Presheaves

This post was machine-translated from the Korean original by Marvin (via Kimi). It may contain errors or awkward phrasing — the Korean original is the source of truth.

Gluing lemma

As we saw in §Subspaces, ⁋Proposition 7, given a continuous function $f:X \rightarrow Y$, restricting it to a family of subsets satisfying one of the two conditions in §Subspaces, ⁋Proposition 5 yields a continuous map. Specifically, these conditions correspond to the following two cases:

$(A_i)$ is an open covering of $X$, or
$(A_i)$ is a locally finite closed covering of $X$.

Conversely, given such a family $(A_i)$ and continuous functions $f_i$ defined on them, we may ask whether they determine a continuous function on $X=\bigcup A_i$.

Lemma 1 Let $X$ be a set and $(A_i)$ a family of subsets satisfying one of the two conditions in §Subspaces, ⁋Proposition 5. If a family of continuous functions $(f_i: A_i \rightarrow Y)$ satisfies

\[f_i\vert_{A_i\cap A_j}=f_j\vert_{A_i\cap A_j}\qquad\text{for all $i,j$}\]

then the function $f:X \rightarrow Y$ obtained by gluing them together is continuous.

Proof

First, the function $f$ is obtained from §Sum of Sets, ⁋Proposition 2. Its continuity follows from §Subspaces, ⁋Proposition 7.

Presheaves of continuous functions

Many structures studied in mathematics, especially in geometry, are spaces equipped with additional data defined over a topological space. To handle such structures, we need tools that manage the above process.

Define the category $\Open(X)$ as the category associated to the ordered set $(\mathcal{T}, \subseteq)$. That is, its objects are open sets, and for each $U\subseteq V$ there is a unique arrow $U\hookrightarrow V$.

Definition 2 For a topological space $X$, a contravariant functor $\mathcal{F}:\Open(X)^\op \rightarrow \Set$ is called a presheaf of sets on $X$.

By convention, presheaves are commonly denoted by $\mathcal{F}$ or $\mathcal{F}$; between these, the calligraphic $\mathcal{F}$ is slightly more natural. However, since we are already using this font for topological structures, we adopt the script font in the topology category.

Since $\mathcal{F}$ is contravariant, each inclusion $U\hookrightarrow V$ of open sets yields a morphism $\rho_{VU}: \mathcal{F}(V)\rightarrow \mathcal{F}(U)$, and because $\mathcal{F}$ preserves composition, whenever $U\hookrightarrow V\hookrightarrow W$ we must have $\rho_{WU}=\rho_{VU}\circ\rho_{WV}$.

Example 3 Let $X, Y$ be two topological spaces, and define $\mathcal{F}$ as follows.

For any open set $U$, set $\mathcal{F}(U)=\Hom_\Top(U, Y)$.
For open sets $U\subseteq V$, let $\rho_{VU}:\mathcal{F}(V) \rightarrow \mathcal{F}(U)$ be the restriction map sending a continuous function on $V$ to its restriction to $U$.

Then $\mathcal{F}$ is a presheaf.

In particular, this definition can be applied when a projection $p:Y \rightarrow X$ is given, by considering the presheaf $\mathcal{F}$ of continuous sections from $X$ to $Y$ (§Retraction and Section, ⁋Definition 2). Generalizing this, we make the following definition.

Definition 4 Let $\mathcal{F}$ be a presheaf defined on a topological space $X$.

For any open set $U\subseteq X$, the elements of $\mathcal{F}(U)$ are called sections over $U$. In particular, elements of $\mathcal{F}(X)$ are called global sections.
For open sets $U\subseteq V$, the map $\rho_{VU}:\mathcal{F}(V) \rightarrow \mathcal{F}(U)$ is called the restriction map from $V$ to $U$.
In particular, for open sets $U\subseteq V$ and $s\in \mathcal{F}(V)$, we write $\rho_{VU}(s)\in \mathcal{F}(U)$ simply as $s\vert_U$.

Meanwhile, in Definition 2 above, $\Set$ can be replaced by an appropriate category, such as $\Ab$. For instance, in Example 3, if $Y=\mathbb{R}$, we could use addition on $\mathbb{R}$ to define addition of continuous functions, and then $\mathcal{F}(U)$ would carry the structure of an abelian group. In such a case, $\mathcal{F}$ is called a presheaf of abelian groups on $X$. For convenience, we shall call a presheaf $\mathcal{F}: \Open(X)^\op \rightarrow \mathcal{A}$ an $\mathcal{A}$-valued presheaf. Among presheaves, those satisfying the gluing condition (Lemma 1) are called sheaves; we define these in the next post.

