colimitCocone — Mathlib

English

The colimit cocone for a functor into a complete lattice is given by pt = iSup F.obj with injections ι_j mapping j to F(j) ≤ iSup F.obj.

Русский

Кокон колимита для функторa в.complete lattice задаётся вершиной pt = iSup F.obj и иньекции ι_j: F(j) ≤ iSup F.obj.

LaTeX

$$$\text{colimitCocone }(F) :\; \text{pt} = \bigvee_j F(j),\; \iota_j: F(j) \to \bigvee_j F(j)$$$

Lean4

/-- The colimit cocone over any functor into a complete lattice.
-/
@[simps]
def colimitCocone (F : J ⥤ α) : ColimitCocone F
    where
  cocone :=
    { pt := iSup F.obj
      ι := { app := fun _ => homOfLE (CompleteLattice.le_sSup _ _ (Set.mem_range_self _)) } }
  isColimit :=
    { desc := fun s => homOfLE (CompleteLattice.sSup_le _ _ (by rintro _ ⟨j, rfl⟩; exact (s.ι.app j).le)) }
      -- see Note [lower instance priority]

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