The Stacks project

Definition 6.21.7. Let $f : X \to Y$ be a continuous map. Let $\mathcal{F}$ be a sheaf of sets on $X$ and let $\mathcal{G}$ be a sheaf of sets on $Y$. An $f$-map $\xi : \mathcal{G} \to \mathcal{F}$ is a collection of maps $\xi _ V : \mathcal{G}(V) \to \mathcal{F}(f^{-1}(V))$ indexed by open subsets $V \subset Y$ such that

\[ \xymatrix{ \mathcal{G}(V) \ar[r]_{\xi _ V} \ar[d]_{\text{restriction of }\mathcal{G}} & \mathcal{F}(f^{-1}V) \ar[d]^{\text{restriction of }\mathcal{F}} \\ \mathcal{G}(V') \ar[r]^{\xi _{V'}} & \mathcal{F}(f^{-1}V') } \]

commutes for all $V' \subset V \subset Y$ open.

Comments (2)

Comment #6037 by Marin Genov on

Hi! Perhaps I am missing something, but what is the raison d'etre of this definition and the short exposition that follows it? More concretely, how is this any different from a morphism of (pre)sheaves G --> f_{*} F ?

If the idea is to give another interpretation of the adjunction b/w the pullback and pushforward of sheaves, perhaps it's reasonable to establish the equality of both sides of the adjunction with the set of collections of maps \phi_{VU}: G(V) \to F(U) compatible with restrictions (as Ravi Vakil does in The Rising Sea FAG, p. 93, Ex. 2.7.B).

What do you think?

Comment #6038 by on


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