Theorems · Definition · category theory

CategoryTheory.Functor.IsCoverDense.restrictHomEquivHom

{C : Type u_1} →
  [inst : CategoryTheory.Category.{v_1, u_1} C] →
    {D : Type u_2} →
      [inst_1 : CategoryTheory.Category.{v_2, u_2} D] →
        {K : CategoryTheory.GrothendieckTopology D} →
          {A : Type u_4} →
            [inst_2 : CategoryTheory.Category.{v_4, u_4} A] →
              {G : CategoryTheory.Functor C D} →
                [G.IsCoverDense K] →
                  [G.IsLocallyFull K] →
                    {ℱ : CategoryTheory.Functor Dᵒᵖ A} →
                      {ℱ' : CategoryTheory.Sheaf K A} → (G.op.comp ℱ ⟶ G.op.comp ℱ'.obj) ≃ (ℱ ⟶ ℱ'.obj)

A locally-full and cover-dense functor G induces an equivalence between morphisms into a sheaf and morphisms over the restrictions via G.

Defined in

Mathlib.CategoryTheory.Sites.DenseSubsite.Basic

Cited by

12 results in Mathlib

Foundations

Depth 51 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Functor.IsCoverDenseCategoryTheory.Functor.IsLocallyFull