Theorems · Theorem · category theory

CategoryTheory.GrothendieckTopology.Point.toPresheafFiber_skyscraperPresheafHomEquiv_symm_assoc

∀ {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.HasProducts A]
  {P : CategoryTheory.Functor Cᵒᵖ A} {M : A} [inst_3 : CategoryTheory.Limits.HasColimitsOfSize.{w, w, v', u'} A]
  (g : P ⟶ Φ.skyscraperPresheaf M) (X : C) (x : Φ.fiber.obj X) {Z : A} (h : M ⟶ Z),
  CategoryTheory.CategoryStruct.comp (Φ.toPresheafFiber X x P)
      (CategoryTheory.CategoryStruct.comp (Φ.skyscraperPresheafHomEquiv.symm g) h) =
    CategoryTheory.CategoryStruct.comp (g.app (Opposite.op X))
      (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.Pi.π (fun t => M) x) h)

Defined in

Mathlib.CategoryTheory.Sites.Point.Skyscraper

Cited by

0 results in Mathlib

Foundations

Depth 57 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Limits.HasProductsCategoryTheory.Limits.HasColimitsOfSize