Theorems · Theorem · algebraic geometry

AlgebraicGeometry.Scheme.Cover.ColimitGluingData.transitionMap_comp

∀ {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 k : 𝒰.I₀} (hij : i ⟶ j) (hjk : j ⟶ k),
  d.transitionMap (CategoryTheory.CategoryStruct.comp hij hjk) =
    CategoryTheory.CategoryStruct.comp
      ((CategoryTheory.MorphismProperty.Over.mapComp ⊤ ⋯ ⋯
              (AlgebraicGeometry.Scheme.Cover.trans 𝒰 (CategoryTheory.CategoryStruct.comp hij hjk)) ⋯).hom.app
        (d.cocone i).pt)
      (CategoryTheory.CategoryStruct.comp ((CategoryTheory.MorphismProperty.Over.map ⊤ ⋯).map (d.transitionMap hij))
        (d.transitionMap hjk))

Defined in

Mathlib.AlgebraicGeometry.ColimitsOver

Cited by

0 results in Mathlib

Foundations

Depth 174 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

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