Examples of presheaves

Next, we examine several examples of presheaves.

Example 5 (Skyscraper sheaf) Fix a topological space $X$ and a point $i_x:\{x\}\hookrightarrow X$, and fix an object $A\in \mathcal{A}$. Then defining

\[(i_x)_\ast A(U)=\begin{cases}A&\text{if $x\in U$,}\\T&\text{if $x\not\in U$,}\end{cases}\qquad \text{$T$ a terminal object in $\mathcal{A}$}\]

and giving restriction maps as $\id_A$ or via the terminal object $T$ defines a presheaf. This is called the skyscraper sheaf.

Example 6 (Constant presheaf) Now fix a topological space $X$ and an object $A\in \mathcal{A}$, assign $A$ to every open set, and let all restriction maps be $\id_A$. This defines a presheaf, called the constant presheaf.

The following examples show ways to obtain new presheaves from an arbitrary presheaf.

Example 7 Given a presheaf $\mathcal{F}$ on $X$, for any open set $U$ we can define $\mathcal{F}\vert_U$ by

\[\mathcal{F}\vert_U(V)=\mathcal{F}(V)\quad\text{for all open $V\subseteq U$}\]

Then $\mathcal{F}\vert_U$ is a presheaf. (§Subspaces, ⁋Lemma 2)

Example 8 (Pushforward) Fix a continuous function $f:X \rightarrow Y$, and let $\mathcal{F}$ be a presheaf on $X$. The pushforward $f_\ast \mathcal{F}$ of $\mathcal{F}$ along $f$ is defined by

\[f_\ast \mathcal{F}(U)=\mathcal{F}(f^{-1}(U))\]

Stalks

Of course, what determines a function defined on $X$ is its value at each point $x\in X$. What distinguishes this from a function on $X$ as a set is that, thanks to the topological structure on $X$, we can also examine what happens in a neighborhood of the point $x$. Based on this intuition, we define the following.

Definition 9 Let $\mathcal{F}$ be a presheaf on a topological space $X$. For any point $x\in X$, the stalk $\mathcal{F}_x$ at $x$ is defined by

\[\mathcal{F}_x=\varinjlim_{x\in U}\mathcal{F}(U)\]

The elements of $\mathcal{F}_x$ are called germs at $x$.

In particular, if $\mathcal{F}$ is a presheaf valued in a complete category, then $\mathcal{F}_x$ is always well-defined. Meanwhile, writing out the limit explicitly in a concrete category, we have

\[\mathcal{F}_x=\{(s,U)\mid x\in U\in\mathcal{T},s\in\mathcal{F}(U)\}/\mathnormal{\sim}\]

where the equivalence relation $\sim$ is defined by

\[(s,U)\sim(t,V)\iff\text{$\exists$ open neighborhood $W\subseteq U\cap V$ of $x$ satisfying $\rho_{UW}(s)=\rho_{VW}(t)$}\]

That is, intuitively, the elements $(s,U)$ of $\mathcal{F}_x$ can be thought of as the value $s(x)$ at $x$ together with additional local information¹ about $s$ near $x$. For convenience, for any $s\in \mathcal{F}(U)$, we write the image of $s$ under $\mathcal{F}(U) \rightarrow \mathcal{F}_x$ as $s_x$.

For now, a presheaf is more of an object with additional algebraic information than a geometric one, but it is possible to make it into a geometric object. Consider a presheaf $\mathcal{F}$ on a topological space $X$ and the set

\[\Spe(\mathcal{F})=\coprod_{x\in X} \mathcal{F}_x=\{(x,\xi)\mid x\in X, \xi\in \mathcal{F}_x\}\]

Then for any open set $U\subseteq X$ and any $s\in \mathcal{F}(U)$, the functions

\[\tilde{s}:U \rightarrow \Spe(\mathcal{F}); \quad x\mapsto (x,s_x)\]

exist. We endow $\Spe(\mathcal{F})$ with the final topology defined by this family of functions (§Initial and Final Topology, ⁋Definition 4), and call this space the étalé space of $\mathcal{F}$.

Morphisms of presheaves

Definition 10 Let $\mathcal{F}, \mathcal{G}:\Open(X) \rightarrow \mathcal{A}$ be two presheaves on a fixed topological space $X$. A natural transformation between them is called a presheaf morphism.

