English
Γ₀ is defined as the composition of Γ₀' with the forgetful functor from Split to SimplicialObject, yielding a functor from ChainComplex to SimplicialObject in C.
Русский
Γ₀ задаётся как композиция Γ₀' с забывающим функтором из Split в SimplicialObject, получая функтор из ChainComplex в SimplicialObject в C.
LaTeX
$$$\Gamma_0 : \mathrm{ChainComplex}(C, \mathbb{N}) \longrightarrow \mathrm{SimplicialObject}(C), \quad \Gamma_0 = \mathrm{Split.forget} \circ \Gamma_0'. $$$
Lean4
/-- The extension of `Γ₀ : ChainComplex C ℕ ⥤ SimplicialObject C`
on the idempotent completions. It shall be an equivalence of categories
for any additive category `C`. -/
@[simps!]
def Γ₂ : Karoubi (ChainComplex C ℕ) ⥤ Karoubi (SimplicialObject C) :=
(CategoryTheory.Idempotents.functorExtension₂ _ _).obj Γ₀
