The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

6.3 Presheaves

Definition 6.3.1. Let $X$ be a topological space.

  1. A presheaf $\mathcal{F}$ of sets on $X$ is a rule which assigns to each open $U \subset X$ a set $\mathcal{F}(U)$ and to each inclusion $V \subset U$ a map $\rho ^ U_ V : \mathcal{F}(U) \to \mathcal{F}(V)$ such that $\rho ^ U_ U = \text{id}_{\mathcal{F}(U)}$ and whenever $W \subset V \subset U$ we have $\rho ^ U_ W = \rho ^ V_ W \circ \rho ^ U_ V$.

  2. A morphism $\varphi : \mathcal{F} \to \mathcal{G}$ of presheaves of sets on $X$ is a rule which assigns to each open $U \subset X$ a map of sets $\varphi : \mathcal{F}(U) \to \mathcal{G}(U)$ compatible with restriction maps, i.e., whenever $V \subset U \subset X$ are open the diagram

    \[ \xymatrix{ \mathcal{F}(U) \ar[r]^\varphi \ar[d]^{\rho ^ U_ V} & \mathcal{G}(U) \ar[d]^{\rho ^ U_ V} \\ \mathcal{F}(V) \ar[r]^\varphi & \mathcal{G}(V) } \]

    commutes.

  3. The category of presheaves of sets on $X$ will be denoted $\textit{PSh}(X)$.

The elements of the set $\mathcal{F}(U)$ are called the sections of $\mathcal{F}$ over $U$. For every $V \subset U$ the map $\rho ^ U_ V : \mathcal{F}(U) \to \mathcal{F}(V)$ is called the restriction map. We will use the notation $s|_ V := \rho ^ U_ V(s)$ if $s\in \mathcal{F}(U)$. This notation is consistent with the notion of restriction of functions from topology because if $W \subset V \subset U$ and $s$ is a section of $\mathcal{F}$ over $U$ then $s|_ W = (s|_ V)|_ W$ by the property of the restriction maps expressed in the definition above.

Another notation that is often used is to indicate sections over an open $U$ by the symbol $\Gamma (U, -)$ or by $H^0(U, -)$. In other words, the following equalities are tautological

\[ \Gamma (U, \mathcal{F}) = \mathcal{F}(U) = H^0(U, \mathcal{F}). \]

In this chapter we will not use this notation, but in others we will.

Definition 6.3.2. Let $X$ be a topological space. Let $A$ be a set. The constant presheaf with value $A$ is the presheaf that assigns the set $A$ to every open $U \subset X$, and such that all restriction mappings are $\text{id}_ A$.


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