Theorems · Definition · commutative algebra

AddCommGroup.DirectLimit.lift

{ι : Type u_2} →
  [inst : Preorder ι] →
    (G : ι → Type u_3) →
      [inst_1 : (i : ι) → AddCommMonoid (G i)] →
        (f : (i j : ι) → i ≤ j → G i →+ G j) →
          [inst_2 : DecidableEq ι] →
            (P : Type u_4) →
              [inst_3 : AddCommMonoid P] →
                (g : (i : ι) → G i →+ P) →
                  (∀ (i j : ι) (hij : i ≤ j) (x : G i), (g j) ((f i j hij) x) = (g i) x) →
                    AddCommGroup.DirectLimit G f →+ P

The universal property of the direct limit: maps from the components to another abelian group 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.Module

Cited by

5 results in Mathlib

Foundations

Depth 92 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

PreorderAddCommMonoidDecidableEqAddCommMonoid