Theorems · Theorem · category theory

CategoryTheory.Pseudofunctor.LocallyDiscreteOpToCat.map_eq_pullHom

∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C]
  {F : CategoryTheory.Pseudofunctor (CategoryTheory.LocallyDiscrete Cᵒᵖ) CategoryTheory.Cat} ⦃X₁ X₂ : C⦄
  ⦃M₁ : ↑(F.obj { as := Opposite.op X₁ })⦄ ⦃M₂ : ↑(F.obj { as := Opposite.op X₂ })⦄ ⦃Y : C⦄ ⦃f₁ : Y ⟶ X₁⦄ ⦃f₂ : Y ⟶ X₂⦄
  (φ : (F.map f₁.op.toLoc).toFunctor.obj M₁ ⟶ (F.map f₂.op.toLoc).toFunctor.obj M₂) ⦃Y' : C⦄ (g : Y' ⟶ Y)
  (gf₁ : Y' ⟶ X₁) (gf₂ : Y' ⟶ X₂) (hgf₁ : CategoryTheory.CategoryStruct.comp g f₁ = gf₁)
  (hgf₂ : CategoryTheory.CategoryStruct.comp g f₂ = gf₂),
  (F.map g.op.toLoc).toFunctor.map φ =
    CategoryTheory.CategoryStruct.comp ((F.mapComp' f₁.op.toLoc g.op.toLoc gf₁.op.toLoc ⋯).inv.toNatTrans.app M₁)
      (CategoryTheory.CategoryStruct.comp
        (CategoryTheory.Pseudofunctor.LocallyDiscreteOpToCat.pullHom φ g gf₁ gf₂ hgf₁ hgf₂)
        ((F.mapComp' f₂.op.toLoc g.op.toLoc gf₂.op.toLoc ⋯).hom.toNatTrans.app M₂))

Defined in

Mathlib.CategoryTheory.Sites.Descent.IsPrestack

Cited by

1 results in Mathlib

Foundations

Depth 31 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.Category