English
For X ⟶ Y and x, x' with x specializes to x', the stalk maps satisfy a specific commutative relation when applied to a function y in the stalk of Y at the specialization image.
Русский
Для X ⟶ Y и x, x', где x специализируется на x', stalkMap удовлетворяет определённому коммутативному соотношению при применении к элементу в расплавленном stalk Y.
LaTeX
$$$$ f.stalkMap x (Y.presheaf.stalkSpecializes (f.base.hom.map_specializes h) y) = (f.stalkMap x') (X.presheaf.stalkSpecializes h y). $$$$
Lean4
theorem stalkSpecializes_stalkMap_apply (x x' : X) (h : x ⤳ x') (y) :
f.stalkMap x (Y.presheaf.stalkSpecializes (f.base.hom.map_specializes h) y) =
(X.presheaf.stalkSpecializes h (f.stalkMap x' y)) :=
DFunLike.congr_fun (CommRingCat.hom_ext_iff.mp (stalkSpecializes_stalkMap f x x' h)) y