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| 1 | +\chapter{Motivation, sheaves and presheaves} |
| 2 | + |
| 3 | +\section{Introduction} |
| 4 | +\todo{Not sure if we want to add the examples etc.} |
| 5 | + |
| 6 | + |
| 7 | +\section{Sheaves and presheaves} |
| 8 | + |
| 9 | +\begin{defn} |
| 10 | + Let $X$ be a topological space. Write $\open(X)$ for the partially ordered set of opens of $X$. A \emph{presheaf} of sets on $X$ is a functor |
| 11 | + \[ |
| 12 | + F: \open(X)\opp \to \catSet. |
| 13 | + \] |
| 14 | +\end{defn} |
| 15 | +By changing the codomain we can obtain, for example, presheaves of abelian groups. In this course, we will focus almost entirely on presheaves of sets and of abelian groups. |
| 16 | +Let $U \subset V $ be open sets of $X$. The inclusion under $F$ gives a \emph{restriction} map $F(V) \to F(U)$. The naming comes from the following example: |
| 17 | + |
| 18 | +\begin{exmp}\label{exmp:ct-maps-presheaf} |
| 19 | + Let $X$ and $Z$ be topological spaces. The assignment $h_Z: \open(X)\opp \to \catSet$ by $U \mapsto \cont(U,Z)$ can be turned into a presheaf: given $U \subseteq V$ opens of $X$, define $$r_{UV}: \cont(V,Z) \to \cont(U,Z)$$ by $f \mapsto f|_U$. |
| 20 | +\end{exmp} |
| 21 | +We generalise the notation of function restriction. For $F(V) \to F(U)$, we denote the map pointwise by $s \mapsto s|_U$. |
| 22 | + |
| 23 | +\begin{defn}\label{defn:sheaf} |
| 24 | + Let $X$ be a topological space and $\mathcal{F}$ a presheaf \[\mathcal{F}: \open(X)\opp \to \catSet. \] We call $\mathcal{F}$ a \emph{sheaf} if |
| 25 | + for every open $U \subset X$ and every open cover $(U_i)_{i \in I}$ of $U$ with $\bigcup_i U_i = U$, the map |
| 26 | + \[ |
| 27 | + \mathcal{F}(U) \to \prod_{i\in I}\mathcal{F}(U_i) |
| 28 | + \] given by $s \mapsto (s|_{U_i})_{i \in I}$: |
| 29 | + \begin{enumerate} |
| 30 | + \item injective, and |
| 31 | + \item its image satisfies a \emph{gluing} condition: it is given by $\setpred{(s_i)_{i\in I}}{s_i |_{U_i \cap U_j} = s_j | _ {U_i \cap U_j} \forall i,j \in I}$. |
| 32 | + \end{enumerate} |
| 33 | +\end{defn} |
| 34 | + |
| 35 | +\begin{rmk}\label{rmk:sheafs-as-equalizers} |
| 36 | + One checks that requiring the conditions i) and ii) above are equivalent to requiring that |
| 37 | +\[\begin{tikzcd} |
| 38 | + {\mathcal{F}(U)} & {\prod_i\mathcal{F}(U_i)} & {\prod_{i,j}\mathcal{F}(U_i \cap U_j)} |
| 39 | + \arrow[from=1-1, to=1-2] |
| 40 | + \arrow["\alpha", shift left, from=1-2, to=1-3] |
| 41 | + \arrow["\beta"', shift right, from=1-2, to=1-3] |
| 42 | +\end{tikzcd}\] |
| 43 | + is an equaliser diagram for all $U$ open in $X$ and for all $(U_i)_{i \in I}$ open covers of $U$ , where $\alpha: (s_i)_{i \in I} \mapsto s_i|_{U_i \cap U_j}$ and $\beta: (s_i)_{i \in I} \mapsto s_j|_{U_i \cap U_j}$. |
| 44 | +\end{rmk} |
| 45 | + |
| 46 | +\begin{lem} \label{lem:ct-maps-sheaf} |
