sheaf

Pre-sheaves

Let be a topological space. A presheaf of algebras on is the data of

such that if is open then there exists an algebra homomorphism

such that

is the identity map and

Sheaves with other categories

We can build sheaves with other objects than just algebras. For example, there can be sheaves of rings, modules, etc…

Sheaves

A sheaf is a pre-sheaf such that if is a collection of opens sets and such that

then there exists a unique such that

This is sometimes known as the “glueing property”.

Intuition

Sheaves are objects that “glue” nicely, so it is ok to describe them locally since we can always glue them together to get the unique global object that restricts to the local version. This is especially nice when describing things globally is hard (for example on a manifold where coordinates are only defined locally.)

Examples

For a topological space:

• The continuous functions are a sheaf
• Bounded continuous functions are a pre-sheaf, not a sheaf
• skyscraper sheaf For a (smooth) manifold
• is a sheaf (not it does NOT glue for closed sets, but it does for open ones.)
• Constant functions is a pre-sheaf not a sheaf
• Locally constant functions is a sheaf
• Smooth vector fields, is a sheaf
• Differential k-forms, is a sheaf

Categorical definition

Let be a topological space. We may define a category such that

• Objects: open sets of
• Morphisms: inclusions at open sets

For a category , a -valued pre-sheaf on is a functor out of the opposite category

that is we get the following diagram

Sheaves defined on topological basis

For a topological space and a basis , then any open set can be written

Suppose given

with coherent restriction maps (i.e. is a pre-sheaf), then we may define for using the inverse limit

Then is a sheaf on if and only if satisfies the sheaf condition. That is, if is a collection of basis open sets such that then

Morphisms

Let be a topological space and and be sheaves on . A morphism

is the data of morphisms

such that for

Pushforward of continuous map

Let be a continuous function and be a sheaf on . Define to be a sheaf on by

This is a pre-sheaf since the restrictions work well with inverse image by set-theory arguments.

This is a sheaf since for

satisfying

then since is a sheaf, then they glue uniquely to

So the pushforward of a sheaf is always a sheaf.