Theorems · Definition · logic and foundations

FirstOrder.Language.DirectLimit.lift

(L : FirstOrder.Language) →
  (ι : Type v) →
    [inst : Preorder ι] →
      (G : ι → Type w) →
        [inst_1 : (i : ι) → L.Structure (G i)] →
          (f : (i j : ι) → i ≤ j → L.Embedding (G i) (G j)) →
            [inst_2 : IsDirectedOrder ι] →
              [inst_3 : DirectedSystem G fun i j h => ⇑(f i j h)] →
                [inst_4 : Nonempty ι] →
                  {P : Type u₁} →
                    [inst_5 : L.Structure P] →
                      (g : (i : ι) → L.Embedding (G i) P) →
                        (∀ (i j : ι) (hij : i ≤ j) (x : G i), (g j) ((f i j hij) x) = (g i) x) →
                          L.Embedding (FirstOrder.Language.DirectLimit G f) 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.ModelTheory.DirectLimit

Cited by

5 results in Mathlib

Foundations

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

Assumes

PreorderFirstOrder.Language.StructureIsDirectedOrderDirectedSystemNonemptyFirstOrder.Language.Structure