instPreorder — Mathlib

English

For a morphism f: X → Y and points x, x' ∈ X with x = x', the stalk maps commute with the canonical stalk identifications.

Русский

Для морфизма f: X → Y и точек x, x' ∈ X, где x = x', композиция на stalk совпадает через канонические идентификации.

LaTeX

$$$ f_{\,*}(x) \\circ (X.presheaf.stalkCongr(\\mathrm{of\\_eq}\\, h_{xx'})).hom = (Y.presheaf.stalkCongr(\\mathrm{of\\_eq}(\\,h_{xx'}\\, \\text{rfl}))).hom \\circ f_{\,*}(x') $$$

Lean4

instance : Preorder X.AffineZariskiSite
    where
  le U V := ∃ f : Γ(X, V.toOpens), X.basicOpen f = U.toOpens
  le_refl U := ⟨1, Scheme.basicOpen_of_isUnit _ isUnit_one⟩
  le_trans := by
    rintro ⟨U, hU⟩ ⟨V, hV⟩ ⟨W, hW⟩ ⟨f, rfl⟩ ⟨g, rfl⟩
    exact hW.basicOpen_basicOpen_is_basicOpen g f

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