Theorems · Theorem · category theory

CategoryTheory.SmallObject.SuccStruct.Iteration.arrowMap_limit

∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {J : Type w} {Φ : CategoryTheory.SmallObject.SuccStruct C}
  [inst_1 : LinearOrder J] [inst_2 : SuccOrder J] [inst_3 : OrderBot J]
  [inst_4 : CategoryTheory.Limits.HasIterationOfShape J C] [inst_5 : WellFoundedLT J] {j : J} (self : Φ.Iteration j)
  (i : J) (hi : Order.IsSuccLimit i) (hij : i ≤ j) (k : J) (hk : k < i),
  CategoryTheory.SmallObject.SuccStruct.arrowMap self.F k i ⋯ hij =
    CategoryTheory.SmallObject.SuccStruct.arrowι (CategoryTheory.SmallObject.restrictionLT self.F hij) hi k hk

The iteration on a limit element identifies to the colimit of the value on smaller elements, see Iteration.isColimit.

Defined in

Mathlib.CategoryTheory.SmallObject.Iteration.Basic

Cited by

1 results in Mathlib

Foundations

Depth 25 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryLinearOrderSuccOrderOrderBotCategoryTheory.Limits.HasIterationOfShapeWellFoundedLT