Theorems · Theorem · algebraic geometry

AlgebraicGeometry.Scheme.Cover.ColimitGluingData.trans_app_left

∀ {P : CategoryTheory.MorphismProperty AlgebraicGeometry.Scheme} [inst : P.IsStableUnderBaseChange]
  [inst_1 : P.IsMultiplicative] {S : AlgebraicGeometry.Scheme} {J : Type u_1}
  [inst_2 : CategoryTheory.Category.{v_1, u_1} J] {D : CategoryTheory.Functor J (P.Over ⊤ S)} {𝒰 : S.OpenCover}
  [inst_3 : CategoryTheory.Category.{v_2, u_2} 𝒰.I₀] [inst_4 : AlgebraicGeometry.Scheme.Cover.LocallyDirected 𝒰]
  (d : AlgebraicGeometry.Scheme.Cover.ColimitGluingData D 𝒰) {i j : 𝒰.I₀} (hij : i ⟶ j) (X : J),
  ((d.trans hij).app X).left =
    CategoryTheory.Limits.pullback.map (D.obj X).hom (𝒰.f i) (D.obj X).hom (𝒰.f j)
      (CategoryTheory.CategoryStruct.id (D.obj X).left) (AlgebraicGeometry.Scheme.Cover.trans 𝒰 hij)
      (CategoryTheory.CategoryStruct.id S) ⋯ ⋯

Defined in

Mathlib.AlgebraicGeometry.ColimitsOver

Cited by

0 results in Mathlib

Foundations

Depth 171 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.MorphismProperty.IsStableUnderBaseChangeCategoryTheory.MorphismProperty.IsMultiplicativeCategoryTheory.CategoryCategoryTheory.CategoryAlgebraicGeometry.Scheme.Cover.LocallyDirected