English
Notes on the structure of Hom(x,y) being a singleton in Δ0 context.
Русский
Заметки об структуре Hom(x,y), являющейся синглтонами в контексте Δ0.
LaTeX
$$Singleton Hom sets in Δ0$$
Lean4
/-- `horn n i` (or `Λ[n, i]`) is the `i`-th horn of the `n`-th standard simplex,
where `i : n`. It consists of all `m`-simplices `α` of `Δ[n]`
for which the union of `{i}` and the range of `α` is not all of `n`
(when viewing `α` as monotone function `m → n`). -/
@[simps -isSimp obj]
def horn (n : ℕ) (i : Fin (n + 1)) : (Δ[n] : SSet.{u}).Subcomplex
where
obj _ := setOf (fun s ↦ Set.range (stdSimplex.asOrderHom s) ∪ { i } ≠ Set.univ)
map φ s hs
h :=
hs
(by
rw [Set.eq_univ_iff_forall] at h ⊢; intro j
apply Or.imp _ id (h j)
intro hj
exact Set.range_comp_subset_range _ _ hj)
