Theorems · Definition · category theory

DirectLimit.map

{ι : Type u_1} →
  [inst : Preorder ι] →
    {F₁ : ι → Type u_2} →
      {F₂ : ι → Type u_3} →
        {T₁ : ⦃i j : ι⦄ → i ≤ j → Sort u_6} →
          (f₁ : (i j : ι) → (h : i ≤ j) → T₁ h) →
            [inst_1 : ⦃i j : ι⦄ → (h : i ≤ j) → FunLike (T₁ h) (F₁ i) (F₁ j)] →
              [inst_2 : DirectedSystem F₁ fun x1 x2 x3 => ⇑(f₁ x1 x2 x3)] →
                {T₂ : ⦃i j : ι⦄ → i ≤ j → Sort u_7} →
                  (f₂ : (i j : ι) → (h : i ≤ j) → T₂ h) →
                    [inst_3 : ⦃i j : ι⦄ → (h : i ≤ j) → FunLike (T₂ h) (F₂ i) (F₂ j)] →
                      [inst_4 : DirectedSystem F₂ fun x1 x2 x3 => ⇑(f₂ x1 x2 x3)] →
                        [inst_5 : IsDirectedOrder ι] →
                          (ih : (i : ι) → F₁ i → F₂ i) →
                            (∀ (i j : ι) (h : i ≤ j) (x : F₁ i), (f₂ i j h) (ih i x) = ih j ((f₁ i j h) x)) →
                              DirectLimit F₁ f₁ → DirectLimit F₂ f₂

To define a function from the direct limit, it suffices to provide one function from each component subject to a compatibility condition.

Defined in

Mathlib.Order.DirectedInverseSystem

Cited by

2 results in Mathlib

Foundations

Depth 14 from the axioms · uses propext, Quot.sound

Assumes

PreorderFunLikeDirectedSystemFunLikeDirectedSystemIsDirectedOrder