Theorems · Theorem · algebraic geometry

AlgebraicGeometry.PresheafedSpace.ColimitCoconeIsColimit.desc_c_naturality

∀ {J : Type u'} [inst : CategoryTheory.Category.{v', u'} J] {C : Type u} [inst_1 : CategoryTheory.Category.{v, u} C]
  [inst_2 : CategoryTheory.Limits.HasColimitsOfShape J TopCat]
  [inst_3 : ∀ (X : TopCat), CategoryTheory.Limits.HasLimitsOfShape Jᵒᵖ (TopCat.Presheaf C X)]
  [inst_4 : CategoryTheory.Limits.HasLimitsOfShape Jᵒᵖ C]
  (F : CategoryTheory.Functor J (AlgebraicGeometry.PresheafedSpace C)) (s : CategoryTheory.Limits.Cocone F)
  {U V : (TopologicalSpace.Opens ↑↑s.pt)ᵒᵖ} (i : U ⟶ V),
  CategoryTheory.CategoryStruct.comp (s.pt.presheaf.map i)
      (AlgebraicGeometry.PresheafedSpace.ColimitCoconeIsColimit.descCApp F s V) =
    CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.PresheafedSpace.ColimitCoconeIsColimit.descCApp F s U)
      (((TopCat.Presheaf.pushforward C
                (CategoryTheory.Limits.colimit.desc (F.comp (AlgebraicGeometry.PresheafedSpace.forget C))
                  ((AlgebraicGeometry.PresheafedSpace.forget C).mapCocone s))).obj
            (AlgebraicGeometry.PresheafedSpace.colimitCocone F).pt.presheaf).map
        i)

Defined in

Mathlib.Geometry.RingedSpace.PresheafedSpace.HasColimits

Cited by

0 results in Mathlib

Foundations

Depth 95 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Limits.HasColimitsOfShapeCategoryTheory.Limits.HasLimitsOfShapeCategoryTheory.Limits.HasLimitsOfShape