English
The counit of nerve₂Adj is natural: for any F: C ⥤ D, the naturality square commutes: nerve₂Adj.counit.app D ∘ ((nerveFunctor₂ ⋙ hoFunctor₂).map F) = F ∘ nerve₂Adj.counit.app C.
Русский
Элемент единицы adjunction нервного двойного преобразования естественен: для любого F: C ⥤ D рисунок натуральности коммутирует.
LaTeX
$$$(nerveFunctor₂ \\gg hoFunctor₂).map F \\gg nerve₂Adj.counit.app D = nerve₂Adj.counit.app C \\gg F$$$
Lean4
/-- The naturality of `nerve₂Adj.counit.app`. -/
theorem naturality {C D : Type u} [SmallCategory C] [SmallCategory D] (F : C ⥤ D) :
(nerveFunctor₂ ⋙ hoFunctor₂).map F ⋙ nerve₂Adj.counit.app D = nerve₂Adj.counit.app C ⋙ F :=
by
apply HomotopyCategory.lift_unique'
change
((oneTruncation₂ ⋙ Cat.freeRefl).map (nerveFunctor₂.map _)) ⋙
HomotopyCategory.quotientFunctor (nerveFunctor₂.obj (Cat.of D)) ⋙ app D =
_
rw [nerve₂Adj.counit.app_eq D]
rw [← Functor.assoc _ _ F, nerve₂Adj.counit.app_eq C]
exact (whiskerRight OneTruncation₂.ofNerve₂.natIso.{u}.hom Cat.freeRefl ≫ ReflQuiv.adj.counit).naturality _
