Theorems · Definition · commutative algebra

Module.DirectLimit.lift

(R : Type u_1) →
  [inst : Semiring R] →
    (ι : Type u_2) →
      [inst_1 : Preorder ι] →
        (G : ι → Type u_3) →
          [inst_2 : (i : ι) → AddCommMonoid (G i)] →
            [inst_3 : (i : ι) → Module R (G i)] →
              (f : (i j : ι) → i ≤ j → G i →ₗ[R] G j) →
                [inst_4 : DecidableEq ι] →
                  {P : Type u_4} →
                    [inst_5 : AddCommMonoid P] →
                      [inst_6 : Module R P] →
                        (g : (i : ι) → G i →ₗ[R] P) →
                          (∀ (i j : ι) (hij : i ≤ j) (x : G i), (g j) ((f i j hij) x) = (g i) x) →
                            Module.DirectLimit G f →ₗ[R] P

The universal property of the direct limit: maps from the components to another module 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

8 results in Mathlib

Foundations

Depth 91 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemiringPreorderAddCommMonoidModuleDecidableEqAddCommMonoidModule