Theorems · Theorem · category theory

CategoryTheory.GrothendieckTopology.Point.toPresheafFiberMap_naturality_apply

∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {D : Type u'} [inst_1 : CategoryTheory.Category.{v', u'} D]
  {J : CategoryTheory.GrothendieckTopology C} (Φ : J.Point) (F : CategoryTheory.Functor C D)
  (K : CategoryTheory.GrothendieckTopology D) [inst_2 : F.IsCocontinuous J K]
  [inst_3 : CategoryTheory.LocallySmall.{w, v', u'} D] {A : Type u''} [inst_4 : CategoryTheory.Category.{v'', u''} A]
  [inst_5 : CategoryTheory.Limits.HasColimitsOfSize.{w, w, v'', u''} A] {P Q : CategoryTheory.Functor Dᵒᵖ A} (g : P ⟶ Q)
  (X : C) (x : Φ.fiber.obj X) {F_1 : A → A → Type uF} {carrier : A → Type w_1}
  {instFunLike : (X Y : A) → FunLike (F_1 X Y) (carrier X) (carrier Y)} [inst_6 : CategoryTheory.ConcreteCategory A F_1]
  (x_1 : carrier (P.obj (Opposite.op (F.obj X)))),
  (CategoryTheory.ConcreteCategory.hom ((Φ.map F K).presheafFiber.map g))
      ((CategoryTheory.ConcreteCategory.hom (Φ.toPresheafFiberMap F K P X x)) x_1) =
    (CategoryTheory.ConcreteCategory.hom (Φ.toPresheafFiberMap F K Q X x))
      ((CategoryTheory.ConcreteCategory.hom (g.app (Opposite.op (F.obj X)))) x_1)

Defined in

Mathlib.CategoryTheory.Sites.Point.Map

Cited by

0 results in Mathlib

Foundations

Depth 64 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Functor.IsCocontinuousCategoryTheory.LocallySmallCategoryTheory.CategoryCategoryTheory.Limits.HasColimitsOfSizeCategoryTheory.ConcreteCategory