Theorems · Definition · category theory

CategoryTheory.Functor.OneHypercoverDenseData.essSurj.presheaf

{C₀ : Type u₀} →
  {C : Type u} →
    [inst : CategoryTheory.Category.{v₀, u₀} C₀] →
      [inst_1 : CategoryTheory.Category.{v, u} C] →
        {F : CategoryTheory.Functor C₀ C} →
          {J₀ : CategoryTheory.GrothendieckTopology C₀} →
            {J : CategoryTheory.GrothendieckTopology C} →
              {A : Type u'} →
                [inst_2 : CategoryTheory.Category.{v', u'} A] →
                  [CategoryTheory.Functor.IsDenseSubsite J₀ J F] →
                    ((X : C) → F.OneHypercoverDenseData J₀ J X) →
                      [CategoryTheory.Limits.HasLimitsOfSize.{w, w, v', u'} A] →
                        CategoryTheory.Sheaf J₀ A → CategoryTheory.Functor Cᵒᵖ A

Let F : C₀ ⥤ C be a dense subsite and data : ∀ X, F.OneHypercoverDenseData J₀ J X be a family. Let G₀ be a sheaf on C₀. This is a presheaf on C which extends G₀ (see OneHypercoverDenseData.essSurj.compPresheafIso) and it is a sheaf (see OneHypercoverDenseData.essSurj.isSheaf).

Defined in

Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense

Cited by

15 results in Mathlib

Foundations

Depth 57 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Functor.IsDenseSubsiteCategoryTheory.Limits.HasLimitsOfSize