Theorems · Definition · category theory

DirectLimit.Algebra.lift

{R : Type u_1} →
  {ι : Type u_2} →
    [inst : Preorder ι] →
      (G : ι → Type u_3) →
        {T : ⦃i j : ι⦄ → i ≤ j → Type u_6} →
          (f : (x x_1 : ι) → (h : x ≤ x_1) → T h) →
            [inst_1 : (i j : ι) → (h : i ≤ j) → FunLike (T h) (G i) (G j)] →
              [inst_2 : DirectedSystem G fun x1 x2 x3 => ⇑(f x1 x2 x3)] →
                [inst_3 : IsDirectedOrder ι] →
                  [inst_4 : CommSemiring R] →
                    [inst_5 : (i : ι) → Semiring (G i)] →
                      [inst_6 : (i : ι) → Algebra R (G i)] →
                        [inst_7 : ∀ (i j : ι) (h : i ≤ j), AlgHomClass (T h) R (G i) (G j)] →
                          [inst_8 : Nonempty ι] →
                            (P : Type u_7) →
                              [inst_9 : Semiring P] →
                                [inst_10 : Algebra R P] →
                                  (g : (i : ι) → G i →ₐ[R] P) →
                                    (∀ (i j : ι) (hij : i ≤ j) (x : G i), (g j) ((f i j hij) x) = (g i) x) →
                                      DirectLimit G f →ₐ[R] P

The universal property of the direct limit: maps from the components to another R-algebra that respect the directed system structure (i.e. make some diagram commute) give rise to a unique map out of the direct limit.

Defined in

Mathlib.Algebra.Colimit.DirectLimit

Cited by

3 results in Mathlib

Foundations

Depth 37 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

PreorderFunLikeDirectedSystemIsDirectedOrderCommSemiringSemiringAlgebraAlgHomClassNonemptySemiringAlgebra