Thus the category of $\mathcal{A}$-valued presheaves on $X$ is the functor category $[\Open(X)^\op, \mathcal{A}]$. We denote this by $\PSh(X, \mathcal{A})$, and when there is no risk of confusion from context, we simply write $\PSh(X)$. As a side note, the $f_\ast$ from Example 8 (Pushforward) is a functor $\PSh(X, \mathcal{A})\rightarrow \PSh(Y, \mathcal{A})$.

Thinking of the intuitive Example 3, for an open set $U$ the map $\phi(U):\mathcal{F}(U) \rightarrow \mathcal{G}(U)$ can be thought of as the function obtained by restricting $\phi:\mathcal{F}\rightarrow \mathcal{G}$ to $U$, so we sometimes write $\phi\vert_U$ instead of $\phi(U)$.

Meanwhile, by the universal property of the limit cone, the following proposition holds.

Proposition 11 Let $\phi:\mathcal{F}\rightarrow\mathcal{G}$ be a morphism of presheaves on a topological space $X$. Then for any $x\in X$, a morphism $\phi_x:\mathcal{F}_x\rightarrow\mathcal{G}_x$ between stalks is naturally induced.

The following examples should have appeared under Examples of presheaves above, but were postponed because we had not yet defined presheaf morphisms.

Example 12 (Sheaf Hom) Fix two presheaves $\mathcal{F}, \mathcal{G}$ and define, for any $U$,

\[\mathcal{Hom}(\mathcal{F},\mathcal{G})(U)=\Hom_{\PSh(U)}(\mathcal{F}\vert_U, \mathcal{G}\vert_U)\]

Example 13 (Product) For a family of presheaves $(\mathcal{F}_i:\Open(X) \rightarrow \Set)_{i\in I}$ on a topological space $X$, their product $\prod_{i\in I} \mathcal{F}_i$ can be defined by

\[\left(\prod_{i\in I} \mathcal{F}_i\right)(U)=\prod_{i\in I} \mathcal{F}_i(U)\]

Using definitions as above, structures defined in a category $\mathcal{A}$—for example products, coproducts, limits, colimits, monoidal products, etc.—can be transferred to $\PSh(X, \mathcal{A})$. In particular, $\PSh(X, \Ab)$ inherits the monoidal structure $(\Ab,\otimes, \mathbb{Z})$, and the monoidal objects here are $\PSh(X, \Ring)$. In the same vein, the following example can be understood.

Example 14 For a $\Ring$-valued presheaf $\mathcal{O}_X$ on a topological space $X$, a left $\mathcal{O}_X$-module object $\mathcal{F}\in\PSh(X,\Ab)$ is simply called an $\mathcal{O}_X$-module.

Abelian presheaves

Until now we have ignored the category in which a presheaf takes values; now we focus on presheaves valued in the category $\Ab$.

Definition 15 For a topological space $X$, a contravariant functor $\Open(X)\rightarrow\Ab$ is called an abelian presheaf.

Definition 16 Let $\phi:\mathcal{F}\rightarrow\mathcal{G}$ be a morphism of abelian presheaves on a topological space $X$. The presheaf kernel $\ker\phi$ of $\phi$ is the data consisting of:

For each open set $U\subseteq X$, the assignment $U\mapsto \ker(\phi(U))$;
For each inclusion $U\subseteq V$ of open sets, the restriction map $\rho_{VU}:\ker(\phi(V))\rightarrow\ker(\phi(U))$ uniquely determined by the following diagram:

$presheaf_kernel-1$

In this definition, $\rho_{VU}$ is the restriction map uniquely determined by the universal property of $\ker(\phi(U))$.

Lemma 17 The $\ker\phi$ defined above is an (abelian) presheaf on $X$.

Proof

This is immediate from the following diagram and the universal property of kernels:

$presheaf_kernel-2$

In the same way, one can define presheaf cokernel, presheaf image, presheaf coimage, presheaf quotient, and so on. Therefore, the category $\PSh(X,\Ab)$ of abelian presheaves on a given topological space $X$ is an abelian category.

References

[Vak] R. Vakil, The rising sea: Foundation of algebraic geometry. Available online.

For example, if $X=\mathbb{R}$, then to define the derivative of a function at a point $x$ of $\mathbb{R}$, it suffices to know the values of $f$ in an arbitrarily small neighborhood of $x$. From this perspective, $f'(x)$ can be regarded as one of the pieces of local information that $x$ carries. ↩

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Gluing lemma

Presheaves of continuous functions

Examples of presheaves

Stalks

Morphisms of presheaves

Abelian presheaves

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