English
For i ∈ {0,1}, the mk.app morphisms respect the δ₂ 0-face via naturality, ensuring compatibility with the δ₂ maps.
Русский
Для i ∈ {0,1} иконная mk.app карта удовлетворяет свойству δ₂ 0 через натуральность, обеспечивая совместимость с δ₂-м empr.
LaTeX
$$$\forall i \in \{0,1\},\; mk_naturality_δ0i(F,i)$$$
Lean4
theorem mk_naturality_δ1i (i : Fin 3) : toNerve₂.mk.naturalityProperty F (δ₂ i) :=
by
ext x
simp only [types_comp_apply]
rw [toNerve₂.mk.app_one]
unfold nerveFunctor₂ truncation SimplicialObject.truncation
simp only [comp_obj, nerveFunctor_obj, Cat.of_α, whiskeringLeft_obj_obj, op_obj, nerve_obj, oneTruncation₂_obj,
ReflQuiv.of_val, Nat.reduceAdd, mk.app_two, Functor.comp_map, op_map, Quiver.Hom.unop_op]
unfold δ₂ inclusion
simp only [ObjectProperty.ι_map]
fin_cases i
· simp only [Fin.zero_eta]
change _ = (nerve C).δ 0 _
rw [nerve.δ₀_mk₂_eq]
fapply ReflPrefunctor.congr_mk₁_map
· unfold ev1₂ ι1₂ δ₂
simp only [← FunctorToTypes.map_comp_apply, ← op_comp]
have := δ_comp_δ (n := 0) (i := 0) (j := 1) (by decide)
dsimp at this
exact congrFun (congrArg X.map (congrArg Quiver.Hom.op this.symm)) x
· unfold ev2₂ ι2₂ δ₂
simp only [← FunctorToTypes.map_comp_apply, ← op_comp]
have := δ_comp_δ (n := 0) (i := 0) (j := 0) (by decide)
dsimp at this
exact congrFun (congrArg X.map (congrArg Quiver.Hom.op this.symm)) x
· aesop
· simp only [Fin.mk_one]
change _ = (nerve C).δ 1 _
rw [nerve.δ₁_mk₂_eq]
rw [← hyp]
fapply ReflPrefunctor.congr_mk₁_map
· unfold ev0₂ ι0₂ δ₂
simp [← FunctorToTypes.map_comp_apply, ← op_comp]
· unfold ev2₂ ι2₂ δ₂
simp [← FunctorToTypes.map_comp_apply, ← op_comp]
· aesop
· simp only [Fin.reduceFinMk]
change _ = (nerve C).δ 2 _
rw [nerve.δ₂_mk₂_eq]
fapply ReflPrefunctor.congr_mk₁_map
· unfold ev0₂ ι0₂ δ₂
simp only [← FunctorToTypes.map_comp_apply, ← op_comp]
have := δ_comp_δ (n := 0) (i := 1) (j := 1) (by decide)
dsimp at this
exact congrFun (congrArg X.map (congrArg Quiver.Hom.op this)) x
· unfold ev1₂ ι1₂ δ₂
simp [← FunctorToTypes.map_comp_apply, ← op_comp]
· aesop
