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*}

6.23 Continuous maps and sheaves of algebraic structures

Let $(\mathcal{C}, F)$ be a type of algebraic structure. For a topological space $X$ let us introduce the notation:

  1. $\textit{PSh}(X, \mathcal{C})$ will be the category of presheaves with values in $\mathcal{C}$.

  2. $\mathop{\mathit{Sh}}\nolimits (X, \mathcal{C})$ will be the category of sheaves with values in $\mathcal{C}$.

Let $f : X \to Y$ be a continuous map of topological spaces. The same arguments as in the previous section show there are functors

\begin{eqnarray*} f_* : \textit{PSh}(X, \mathcal{C}) & \longrightarrow & \textit{PSh}(Y, \mathcal{C}) \\ f_* : \mathop{\mathit{Sh}}\nolimits (X, \mathcal{C}) & \longrightarrow & \mathop{\mathit{Sh}}\nolimits (Y, \mathcal{C}) \\ f_ p : \textit{PSh}(Y, \mathcal{C}) & \longrightarrow & \textit{PSh}(X, \mathcal{C}) \\ f^{-1} : \mathop{\mathit{Sh}}\nolimits (Y, \mathcal{C}) & \longrightarrow & \mathop{\mathit{Sh}}\nolimits (X, \mathcal{C}) \end{eqnarray*}

constructed in the same manner and with the same properties as the functors constructed for abelian (pre)sheaves. In particular there are commutative diagrams

\[ \xymatrix{ \textit{PSh}(X, \mathcal{C}) \ar[r]^{f_*} \ar[d]^ F & \textit{PSh}(Y, \mathcal{C}) \ar[d]^ F & \mathop{\mathit{Sh}}\nolimits (X, \mathcal{C}) \ar[r]^{f_*} \ar[d]^ F & \mathop{\mathit{Sh}}\nolimits (Y, \mathcal{C}) \ar[d]^ F \\ \textit{PSh}(X) \ar[r]^{f_*} & \textit{PSh}(Y) & \mathop{\mathit{Sh}}\nolimits (X) \ar[r]^{f_*} & \mathop{\mathit{Sh}}\nolimits (Y) \\ \textit{PSh}(Y, \mathcal{C}) \ar[r]^{f_ p} \ar[d]^ F & \textit{PSh}(X, \mathcal{C}) \ar[d]^ F & \mathop{\mathit{Sh}}\nolimits (Y, \mathcal{C}) \ar[r]^{f^{-1}} \ar[d]^ F & \mathop{\mathit{Sh}}\nolimits (X, \mathcal{C}) \ar[d]^ F \\ \textit{PSh}(Y) \ar[r]^{f_ p} & \textit{PSh}(X) & \mathop{\mathit{Sh}}\nolimits (Y) \ar[r]^{f^{-1}} & \mathop{\mathit{Sh}}\nolimits (X) } \]

The main formulas to keep in mind are the following

\begin{eqnarray*} f_*\mathcal{F}(V) & = & \mathcal{F}(f^{-1}(V)) \\ f_ p\mathcal{G}(U) & = & \mathop{\mathrm{colim}}\nolimits _{f(U) \subset V} \mathcal{G}(V) \\ f^{-1}\mathcal{G} & = & (f_ p\mathcal{G})^\# \\ (f_ p\mathcal{G})_ x & = & \mathcal{G}_{f(x)} \\ (f^{-1}\mathcal{G})_ x & = & \mathcal{G}_{f(x)} \end{eqnarray*}

Each of these formulas has the property that they hold in the category $\mathcal{C}$ and that upon taking underlying sets we get the corresponding formula for presheaves of sets. In addition we have the adjointness properties

\begin{eqnarray*} \mathop{Mor}\nolimits _{\textit{PSh}(X, \mathcal{C})}(f_ p\mathcal{G}, \mathcal{F}) & = & \mathop{Mor}\nolimits _{\textit{PSh}(Y, \mathcal{C})}(\mathcal{G}, f_*\mathcal{F}) \\ \mathop{Mor}\nolimits _{\mathop{\mathit{Sh}}\nolimits (X, \mathcal{C})}(f^{-1}\mathcal{G}, \mathcal{F}) & = & \mathop{Mor}\nolimits _{\mathop{\mathit{Sh}}\nolimits (Y, \mathcal{C})}(\mathcal{G}, f_*\mathcal{F}). \end{eqnarray*}

To prove these, the main step is to construct the maps

\[ i_\mathcal {G} : \mathcal{G} \longrightarrow f_*f_ p\mathcal{G} \]

and

\[ c_\mathcal {F} : f_ p f_* \mathcal{F} \longrightarrow \mathcal{F} \]

which occur in the proof of Lemma 6.21.3 as morphisms of presheaves with values in $\mathcal{C}$. This may be safely left to the reader since the constructions are exactly the same as in the case of presheaves of sets.

Given a continuous map $f : X \to Y$ and sheaves of algebraic structures $\mathcal{F}$ on $X$, $\mathcal{G}$ on $Y$, the notion of an $f$-map $\mathcal{G} \to \mathcal{F}$ of sheaves of algebraic structures makes sense. We can just define it exactly as in Definition 6.21.7 (replacing maps of sets with morphisms in $\mathcal{C}$) or we can simply say that it is the same as a map of sheaves of algebraic structures $\mathcal{G} \to f_*\mathcal{F}$. We will use this notion freely in the following. The set of $f$-maps between $\mathcal{G}$ and $\mathcal{F}$ will be in canonical bijection with the sets $\mathop{Mor}\nolimits _{\mathop{\mathit{Sh}}\nolimits (X, \mathcal{C})}(f^{-1}\mathcal{G}, \mathcal{F})$ and $\mathop{Mor}\nolimits _{\mathop{\mathit{Sh}}\nolimits (Y, \mathcal{C})}(\mathcal{G}, f_*\mathcal{F})$.

Composition of $f$-maps is defined in exactly the same manner as in the case of $f$-maps of sheaves of sets. In addition, given an $f$-map $\mathcal{G} \to \mathcal{F}$ as above, the induced maps on stalks

\[ \varphi _ x : \mathcal{G}_{f(x)} \longrightarrow \mathcal{F}_ x \]

are homomorphisms of algebraic structures.

Lemma 6.23.1. Let $f : X \to Y$ be a continuous map of topological spaces. Suppose given sheaves of algebraic structures $\mathcal{F}$ on $X$, $\mathcal{G}$ on $Y$. Let $\varphi : \mathcal{G} \to \mathcal{F}$ be an $f$-map of underlying sheaves of sets. If for every $V \subset Y$ open the map of sets $\varphi _ V : \mathcal{G}(V) \to \mathcal{F}(f^{-1}V)$ is the effect of a morphism in $\mathcal{C}$ on underlying sets, then $\varphi $ comes from a unique $f$-morphism between sheaves of algebraic structures.

Proof. Omitted. $\square$


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