Topological Spaces, Abelian Groups and Sheaves

Topology is the study of shapes. A topological space is a set equipped with a collection of, what is called, open sets (subject to certain mild axioms). These data of open sets give the set its shape as open sets specify the proximity between points. For example, if you fix a set S and specify its open sets as its all subsets, that would give the set S its most discrete form. If you specify its open subsets as minimal as possible (i.e. only the empty subset and S itself), this would give the set S its most indiscrete form.
Let’s consider a more specific example for further reference. Consider the set of complex numbers (1, e, 2+i etc). Equip this set by specifying its open subsets as any subset that misses only finitely many points (i.e. whose complement is a finite set). Note that these open subsets are “big and rare”, so the topology they induce will be quite rigid.

An abelian group is a set equipped with, what is called, a binary operation (subject to certain axioms). Elements of this set can be “added” and “subtracted” where the “addition” operation does not care about the order of the elements.

This much was only a warm-up. The star notion of this writing is Sheaves. Sheaves are basic objects used in different kinds of geometry in mathematics. A sheaf (of abelian groups) is simply a topological space equipped with a collection of functions. More precisely, a (pre)sheaf on a topological space X is an assignment of abelian groups to each open subset of X that should be considered as functions on the corresponding open subset such that among these abelian groups, there are restriction maps for every inclusion of open subsets. That is, “a function on a bigger open subset restricts to a function on a smaller open subset”. Moreover, if one has a coherent collection of functions that are defined over a bunch of open subsets, these functions uniquely glue to a function that is defined on the union of these open subsets. This is called the glueing axiom and we will not worry about the uniqueness axiom for brevity.

Let’s begin with an example. Take a topological space X and for each open subset U of X, consider the collection C(U) of real valued functions (f:U—> R) defined on U. This collection has the structure of an abelian group as functions can be added up pointwise.

Let’s consider a more algebraic example. Recall the topological space we considered above: The set of complex number with the described topology. The collection of continuous functions on this set (and its open subsets) might be huge. We would like to consider a smaller collection of functions (yielding a more rigid geometry): regular functions. A regular function on an open subset U is a fraction f / g of two polynomials f and g where g has no zeroes on U. For example, 1 / x is a regular function on C-{0} (imagine a punctured plane) whereas 1/(x-2) is not. Of course, every polynomial (x, 5x^2+1, x^7+x) is a regular function on every (nonempty) open subset. Moreover, the global functions (i.e. those regular functions defined on all C are all polynomials. The smaller your open set is, the larger the collection of regular functions on it.

In the way we presented the second example, first comes the topological space and then the sheaf of regular functions on it. In fact, it should have been presented in the opposite direction. One first takes an algebraic structure called a commutative ring R, considers a topological space X attach to it (called the prime spectrum of R) and then considers the set X with the sheaf of regular functions on it as a whole then arriving at the most beautiful structures of modern mathematics: Schemes.

To be continued..

Author: Canberk İrimağzı

References: Liu, Qing. Algebraic Geometry and Arithmetic Curves. https://books.google.com/books/about/Algebraic_Geometry_and_Arithmetic_Curves.html?id=ePpzBAAAQBAJ&printsec=frontcover&source=kp_read_button