Re: Sheaves and the empty set

On May 18, 9:49 pm, Kira Yamato <kira...@xxxxxxxxxxxxx> wrote:

Oh, is the requirement that
F(emptyset) = zero ring
be part of the definition for a sheaf? Otherwise, it seems that my
example satisfies the definition for a sheaf.

No, it's not part of the definition. It can be deduced. In your
example you said for *all* open set U.. did you meant that
F(emptyset)=0 .. besides what are you restriction maps? If your
restriction maps is the identity for nontrivial restriction and if
F(emptyset)=0 your example wouldn't work if your topological space had
two disjoint open set.


About your exact sequence

F(U) --> \prod_i F(U_i) ==> \prod_i, \prod_j F(U_i \cap U_j)

isn't the emptyset a covering (of a single open set) of the emptyset
itself? So, the products in the exact sequence are not empty products.


it is , but I didnt take the emptyset as my covering.. I took the
empty covering (which is something else ;) )..

What I took is \/_{i\in\emptyset} U_i = emptyset
Which forces the product to be an empty product, making in the zero
ring.

.

Loading