Theorems · Theorem · category theory

CategoryTheory.Functor.IsCoverDense.Types.appHom_restrict

∀ {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}
  {G : CategoryTheory.Functor C D} {ℱ : CategoryTheory.Functor Dᵒᵖ (Type v)} {ℱ' : CategoryTheory.Sheaf K (Type v)}
  (α : G.op.comp ℱ ⟶ G.op.comp ℱ'.obj) [inst_2 : G.IsCoverDense K] [inst_3 : G.IsLocallyFull K] {X : D} {Y : C}
  (f : Opposite.op X ⟶ Opposite.op (G.obj Y)) (x : (fun X => X) (ℱ.obj (Opposite.op X))),
  (CategoryTheory.ConcreteCategory.hom (ℱ'.obj.map f))
      ((CategoryTheory.ConcreteCategory.hom (CategoryTheory.Functor.IsCoverDense.Types.appHom α X)) x) =
    (CategoryTheory.ConcreteCategory.hom (α.app (Opposite.op Y))) ((CategoryTheory.ConcreteCategory.hom (ℱ.map f)) x)

Defined in

Mathlib.CategoryTheory.Sites.DenseSubsite.Basic

Cited by

1 results in Mathlib

Foundations

Depth 39 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

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