Theorems · Theorem · category theory

CategoryTheory.Functor.OneHypercoverDenseData.essSurj.presheafObjObjIso.hom_mapPreimage

∀ {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]
  [inst_3 : CategoryTheory.Functor.IsDenseSubsite J₀ J F] (data : (X : C) → F.OneHypercoverDenseData J₀ J X)
  [inst_4 : CategoryTheory.Limits.HasLimitsOfSize.{w, w, v', u'} A] (G₀ : CategoryTheory.Sheaf J₀ A) {X₀ W₀ : C₀}
  (a : F.obj W₀ ⟶ F.obj X₀) {i : (data (F.obj X₀)).I₀} (p : F.obj W₀ ⟶ F.obj ((data (F.obj X₀)).X i)),
  CategoryTheory.CategoryStruct.comp p ((data (F.obj X₀)).f i) = a →
    CategoryTheory.CategoryStruct.comp
        (CategoryTheory.Functor.OneHypercoverDenseData.essSurj.presheafObjObjIso.hom data G₀ X₀)
        (CategoryTheory.Functor.IsDenseSubsite.mapPreimage J F G₀ a) =
      CategoryTheory.CategoryStruct.comp
        (CategoryTheory.Functor.OneHypercoverDenseData.essSurj.presheafObjπ data G₀ (F.obj X₀) i)
        (CategoryTheory.Functor.IsDenseSubsite.mapPreimage J F G₀ p)

Defined in

Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense

Cited by

1 results in Mathlib

Foundations

Depth 61 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

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