English
The boundary ∂Δ[n] is the subcomplex of the nth standard simplex consisting exactly of those m-simplices whose associated order-preserving map m → n is not surjective.
Русский
Граница ∂Δ[n] является подвектором n-й стандартной простоты, состоящим ровно из тех m-симпликалий, чья монотонная отображение m → n не сюрьективно.
LaTeX
$$$ \\partial \\Delta[n] = \\{ s ∈ \\Delta[n]_m \\mid \\operatorname{Surjective}(s) \\text{ is false} \\}$, приведённое как подсубкомплекс.$$
Lean4
/-- The boundary `∂Δ[n]` of the `n`-th standard simplex consists of
all `m`-simplices of `stdSimplex n` that are not surjective
(when viewed as monotone function `m → n`). -/
def boundary (n : ℕ) : (Δ[n] : SSet.{u}).Subcomplex
where
obj _ := setOf (fun s ↦ ¬Function.Surjective (stdSimplex.asOrderHom s))
map _ _ hs h := hs (Function.Surjective.of_comp h)
