The Stacks project

Lemma 6.30.16. Let $f : X \to Y$ be a continuous map of topological spaces. Let $(\mathcal{C}, F)$ be a type of algebraic structures. Let $\mathcal{F}$ be a sheaf with values in $\mathcal{C}$ on $X$. Let $\mathcal{G}$ be a sheaf with values in $\mathcal{C}$ on $Y$. Let $\mathcal{B}_ Y$ be a basis for the topology on $Y$. Let $\mathcal{B}_ X$ be a basis for the topology on $X$. Suppose given for every $V \in \mathcal{B}_ Y$, and $U \in \mathcal{B}_ X$ such that $f(U) \subset V$ a morphism

\[ \varphi _ V^ U : \mathcal{G}(V) \longrightarrow \mathcal{F}(U) \]

of $\mathcal{C}$ compatible with restriction mappings. Then there is a unique $f$-map (see Definition 6.21.7 and the discussion of $f$-maps in Section 6.23) $\varphi : \mathcal{G} \to \mathcal{F}$ recovering $\varphi _ V^ U$ as the composition

\[ \mathcal{G}(V) \xrightarrow {\varphi _ V} \mathcal{F}(f^{-1}(V)) \xrightarrow {\text{restr.}} \mathcal{F}(U) \]

for every pair $(U, V)$ as above.

Proof. Let us first proves this for sheaves of sets. Fix $V \subset Y$ open. Pick $s \in \mathcal{G}(V)$. We are going to construct an element $\varphi _ V(s) \in \mathcal{F}(f^{-1}V)$. We can define a value $\varphi (s)_ x$ in the stalk $\mathcal{F}_ x$ for every $x \in f^{-1}V$ by picking a $U \in \mathcal{B}_ X$ with $x \in U \subset f^{-1}V$ and setting $\varphi (s)_ x$ equal to the equivalence class of $(U, \varphi _ V^ U(s))$ in the stalk. Clearly, the family $(\varphi (s)_ x)_{x \in f^{-1}V}$ satisfies condition $(*)$ because the maps $\varphi _ V^ U$ for varying $U$ are compatible with restrictions in the sheaf $\mathcal{F}$. Thus, by the proof of Lemma 6.30.6 we see that $(\varphi (s)_ x)_{x \in f^{-1}V}$ corresponds to a unique element $\varphi _ V(s)$ of $\mathcal{F}(f^{-1}V)$. Thus we have defined a set map $\varphi _ V : \mathcal{G}(V) \to \mathcal{F}(f^{-1}V)$. The compatibility between $\varphi _ V$ and $\varphi _ V^ U$ follows from Lemma 6.30.5.

We leave it to the reader to show that the construction of $\varphi _ V$ is compatible with restriction mappings as we vary $v \in \mathcal{B}_ Y$. Thus we may apply Lemma 6.30.14 above to “glue” them to the desired $f$-map.

Finally, we note that the map of sheaves of sets so constructed satisfies the property that the map on stalks

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

is the colimit of the system of maps $\varphi _ V^ U$ as $V \in \mathcal{B}_ Y$ varies over those elements that contain $f(x)$ and $U \in \mathcal{B}_ X$ varies over those elements that contain $x$. In particular, if $\mathcal{G}$ and $\mathcal{F}$ are the underlying sheaves of sets of sheaves of algebraic structures, then we see that the maps on stalks is a morphism of algebraic structures. Hence we conclude that the associated map of sheaves of underlying sets $f^{-1}\mathcal{G} \to \mathcal{F}$ satisfies the assumptions of Lemma 6.23.1. We conclude that $f^{-1}\mathcal{G} \to \mathcal{F}$ is a morphism of sheaves with values in $\mathcal{C}$. And by adjointness this means that $\varphi $ is an $f$-map of sheaves of algebraic structures. $\square$


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