isColimit — Mathlib

English

For limit i, the isColimit is defined via restrictionLT and is isomorphic to the colimit of F.

Русский

Для предела i издерживается isColimit через restrictionLT и эквивалентен колимиту F.

LaTeX

$$$ \\text{IsColimit}(\\text{coconeOfLE} iter.F hij) $$$

Lean4

/-- The iteration on a limit element identifies to the colimit of the
value on smaller elements. -/
noncomputable def isColimit (i : J) (hi : Order.IsSuccLimit i) (hij : i ≤ j) : IsColimit (coconeOfLE iter.F hij) :=
  by
  letI := hasColimitsOfShape_of_isSuccLimit C i hi
  refine
    IsColimit.ofIsoColimit (colimit.isColimit (restrictionLT iter.F hij))
      (Cocones.ext (eqToIso (iter.obj_limit i hi hij).symm) ?_)
  rintro ⟨k, hk⟩
  apply Arrow.mk_injective
  dsimp
  rw [← arrowMap]
  simp [iter.arrowMap_limit i hi hij k hk, arrowι_def]

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