English
The construction of the 2-coskeletal map from a refl prefunctor realizes the underlying data as components of a map into the nerve.
Русский
Конструкция 2-коскрельной карты из ребра отражающего префунктера реализует лежащие данные как компоненты отображения в нерв.
LaTeX
$$$\mathrm{app}: X_{op} \to (\mathrm{nerveFunctor}_2(\mathrm{Cat.of}\mathcal{C}))._{op}$ defined levelwise by polynomial formulas in mk₂, with compatibility conditions.$$
Lean4
/-- Because nerves are 2-coskeletal, the components of a map of 2-truncated simplicial sets valued
in a nerve can be recovered from the underlying ReflPrefunctor. -/
def app (n : SimplexCategory.Truncated 2) : X.obj (op n) ⟶ (nerveFunctor₂.obj (Cat.of C)).obj (op n) :=
by
obtain ⟨n, hn⟩ := n
induction n using SimplexCategory.rec with
| _ n
match n with
| 0 => exact fun x => .mk₀ (F.obj x)
| 1 => exact fun f => .mk₁ (F.map ⟨f, rfl, rfl⟩)
| 2 => exact fun φ => .mk₂ (F.map (ev01₂ φ)) (F.map (ev12₂ φ))
