Theorems · Definition · category theory

CategoryTheory.Pseudofunctor.DescentData.isoMk

{C : Type u} →
  [inst : CategoryTheory.Category.{v, u} C] →
    {F : CategoryTheory.Pseudofunctor (CategoryTheory.LocallyDiscrete Cᵒᵖ) CategoryTheory.Cat} →
      {ι : Type t} →
        {S : C} →
          {X : ι → C} →
            {f : (i : ι) → X i ⟶ S} →
              {D₁ D₂ : F.DescentData f} →
                (e : (i : ι) → D₁.obj i ≅ D₂.obj i) →
                  autoParam
                      (∀ ⦃Y : C⦄ (q : Y ⟶ S) ⦃i₁ i₂ : ι⦄ (f₁ : Y ⟶ X i₁) (f₂ : Y ⟶ X i₂)
                        (hf₁ : CategoryTheory.CategoryStruct.comp f₁ (f i₁) = q)
                        (hf₂ : CategoryTheory.CategoryStruct.comp f₂ (f i₂) = q),
                        CategoryTheory.CategoryStruct.comp ((F.map f₁.op.toLoc).toFunctor.map (e i₁).hom)
                            (D₂.hom q f₁ f₂ ⋯ ⋯) =
                          CategoryTheory.CategoryStruct.comp (D₁.hom q f₁ f₂ ⋯ ⋯)
                            ((F.map f₂.op.toLoc).toFunctor.map (e i₂).hom))
                      CategoryTheory.Pseudofunctor.DescentData.isoMk._auto_1 →
                    (D₁ ≅ D₂)

Constructor for isomorphisms in Pseudofunctor.DescentData.

Defined in

Mathlib.CategoryTheory.Sites.Descent.DescentData

Cited by

4 results in Mathlib

Foundations

Depth 40 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.Category