---- > [!definition] Definition. ([[morphism of (pre)sheaves]]) > > A **morphism of [[presheaf|presheaves]] $\mathcal{F}$, $\mathcal{G}$ on $X$**, denote $f:\mathcal{F} \to \mathcal{G}$, is a [[natural transformation]] of the [[contravariant functor|functors]]. Concretely, $f$ is the data of, for each open $U \subset X$, a morphism $f_{U}:\mathcal{F}(U) \to \mathcal{G}(U)$ that is compatible with restriction, in the sense that whenever $V \subset U$ we get a commutative diagram (naturality square): > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAB12BbOnACwDGjYADEAvgAoAqgEoQY0uky58hFAEZyVWoxZtOPfkIajJANTkKl2PASJl12+s1aIO3XoOEBxSbPmKIBg2qkSajtTOem4GnsbAvhIW8towUADm8ESgAGYAThBcSGQgOBBIAEyRuq4gOQD6UiDUDHQARjAMAArKtmogeVjpfDgBuQVFiJql5YgAzNUubA1mY3UTxdRlSNNRtbFGwuL1wFJmYmv5hZVbsws6SzEehya+J2cXLe2dPSF2boNhqMxBQxEA > \begin{tikzcd} > \mathcal{F}(U) \arrow[d, "f_U"'] \arrow[r, "\mathcal{F}_{UV}"] & \mathcal{F}(V) \arrow[d, "f_V"] \\ > \mathcal{G}(U) \arrow[r, "\mathcal{G}_{UV}"'] & \mathcal{G}(V) > \end{tikzcd} > \end{document} > ``` > > A **morphism of [[sheaf|sheaves]]** is just a morphism of the underlying presheaves. ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```