Category: mathematics

An Accessible Introduction to Sheaves

An Accessible Introduction to Sheaves

Below is reproduced a post that will be featured on the blog cozilikethinking at https://cozilikethinking.wordpress.com/ as a part of a series of posts on introductory algebraic geometry. This reproduction is partly for the sake of ensuring there are no \LaTeX errors and partly because I would like to soon begin blogging again, for the first time here. The aforementioned blog, cozilikethinking, is run by Ayush Khaitan, who I have had great pleasure communicating with and who writes exceptionally about a wide variety of mathematical topics, which are typically within the realm of undergraduate interest. Without further ado, below is the post.

Let me just preface this by saying that I look forward to writing, in an accessible way, about the realm of algebraic geometry with Ayush on this blog.

In the study of algebraic geometry, one often hopes to delve into abstract notions, research tools, and topics such as those of cohomology, schemes, orbifolds, and stacks (or maybe you just want to prove that there are 27 lines on a cubic surface). But, all of these things probably seem very far away, because algebraic geometry is a very rich, very technical field. It is also one that was inaccessible to most until recently, despite often being concerned with rather simple ideas and objects. Luckily, today we all need not read the SGA and EGA. While I think it unwise to jump ahead to something as modern as schemes just yet, it should be possible to study one of algebraic geometry’s most important and deceivingly intuitive tools: sheaves.

In not-so-technical terms, a sheaf is a device used to track locally defined information attached to the open sets of some (topological) space. Basically, this is a nice tool to organize information. To get a bit more formal, we’ll need to first define what a presheaf is.

Presheaf: A presheaf F on a topological space X is a functor with values in some category \mathbf{C} defined as follows:

  • For each open set U \subset X, there is a corresponding object F(U) in \mathbf{C}
  • For each inclusion of sets S \subseteq U there exists a corresponding morphism, called the restriction morphism, \mathrm{res}_{S, U}: F(U) \to F(S) in \mathbf{C}
  • \mathrm{res}_{S, U} must satisfy the following:
    • For all U \subset X, \mathrm{res}_{U,U}:F(U) \to F(U) is the identity morphism on F(U)
    • For open sets S \subseteq T \subseteq U, \mathrm{res}_{S,T} \circ \mathrm{res}_{T,U} = \mathrm{res}_{S,U}, i.e. restriction can be done all at once or in steps.

Presheaves are certainly important, but I will stay focused on our goal to understand sheaves instead of dealing with details about presheaves. With that having been said, recall the property of locality from our loose definition of a sheaf, as it is also our first axiom for sheaves, with the other being concatenation or gluing. These two axioms may also be thought of as ensuring existence and uniqueness.

Sheaf: A presheaf satisfying the following:

  • (Locality) If \{U_i\} is an open covering of the open U, and if x,y \in F(U) such that \mathrm{res}_{U_i,U}(x)=\mathrm{res}_{U_i,U}(y) for each U_i, then x=y.
  • (Gluing) Suppose \{ U_i \} is an open cover of U. Further suppose that if for all i a section s_i \in F(U_i) is given such that for each pairing U_i, U_k of the covering sets \mathrm{res}_{U_i \cap U_k, U}(s_i)=\mathrm{res}_{U_i \cap U_k, U}(s_j), then there exists s \in F(U) where \mathrm{res}_{U_i,U}(s)=s_i \; \forall i.

In other words, a sheaf is a presheaf if we can uniquely “glue” pieces together.

For the sake of brevity and simplicity, we will ignore some technical details henceforth. We will also focus on functions and let the reader work out the details for the same reason.

For our first example, consider topological spaces X and Y with a “rule” \mathscr{F} such that open U \subset X is associated with

\mathscr{F}(U)=\{f: U \to Y : f \; \mathrm{is \; continuous} \}

This is a sheaf. It is actually a pretty well-known example too. (Hint: In justifying that this is a sheaf, it may be a good strategy to begin by considering restriction maps.)

For our second and final example, we shall consider the sheaf of infinitely differentiable (\mathbb{C}^{\infty} smooth) functions. The sheaf of infinitely differentiable functions of a differential manifold M has two properties:

  • For all open sets U the ring of C^{\infty} functions is associated from U to \mathbb{R}
  • The restriction map is a function restriction

Again, it’s worth playing around with this sheaf a bit.

So, to review a sheaf is a tool used in algebraic geometry and other fields, and it serves as a sort of data collection method such that the data is all put in one place, i.e. a point.

Notes

  • It’s worth noting that there are tons of different definitions of presheaves and sheaves, but I think the provided one is most intuitive and works well for many instances.
  • An alternate notation for \mathrm{res}_{V,U}(s) is s|_{V}; keep this in mind if you plan to read more on this topic.
  • Be sure to keep functions in mind while you’re developing an understanding of sheaves. Another example to consider is the sheaf of regular functions.