Proposition. The canonical morphism is universal in the sense that if was another morphism to a sheaf , then it factors through . That is, there exists a such that .

Proposition. If is a sheaf, then the canonical morphism is an isomorphism of sheaves.

Proposition. If and is the canonical morphism,  then induced map is an isomorphism.

Remarks:

1. I may be incorrect about the definition of .
2. Proofs are forthcoming!

sheaf associated to

1. for all ;
2. For all , there exists an open neighborhood of and such that for all .

Proposition. The correspondence is a sheaf.

Proof. To do!

Proof. If is an isomorphism of sheaves, then because the construction of a stalk at a point is functorial, then is an isomorphism of stalks for all .

Now suppose is an isomorphism of stalks for all . It is sufficient to show is an isomorphism for open.

To see it is injective, let such that . Then passing to stalks, we have for all . But is an isomorphism so that .  Then there exists some open neighborhood of such that .Then the form an open neighborhood of and since is a sheaf, this implies .

For surjectivity, let . Again passing to stalks, we have for some for all . Then is the image of some for some open neighborhood   of ; and without loss of generality, we may assume . We may assume . Then

and so by injectivity: . Then since is a sheaf, there exists some such that . Then .

Remarks

1. I’m not really sure about proof of surjectivity. It needs to be cleaned up.

]]> http://taggablemath.wordpress.com/2009/11/03/isomorphism-of-sheaves-vs-isomorphism-of-stalks/feed/ 0 K http://taggablemath.wordpress.com/2009/11/03/isomorphism-of-presheaves/ http://taggablemath.wordpress.com/2009/11/03/isomorphism-of-presheaves/#comments Tue, 03 Nov 2009 16:32:55 +0000 K http://taggablemath.wordpress.com/?p=137 ]]> Proposition: A morphism of presheaves is an isomorphism (in the categorical sense) if and only if is an isomorphism for every open.

To see this is well defined, suppose . Then there is some containg such that . Then . But and . That is, .

stalk of at

where the inductive limit is taken over all open in containing .

Remarks

1. Explicity, this is the collection of sections for   where we identify and whenever there is some open containing such that
2. This endows with well-defined algebraic operations. For example

   .

1. If whenever there exists such that for all , then .
2. If whenever there exists a collection for all such that

   ,

   then there exists some such that for all .

Remarks

1. In this sense, in a sheaf sections are determined locally, i.e. on their restrictions to open sets.
2. We may define morphisms on sheaves as morphisms on presheaves which are sheaves. Doing so, we get a category of sheaves on .

such that

for all open sets and .

Remarks

1. I don’t know how to put commutative diagrams into wordpress.
2. If we are considering presheaves over some category, we require the maps to be morphisms in the appropriate category.