Theorems · Theorem · category theory

CategoryTheory.GrothendieckTopology.Point.presheafFiberDesc.congr_simp

∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {J : CategoryTheory.GrothendieckTopology C} (Φ : J.Point)
  {A : Type u'} [inst_1 : CategoryTheory.Category.{v', u'} A]
  [inst_2 : CategoryTheory.Limits.HasColimitsOfSize.{w, w, v', u'} A] {P : CategoryTheory.Functor Cᵒᵖ A} {T : A}
  (φ φ_1 : (X : C) → Φ.fiber.obj X → (P.obj (Opposite.op X) ⟶ T)) (e_φ : φ = φ_1)
  (hφ :
    ∀ ⦃X Y : C⦄ (f : X ⟶ Y) (x : Φ.fiber.obj X),
      CategoryTheory.CategoryStruct.comp (P.map f.op) (φ X x) =
        φ Y ((CategoryTheory.ConcreteCategory.hom (Φ.fiber.map f)) x)),
  Φ.presheafFiberDesc φ hφ = Φ.presheafFiberDesc φ_1 ⋯

Defined in

Mathlib.CategoryTheory.Sites.Point.Basic

Cited by

2 results in Mathlib

Foundations

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

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Limits.HasColimitsOfSize