a in Obj(C) there is exactly one C-morphism a --> c )
We thus take the empty covering, i.e. with the index set I=emptyset..
the products become empty product and so they are the terminal object
in our category (see http://groups.google.com/group/sci.math/msg/c9edc82a292b50e9
)
and so because of the definition of equalizer F(emptyset)=terminal
object
how is my argumentation? I just wanted to share :) .. well I havent
seen this being stated in books about sheaf theory and I thought its
worth mentioning :)
Sincerely,
Jose Capco
PS: Of course if F is just a presheaf, then F(emptyset) could be
anything! Anyone care to give some examples?
What if C is the category of rings with unity and ring homomorphisms
preserving unity?
.
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