English
There is a natural reflexive quiver structure on the underlying representation of a 2-truncated simplicial set, given by the Hom sets between objects and the identity edges.
Русский
Образ подстановки 2-трuncation даёт естественную рифлексивную структуру реберной пары, где множества гомоморфизмов и т.д.
LaTeX
$$$$\text{ReflQuiv} \text{ structure on } SSet.OneTruncation₂,$$$$
Lean4
/-- A 2-truncated simplicial set `S` has an underlying refl quiver `SSet.OneTruncation₂ S`. -/
instance (S : SSet.Truncated 2) : ReflQuiver (OneTruncation₂ S)
where
Hom X Y := SSet.OneTruncation₂.Hom X Y
id
X :=
{ edge := S.map (σ₂ (n := 0) 0).op X
src_eq := by
simp only [← FunctorToTypes.map_comp_apply, ← op_comp, δ₂_one_comp_σ₂_zero, op_id, FunctorToTypes.map_id_apply]
tgt_eq := by
simp only [← FunctorToTypes.map_comp_apply, ← op_comp, δ₂_zero_comp_σ₂_zero, op_id,
FunctorToTypes.map_id_apply] }
