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216fb31 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 | import Mathlib.CategoryTheory.Functor.Basic
import Mathlib.CategoryTheory.NatIso
import Mathlib.Logic.Function.Iterate
namespace OAI
noncomputable section
open CategoryTheory
namespace LittleFinitistic
universe u v
variable {C : Type u} [Category.{v} C] {F G : C ⥤ C}
/-- An actual natural isomorphism identifies every objectwise iterate. -/
def functorObjectIteration (e : F ≅ G) (r : ℕ) (Y : C) :
(F.obj^[r]) Y ≅ (G.obj^[r]) Y := by
induction r with
| zero => exact Iso.refl _
| succ r ih =>
rw [Function.iterate_succ_apply', Function.iterate_succ_apply']
exact F.mapIso ih ≪≫ e.app _
/-- Transport an object isomorphism through an actual repeated functor. -/
def mapObjectIteration (F : C ⥤ C) {X Y : C} (e : X ≅ Y) (r : ℕ) :
(F.obj^[r]) X ≅ (F.obj^[r]) Y := by
induction r with
| zero => exact e
| succ r ih =>
rw [Function.iterate_succ_apply', Function.iterate_succ_apply']
exact F.mapIso ih
end LittleFinitistic
end
end OAI
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