Why are sheafs defined using abelian groups?

[dx]

TL;DR Summary

why are sheafs defined using abelian groups?

Let M be a manifold. The space of cotangent spaces to M is called the cotangent bundle T*M. a function on M can be lifted into the cotangent bundle. On the manifold T*M we can define a 1-form θ which describes the natural lifts and so on. A vector field on M is a section of the tangent bundle. The space of sections of T*M is denoted Γ(T*M) and is an example of a sheaf. The set of k-forms at the point p is denoted ∧^kT*M_p. the sheaf of p-forms on M is denoted Ω^p, which is equal to Γ(∧^pT*M). Restrictions of the sections Γ to an open set U has the natural structure of a vector space. If we want to talk about a sheaf of groups, or a sheaf of rings or a sheaf of some other type of objects, we have to assign abelian groups to the restrictions Γ(U) rather than vector spaces. how is this idea motivated or understood?

 

[Infrared]

First of all, $\Gamma(T^*M)$ usually denotes the space of global sections of $T^*M$ and so it not a sheaf (which should include the data of sections of $T^*M$ over any open subset of $M$).

If we want to talk about a sheaf of groups, or a sheaf of rings or a sheaf of some other type of objects, we have to assign abelian groups to the restrictions Γ(U) rather than vector spaces.

Why do you think this is true? You can have a sheaf of sets with no such structure.

Also, the plural of "sheaf" is "sheaves".

 

[martinbn]

Even the wikipedia article has enough information.

 

[dx]

In the book "A Panorama of Pure Mathematics" on page 243, Jean Dieudonne says that a sheaf is a "family of objects" that is "continuously parametrized by the points of an underlying space."

I was just wondering how the definition of sheaf achieves this idea, whatever it means. How does one motivate the definition of sheaf using the above mentioned vague idea of a family of objects varying as a function of a parameter?

 

[dx]

This same statement is made in the book "Differential Topology and Quantum Field Theory" by Charles Nash. He says "loosely speaking, one can think of a sheaf as a kind of parameterized family of functions." I'm not entirely sure what they are imagining when they say this.

 

[martinbn]

Do you understand the definition of a sheaf? And have seen some examples? If not, it would be impossible to learn from short statements meant for people that have already seen the definitions.

 

[dx]

Do you understand the definition of a sheaf? And have seen some examples? If not, it would be impossible to learn from short statements meant for people that have already seen the definitions.

Yes, I have seen the definition. I also know an example, the sheaf of p-forms. I would like to understand the idea of sheaf by analogy to fiber bundles. I have read that they are closely related notions. From the statements above, it seems like a sheaf of groups on a space X is somehow connected with the idea of a family of groups G_x which are parametrised in a suitably continuous way by the points x ∈ X. So it looks like G_x is like the fiber at the point x. But the technical definition of a sheaf, and also of things like germ, stalk, gerbe etc. are rather abstract. I would like to gain some intuition into these things so that the technical definitions are more digestible.

 

[dx]

The thought above is a direct quotation from Saunders MacLane

Roughly speaking, a sheaf A of abelian groups on a topological space X is a family of abelian groups A_x, parametrized by the points x ∈ X in a suitably "continuous" way. This means in particular that the disjoint union of all these groups is a space, so topologized that the projection of this space into X (sending each group A_x to the point x) is continuous and also etale

 

[jbergman]

The thought above is a direct quotation from Saunders MacLane

It's an interesting question and I would like to know how to intuitively think about sheaves. The technical definition seems pretty straightforward on wikipedia.

It just associates data/objects/functions/sections to your open sets of your manifold that respect restriction.

The simplest example would be continuous functions on $U$. This is an abelian group under restriction, (or vector space or a ring) and the satisfy the pre-sheaf requirements,

$$f: U \rightarrow \mathbb{R}, U \subset V \subset W, f_{|W} = (f_{|V})_{|W}$$

and the sheaf requirements which are just about the ability to glue together sheaves that agree on overlap.

You can see that your examples say of a vector field are also abelian group sheaves. You can add vector fields and restrict their domains to open sets.

From what I gather the POV isn't that useful for differential geometry because smooth functions can be extended to global smooth functions via partitions of unity. So everything can be considered as global sections.

But in other spaces, you can't do that so you need sheaves to talk about what is defined where.