English
A category is idempotent complete iff for every object and idempotent endomorphism, the equalizer of the identity and the endomorphism exists.
Русский
Категория идемпотентна тогда и только тогда, когда для каждого объекта и_Idempotent конца, равножелезный эквалайзер существующего граничит.
LaTeX
$$$ \\mathrm{IsIdempotentComplete} \\ C \\iff \\forall X \\in C, \\forall p: X \\to X, p^2 = p \\Rightarrow HasEqualizer(\\mathrm{Id}_X, p) $$$
Lean4
/-- A category is idempotent complete iff for all idempotent endomorphisms,
the equalizer of the identity and this idempotent exists. -/
theorem isIdempotentComplete_iff_hasEqualizer_of_id_and_idempotent :
IsIdempotentComplete C ↔ ∀ (X : C) (p : X ⟶ X), p ≫ p = p → HasEqualizer (𝟙 X) p :=
by
constructor
· intro _ X p hp
rcases IsIdempotentComplete.idempotents_split X p hp with ⟨Y, i, e, ⟨h₁, h₂⟩⟩
exact
⟨Nonempty.intro
{ cone := Fork.ofι i (show i ≫ 𝟙 X = i ≫ p by rw [comp_id, ← h₂, ← assoc, h₁, id_comp])
isLimit := by
apply Fork.IsLimit.mk'
intro s
refine ⟨s.ι ≫ e, ?_⟩
constructor
· erw [assoc, h₂, ← Limits.Fork.condition s, comp_id]
· intro m hm
rw [Fork.ι_ofι] at hm
rw [← hm]
simp only [assoc, h₁]
exact (comp_id m).symm }⟩
· intro h
refine ⟨?_⟩
intro X p hp
haveI : HasEqualizer (𝟙 X) p := h X p hp
refine
⟨equalizer (𝟙 X) p, equalizer.ι (𝟙 X) p, equalizer.lift p (show p ≫ 𝟙 X = p ≫ p by rw [hp, comp_id]), ?_,
equalizer.lift_ι _ _⟩
ext
simp only [assoc, limit.lift_π, Fork.ofι_pt, Fork.ofι_π_app, id_comp]
rw [← equalizer.condition, comp_id]
