# Presheaves

One of the themes that pervade mathematics is the study of sets and functions between sets. In previous posts we have seen some interesting things that we can do with sets; for instance we can put an “arrangement” or “organization” on them using the concept of a topology (see Basics of Topology and Continuous Functions), or we can define a law of composition on them (see Groups), so that we can say that they are closed under some kind of operation (see also Rings, Fields, and IdealsMore on Ideals, and Vector Spaces, Modules, and Linear Algebra).

We will now construct yet another interesting thing from sets. We will need two kinds; the first is a set equipped with a topology, or topological space; the other is either just an ordinary set, or equipped with a law of composition, either groups, or rings, or modules. Note that functions between sets also themselves form sets; we now use this to construct a motivating example for the concepts that we are about to introduce.

Consider the set of all complex numbers $\mathbb{C}$ (also called the complex plane) equipped with the topology where we declare the closed sets to be the finite sets of complex numbers (which can be imagined as a finite number of points in the complex plane). The open sets are therefore the complements of these closed sets. This is a special case of what is called the Zariski topology.

Consider also the functions from the complex numbers to the complex numbers which are of the form $\frac{f}{g}$ where $f$ and $g$ are polynomials (the functions we refer to in the rest of this post will be of this form). Examples of these functions are $x$, $x+1$, $x^2$, and so on. Consider now the function

$\displaystyle \frac{1}{x}$.

It is not actually a function from the complex numbers to the complex numbers. Why? Because it does not send the complex number $0$ anywhere, and a function must send every element of its domain to some element in its range. But we can say that it is a function from the set of complex numbers except for the complex number $0$, written $\mathbb{C}-\{0\}$ or $\mathbb{C}\backslash \{0\}$, to the complex numbers. This set $\mathbb{C}-\{0\}$ is of course a subset, and actually an open subset, of the set of all complex numbers $\mathbb{C}$.

For ease of notation, we keep the range of our functions fixed (in this case it is the complex numbers) and speak informally of functions “on”, or sometimes “living on” their respective domains.

Although $\mathbb{C}-\{0\}$ is “smaller” than $\mathbb{C}$, there are actually more functions on $\mathbb{C}-\{0\}$ than on $\mathbb{C}$. Aside from $\frac{1}{x}$, we also have $\frac{1}{x^2}$, $\frac{1}{x^3}$, and all other functions whose denominator otherwise would have been $0$ on the complex number $0$ and nowhere else on the complex plane $\mathbb{C}$. If we take an even “smaller” open subset of $\mathbb{C}$, such as $\mathbb{C}-\{0,1\}$, we will obtain even more functions on this open subset, such as $\frac{1}{x-1}$ and $\frac{1}{x^{2}-x}$.

At the same time, for every function $f$ on $\mathbb{C}$ there is also a corresponding function $f|_{\mathbb{C}-\{0\}}$ on the open subset $\mathbb{C}-\{0\}$ which assigns to every element of $\mathbb{C}-\{0\}$ the same element that the function $f$ on $\mathbb{C}$ assigns to every element of $\mathbb{C}-\{0\}$. Technically $f|_{\mathbb{C}-\{0\}}$ is a different function from $f$ because it has a different domain. It is called the restriction of $f$ to $\mathbb{C}-\{0\}$. For every function on $\mathbb{C}-\{0\}$ there are also corresponding restrictions to $\mathbb{C}-\{0,1\}$.

In order to formalize this, we note that if $V$ is a subset of $U$, usually written $V\subseteq U$, then we have a function called an inclusion function, or inclusion, from $V$ to $U$, which sends every element of $V$ to the same element in $U$. If we write the set of functions on $V$ as $\mathcal{F}(V)$,  and the set of functions on $U$ as $\mathcal{F}(U)$, we obtain a map from $\mathcal{F}(U)$ to $\mathcal{F}(V)$ that assigns to every function on $U$ its restriction to $\mathcal{F}(V)$. This mapping from $\mathcal{F}(U)$ to $\mathcal{F}(V)$ is called a restriction function, or restriction map.

We can summarize and generalize our discussion above as the condition that whenever we have an inclusion from $V$ to $U$, then we also have a restriction map from $\mathcal{F}(U)$ to $\mathcal{F}(V)$. We now obtain the “classical” notion of a presheaf. For the more rigorous definition, we quote from the book Algebraic Geometry by Robin Hartshorne:

Let $X$ be a topological space. A presheaf $\mathcal{F}$of abelian groups on $X$ consists of the data

(a) for every open subset $U\subseteq X$, an abelian group $\mathcal{F}(U)$, and

(b) for every inclusion $V\subseteq U$ of open subsets of $X$, a morphism of abelian groups $\rho_{UV}:\mathcal{F}(U)\rightarrow \mathcal{F}(V)$,

subject to the conditions

(0) $\mathcal{F}(\varnothing)$, where $\varnothing$ is the empty set,

(1) $\rho_{UU}$ is the identity map, $\mathcal{F}(U)\rightarrow \mathcal{F}(U)$,

(2) if $W\subseteq V \subseteq U$ are three open subsets, then $\rho_{UW}=\rho_{VW}\circ\rho_{UV}$.

Since this particular definition in the book of Hartshorne only defines presheaves of abelian groups, the functions $\rho_{UV}$ are required to be morphisms, which means that they respect the abelian group structure on $\mathcal{F}(U)$ and $\mathcal{F}(V)$, i.e. if we write the law of composition of the abelian group using “$+$“, and we have $a+b=c$ in the domain, then a morphism of abelian groups is a function that satisfies $f(a)+f(b)=f(c)$. However, presheaves can be defined for more general sets and functions.

We quote some more useful terminology from the book of Hartshorne:

If $\mathcal{F}$ is a presheaf on $X$, we refer to $\mathcal{F}(U)$ as the sections of the presheaf $\mathcal{F}$ over the open set $U$, and we sometimes  use the notation $\Gamma(U,\mathcal{F})$ to denote the group $\mathcal{F}(U)$. We call the maps $\rho_{UV}$ restriction maps, and we sometimes write $s|_{V}$ instead of $\rho_{UV}(s)$, if $s\in\mathcal{F}$.

The concept of presheaf can be generalized even further so that the functions from $V$ to $U$ need not be inclusion functions. Together with the generalization of the concept of open covers in topology, and the concept of a sheaf, this leads to the concepts of site and topos.

References:

Sheaf on Wikipedia

Presheaf on Wikipedia

Algebraic Geometry by Andreas Gathmann

The Rising Sea: Foundations of Algebraic Geometry by Ravi Vakil

Algebraic Geometry by Robin Hartshorne

Sheaves in Geometry and Logic: A First Introduction to Topos Theory by Saunders Mac Lane and Ieke Moerdijk