| 47 | + The presheaf $h_Z$ from \cref{exmp:ct-maps-presheaf} is a sheaf. |
| 48 | +\end{lem} |
| 49 | +\begin{proof} |
| 50 | + If two functions agree on every open of a cover of $U$ they agree on $U$, this gives \cref{defn:sheaf} i). For ii), we use the pasting lemma. |
| 51 | +\end{proof} |
| 52 | + |
| 53 | +\begin{exmp} |
| 54 | + Let $Z$ be a discrete topological space, let $X$ be a topological space. Given an open subset $U$ of $X$, a map $f: U \to Z$ is continuous if and only if it is locally constant. The sheaf $h_Z$ is called the \emph{constant sheaf} on the set $Z$, labelled $\constSheaf{Z}$ or $\constSheaf{Z}_X$. |
| 55 | +\end{exmp} |
| 56 | + |
| 57 | +\begin{exmp} |
| 58 | + If $X$ is a manifold, then the assignment $U \mapsto C^\infty(U, \mathbb{R})$ is a sheaf of $\mathbb{R}$ vector spaces. One can show that the assignment $U \mapsto \Omega^k(U)$ (smooth differential $k$-forms) is a sheaf. |
| 59 | +\end{exmp} |
| 60 | + |
| 61 | +\begin{lem}[name=Sheaf of sections]\label{exmp:sheaf-of-sections} |
| 62 | + Let $f: Y \to X$ be a continuous map of topological spaces. The assignment on opens of $X$ given by |
| 63 | + \[ |
| 64 | + h_{Y/X}: U \mapsto \setpred{s: U \to f^{-1}(U) }{ f \circ s = \id_U} =: \cont[X](U, Y) |
| 65 | + \] |
| 66 | + is a sheaf. |
| 67 | +\end{lem} |
| 68 | +\begin{proof} |
| 69 | + One can prove the above lemma in a similar way we proved \cref{lem:ct-maps-sheaf}. |
| 70 | + Alternatively, consider the diagram |
| 71 | + \[\begin{tikzcd} |
| 72 | + {\cont[X](U,Y)} & {\cont(U,Y)} \\ |
| 73 | + {\{U \to X\}} & {\cont(U,X)} |
| 74 | + \arrow["{f \circ -}", from=1-2, to=2-2] |
| 75 | + \arrow[from=1-1, to=2-1] |
| 76 | + \arrow[from=2-1, to=2-2] |
| 77 | + \arrow[from=1-1, to=1-2] |
| 78 | + \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=1-1, to=2-2] |
| 79 | +\end{tikzcd}\] |
| 80 | + and check that it is a pullback. We will come back to this in more detail in later lectures. |
| 81 | +\end{proof} |
| 82 | + |
| 83 | +The example in the lemma above is why the elements of $\mathcal{F}(U)$ are often called sections. |
| 84 | + |
| 85 | +\begin{exmp} |
| 86 | + Let $Y \xrightarrow{f} X$ be the $2:1$ cover of the circle: $f: z \mapsto z^2$, with $X := S^1$ and $Y: S^1$. |
| 87 | + On small intervals $U \subset X$ we get $f^{-1}(U) \cong U \times \{1,2\}$. We thus have two sections: $U \mapsto (U, 1)$ and $U \mapsto (U, 2)$. So $h_{Y/X}(U)$ has two elements |
| 88 | + On $V$ a union of small intervals, we get $2^{|\pi_0(V)|}$ elements, where $|\pi_0(V)|$ is the number of path components of $V$. |
| 89 | + On $W = X$, we get no sections. If $s: X \to Y$ is a section, then the induced map $s_*: \pi_1(X) \to \pi_1(Y)$ is a section to the map $f_*: \pi_1(Y) \to \pi_1(X)$. But this induced map is multiplication by $2$, and it does not have a section. |
| 90 | +\end{exmp} |
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