Theorems · Definition · algebraic geometry

AlgebraicGeometry.PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit

{J : Type u'} →
  [inst : CategoryTheory.Category.{v', u'} J] →
    {C : Type u} →
      [inst_1 : CategoryTheory.Category.{v, u} C] →
        [CategoryTheory.Limits.HasColimitsOfShape J TopCat] →
          [∀ (X : TopCat), CategoryTheory.Limits.HasLimitsOfShape Jᵒᵖ (TopCat.Presheaf C X)] →
            [inst_4 : CategoryTheory.Limits.HasLimitsOfShape Jᵒᵖ C] →
              (F : CategoryTheory.Functor J (AlgebraicGeometry.PresheafedSpace C)) →
                [inst_5 : CategoryTheory.Limits.HasColimit F] →
                  (U : TopologicalSpace.Opens ↑↑(CategoryTheory.Limits.colimit F)) →
                    (CategoryTheory.Limits.colimit F).presheaf.obj (Opposite.op U) ≅
                      CategoryTheory.Limits.limit (AlgebraicGeometry.PresheafedSpace.componentwiseDiagram F U)

The components of the colimit of a diagram of PresheafedSpace C is obtained via taking componentwise limits.

Defined in

Mathlib.Geometry.RingedSpace.PresheafedSpace.HasColimits

Cited by

2 results in Mathlib

Foundations

Depth 102 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Limits.HasColimitsOfShapeCategoryTheory.Limits.HasLimitsOfShapeCategoryTheory.Limits.HasLimitsOfShapeCategoryTheory.Limits.HasColimit